Nautical Almanac Research Articles

Course Performance of Marine Transportation in Celestial Navigation (NAV4 233L3) Using Mathematical Processes: Basis for Course Enhancement

The study determined the course performance of the 2nd Year Bachelor of Science in Marine Transportation students of Asian Institute of Maritime Studies enrolled during the 3rd Trimester of School Year 2021-2022 in Celestial Navigation (NAV4 233L3) course. To know the course performance, mathematical processes was evaluated by answering a self-assessment questionnaire. The content of the survey was inspired by the mathematical framework created by of the Department of Science and Technology with the Philippine Council of Mathematics Teacher Education, Inc. .The mathematical processes assessed in the study were: Problem Solving Skills, Communicating Mathematically, Math Reasoning and Making Math Connections. There were 173 samples taken using Slovin’s formula with 5% level of significance. Embedded mixed method was applied to conduct the study. The grades of the respondents in NAV 3 and NAV4 233L3 is being correlated to find if the grades have significant relationship using Pearson correlation by running SPSS version 29. It yielded that the grades in NAV 3 and NAV4 233L3 has positive weak correlation, r=.244, p<.001. Thus, there is a significant relationship between the two grades, therefore, the null hypothesis was rejected. The mean of the course performances applying math processes was compared when grouped according to the profile of the respondents. The Analysis of Variance was the statistical treatment applied utilizing SPSS. There is no significant differences with the means of the course performances applying mathematical processes when grouped according to age, gender and Senior High School strand because the p value are all greater than .05 so null hypothesis was accepted. Under the thematic qualitative method, the sharing of interests and of difficulties of the respondents were analyzed. The answers under the interests of the respondents were grouped into three themes: Problem Solving is interesting; NAV4 233L3 has interesting topic/s; and Interesting Laboratory Activities: Using Nautical Almanac, Marine Sextant and Sight Reduction Table. On the other hand, the difficulties the respondents had when studying NAV 4 233L3 was also summarized into three themes: NAV4 233L3 is not difficult: the Instructors discussed the topics well; Crucial in NAV4 233L3:Problem Solving; and NAV4 233L3 Mathematics Connections is Hard: Plotting, Completing Tables and Math Equations.

Influence of chronometer error uncertainties on the Longitude of Shackleton's vessel, Endurance

AbstractIn 1915 while the Imperial Trans-Antarctic Expedition's vessel Endurance was icebound in the Weddell Sea, lunar occultation timings were carried out in order to rate the chronometers and thereby find longitude. The original observations have been re-analysed using modern lunar ephemerides and catalogues of star positions. The times derived in this way are found to differ by an average of 20 s from those obtained during the expedition using positions given from the Nautical Almanac and introduces an additional offset of the true positions to the east of those recorded in the log.

A (NOT SO) BRIEF HISTORY OF LUNAR DISTANCES: LUNAR LONGITUDE DETERMINATION AT SEA BEFORE THE CHRONOMETER

Longitude determination at sea gained increasing commercial importance in the late Middle Ages, spawned by a commensurate increase in long-distance merchant shipping activity. Prior to the successful development of an accurate marine timepiece in the late-eighteenth century, marine navigators relied predominantly on the Moon for their time and longitude determinations. Lunar eclipses had been used for relative position determinations since Antiquity, but their rare occurrences precludes their routine use as reliable way markers. Measuring lunar distances, using the projected positions on the sky of the Moon and bright reference objects—the Sun or one or more bright stars—became the method of choice. It gained in profile and importance through the British Board of Longitude's endorsement in 1765 of the establishment of a <italic>Nautical Almanac</italic>. Numerous 'projectors' jumped onto the bandwagon, leading to a proliferation of lunar ephemeris tables. Chronometers became both more affordable and more commonplace by the mid-nineteenth century, signaling the beginning of the end for the lunar distance method as a means to determine one's longitude at sea.

Mathematicians on board: introducing lunar distances to life at sea.

Nevil Maskelyne, the Cambridge-trained mathematician and later Astronomer Royal, was appointed by the Royal Society to observe the 1761 transit of Venus from the Atlantic island of St Helena, assisted by the mathematical practitioner Robert Waddington. Both had experience of measurement and computation within astronomy and they decided to put their outward and return voyages to a further use by trying out the method of finding longitude at sea by lunar distances. The manuscript and printed records they generated in this activity are complemented by the traditional logs and journals kept by the ships' officers. Together these records show how the mathematicians came to engage with the navigational practices that were already part of shipboard routine and how their experience affected the development of the methods that Maskelyne and Waddington would separately promote on their return. The expedition to St Helena, in particular the part played by Maskelyne, has long been regarded as pivotal to the introduction of the lunar method to British seamen and to the establishment of the Nautical Almanac. This study enriches our understanding of the episode by pointing to the significant role played by the established navigational competence among officers of the East India Company.

Stand-alone Celestial Navigation Positioning Method

Finding one's geographical position (fix) without the use of a Global Navigation Satellite System (GNSS), which was common place before the establishment of the latter, could be tedious and/or inaccurate. Apart from sound knowledge of spherical trigonometry and navigational methods, it also requires the knowledge of the navigator's approximate or assumed position, the use of the current year's celestial bodies' ephemeris (Nautical Almanac), and graphical methods (Lines of Position – LOP) which sometimes can prove wanting in accuracy and/or challenging for the unaccustomed user. The method proposed here is based on sight reduction from two celestial bodies, and directly calculates the geographical position, both for stationary and moving observers (“running fix”) using easily available modern programmable calculating devices, without the need of the assumed position, graphical methods (LOP) or the current year's ephemeris, hence the term “stand-alone”. This self-contained method is implemented by the author in a software application, which can be easily used in a portable computer (for example, a smartphone). The results are considered satisfactorily accurate.

Life and Statistical Legacy of Charles Hugo Kummell

SummaryEuropean statistical research flourished in the years 1770 to 1850 whereas statistical research in the United States did not develop fully until the latter half of the nineteenth century. The establishment of the United States Coast Survey, the Lake Survey, and the Nautical Almanac in 1807, 1841, and 1849, respectively, encouraged the rapid advancement of interest in statistics and the development of statistical methods. It was pioneered by Robert Adrain, Benjamin Peirce, his son Charles Sanders Peirce, Simon Newcomb, and Erastus Lyman De Forest, whose work is well researched. In this article we focus on life and the statistical accomplishments of Charles Hugo Kummell, a statistician for the Lake Survey and the U.S. Coast and Geodetic Survey, and an active member of the Philosophical Society of Washington. We describe his research into laws of errors of observations and his contributions to the development of the least-squares method.

Brown’s Nautical Almanac – skarbnica dzisiejszej wiedzy nawigacyjnej

Brown’s Nautical Almanac – skarbnica dzisiejszej wiedzy nawigacyjnej

Astro Navigation Remembered

The authors give a personal account of the use of astro navigation during the years before GPS became the primary aid for oceanic navigation. The paper also contains an account of the support provided by The Nautical Almanac (NA) together with its associated tables and computer programs which ease the tedium of sight reduction while leaving the satisfaction to be found in successful astro observations at sea.An early version of Section 2 of this paper formed part of a presentation made at the Symposium After Longitude – Modern Navigation in Context which took place at the UK National Maritime Museum from 22–23 March 2012.

On the transit of Venus 2012: Method of computation for prediction of contact timings

Transit of Venus over the solar disc is an extremely rare event. The phenomenon occurred last time on June 8, 2004 when the entire event was visible from all parts of India. Another Transit of Venus is going to occur on June 6, 2012, though the entire event will not be visible from India. The Positional Astronomy Centre publishes data on Transit of Mercury and Venus in its annual publication ‘The Indian Astronomical Ephemeris’. In this paper an attempt has been made to provide documentation on the methodology for computation of contact timings of the event. Using the methodology, the geocentric contact timings and local contact timings for important places of India for the event of Transit of Venus of June 6, 2012 have been predicted. The result thus obtained for different geocentric phases of the event has been compared with the predicted timings published by The Nautical Almanac Office, United States Naval Observatory and NASA.

Essay Review: Tobias Mayer's Lunar Theory: Between Theory and Observations: Tobias Mayer's Explorations of Lunar Motion, 1751–1755

(ProQuest: ... denotes formulae omitted.)TOBIAS MAYER'S LUNAR THEORY Between Theory and Observations: Tobias Mayer's Explorations of Lunar Motion, 1751-1755. Steven A. Wepster (Springer Verlag, Berlin, 2010). Pp. xiv + 246. euro96. ISBN 978-1-4419-1313-5.Tobias Mayer (1723-62), professor of practical mathematics at the Georg-August Academy in Gottingen from 1751 till his death in 1762, published in 1753 tables of the Moon's motion which, in comparisons with 200 observations, differed in all but a dozen cases by less than arcminutes. Consequently, deteimining longitude at sea by the method of lunar distances looked to be within reach: an accuracy of 2 arcminutes would enable the longitude to be computed to Io. Leonhard Euler among others encouraged the modest young Mayer to compete for the Longitude Prize set up by the British Parliament, promising rewards of £20,000, £15,000, and £10,000 for procedures giving the longitude to \°, §°, and Io, respectively. Mayer's final tables, submitted by his widow in 1763, became the initial basis for the British Nautical almanac and astronomical ephemeris, which began publication in 1767 and has appeared annually ever since.How did Mayer obtain lunar tables good to within 2 arcminutes, when the theories derived from Newton's gravitational law by the leading mathematicians of the day - Euler, Clairaut, and d'Alembert - erred frequently by 3 to 5 arcminutes? Mayer did not conceal that he had honed the accuracy of his tables by comparisons with observations, but of his process he revealed no details. Conjectures about it put forward later have remained unsupported by solid evidence. Gauss, believing the method of least squares the obvious way to go, suggested that Mayer had used it (he would thus have been the first to do so), but on looking into Mayer's papers found only home-baked combinations [hausbackene Combinationen]. Evidently, any understanding of Mayer's success must be extracted from Mayer's worksheets, extant in some twenty-four manuscript codices now preserved in Gottingen. But can these manuscripts be made to divulge their story? This is Steven A. Wepster's project in the study here reviewed.An initial step was to sort out the different tasks accomplished in the worksheets, and to order them chronologically. Numerous among the worksheets are position computations: lunar positions computed from lunar tables for specified times and compared with observations. Codex //*,, for example, contains 366 such computations. The ostensible aim is to check the accuracy of lunar tables against observations, assuming the accuracy of the latter. Since during the years 175 1-55, the period of Mayer's most intense work on the tables, the new Gottingen observatory was incomplete, lacking its mural quadrant, Mayer had to depend chiefly on observations recorded by other astronomers. He sought observations of high quality, rejecting lunar longitudes derived from meridian transits, which were frequently in error because of erroneous stellar longitudes assumed in their derivation. In the determinations most crucial for the tabular accuracy he achieved, Mayer employed occultations of stars by the Moon. The winking out of a star as it is eclipsed by the Moon and its reappearance when uncovered are among the most sharply defined of celestial events. At the moment of immersion or emersion, a point on the Moon's limb has the same longitude as the star, and this fact could be used to investigate the Moon's without adopting any pre-assigned value for the star's longitude.To identify the theory under test in Mayer's position computation sheets, Wepster had to depend on numbers that the sheet contained: 'mean motions' in the values of longitude, apogee, or ascending node; 'epochs', the values of these parameters for fixed dates; and periodic departures from the mean motions. Traditionally, astronomers called the periodic departures inequalities and modelled them by circular functions called equations. …

Time references in US and UK astronomical and navigational almanacs

An astronomical or navigational almanac can best be thought of as a device for connecting an observer with celestial objects. For an observer with a known position and time the almanac allows the observer to identify the celestial objects. Conversely, if the observer knows what objects he is observing and when, the almanac allows him to determine his position. In either case, knowledge of the time is crucial in providing this link between the celestial objects and the observer through the almanac. This paper summarizes the various time scales used in the astronomical and nautical almanacs produced jointly by the US Naval Observatory and HM Nautical Almanac Office.

The Development and Study of Calculating Compass Error Software

With the principle, methods and steps of calculating compass error from observing the low height sun azimuth, the software program about calculating compass error based on VB (Visual Basic) environment has been given out. By analyzing the data regularity of Nautical Almanac, get the result that the data of Nautical Almanac shows the pattern of an arithmetic series, then with the data regularity and Compass Error’s calculating formula, a perfect software of calculating Compass Error is designed with Computer language Visual Basic. The data from observation and given can be put into the software program directly, avoid the complex of table lookup and data calculation, the software program has great effect to reduce the amount of calculation and raise the speed, precision and accuracy of calculating compass error.

COMPUTATION PROGRAMS OF THE ASTRONOMICAL VESSEL POSITION

In open sea sailing, as opposed to sailing with satellite navigation systems, the astronomical vessel position (AVP) is not limited by military codes. If programs can be developed to solve the AVP directly, drawbacks of current methods for performing marine operations can be greatly improved. Complete AVP computations have two necessary points: compute the observed altitude and compute the AVP for two bodies. Thus, this paper utilizes Matlab ® programming language to develop the observed altitude (ObsAltPro) and the AVP by using intercept method (IMPro-2) programs, respectively. Adopting the ObsAltPro can avoid the use of nautical almanacs and directly obtain the observed altitudes of various celestial bodies between 1986 and 2050. As for merits of using the IMPro-2, they can skip limits of tabular methods and directly use the dead reckoning (DR) as the input variable to replace the assumed position (AP). To address the deficiencies of the intercept method (IM), iteration computation is added into the latter program as a verification function. The above two programs, through demonstrated examples, have shown advantages of being simple, fast, and accurate.

Daytime Celestial Navigation for the Novice

What kinds of astronomical lab activities can high school and college astronomy students carry out easily in daytime? The most impressive is the determination of latitude and longitude from observations of the Sun. The “shooting of a noon sight” and its “reduction to a position” grew to become a daily practice at the start of the 19th century1 following the perfection of the marine chronometer by John Harrison and its mass production.2 This technique is still practiced by navigators in this age of GPS. Indeed, the U.S. Coast Guard exams for ocean-going licenses include celestial navigation.3 These techniques continue to be used by the military and by private sailors as a backup to all-too-fallible and jammable electronic navigation systems. A sextant, a nautical almanac,4 special sight reduction tables,5 and involved calculations are needed to determine position to the nearest mile using the Sun, Moon, stars, or planets. Yet, finding latitude and longitude to better than 30 miles from measurements of the Sun's altitude is easily within the capability of those taking astronomy or physics for the first time by applying certain basic principles. Moreover, it shows a practical application of astronomy in use the world over. The streamlined method described here takes advantage of the similar level of accuracy of its three components: 1.Observations using a homemade quadrant6 (instead of a sextant), 2. Student-made graphs of the altitude of the Sun over a day7 (replacing lengthy calculation using sight reduction tables), and 3. An averaged 20-year analemma used to find the Sun's navigational coordinates8,9 (rather than the 300+ page Nautical Almanac updated yearly).

THE (ALMOST) UNSEEN TOTAL ECLIPSE OF 1831

The total eclipse of August 1831 began at sunrise in Australia, swept across the western South Pacific Ocean, and ended at sunset in the central South Pacific. As a result of the eclipse’s path over mostly uninhabited ocean, the region’s sparse European (British) population, and near-useless local predictions of the event at Hobart and Sydney in almanacs sold to the general public, almost no one witnessed its passage. In an attempt to document the eclipse, journals of naive observers—those having no access to a prediction—were examined. Thus far, the sole record is in the Pitcairn Island Register Book. Considering the Pitcairners’ extreme isolation and the rather modest partial eclipse that occurred there, the entry is a surprising one; however, it can be explained in terms of events associated with their initial removal to Tahiti in March 1831 followed by their return home in June. Further, an authoritative means to identify any issues associated with eclipse predictions compiled for private-sector almanacs came in 1833 when sweeping changes in the British Nautical Almanac’s section on eclipses were instituted..

Simon Newcomb, astronomer and personality

Expending the time and effort to write a book review is a selfless service that often goes unrecognized. We thank Bradley Schaefer for his knowledgeable and incisive review of Simon Newcomb: America’s Unofficial Astronomer Royal (Physics Today, February 2007, page 66 ). However, one of our primary goals in writing the book was to dispel the notion that Newcomb was “cold, ruthless, and more feared than liked,” a characterization that we found little basis for in the extensive collection of correspondence archived at the US Naval Observatory Library and the Library of congress. Newcomb knew and corresponded with virtually every leading astronomer of his era, not to mention many of the physicists, mathematicians, and economists. Colleagues regularly stayed at his home when visiting Washington, DC, and he and his family reciprocated in visiting colleagues across Europe and South Africa. It should not be surprising to learn that when Newcomb left the Naval Observatory for the Nautical Almanac Office, he left behind more than a few jealous colleagues, who would await their opportunities to “get even” for any number of imagined slights. But the world-class astronomers of the day recognized his achievements with nearly every prestigious award, and dozens of universities conferred honorary degrees on him as well.We did not mention in our book that some have suggested Newcomb as the “intentional prototype for Arthur Conan Doyle’s arch-villain Professor James Moriarty” or as the “learn’d astronomer” in Walt Whitman’s famous “When I heard the Learn’d Astronomer,” because we found only speculations 1 1. B. E. Schaefer, J. Br. Astron. Assoc. 103(1), 30 (1993). but no credible evidence that either claim is true. In fact, several scientists have been suggested as possible models for Professor Moriarty, among them Carl Friedrich Gauss. But the model that Doyle seems to imply in The Valley of Fear 2 2. A. C. Doyle, The Valley of Fear, originally published by A. L. Burt, New York, 1914; available at http://www.gutenberg.org/etext/3776. is the London arch-criminal of the 18th century, Jonathan Wild. Why would Doyle choose as his prototype for Moriarty a relatively remote American astronomer rather than Wild, the locally well-known British villain and criminal?Historians who do not practice science seem often to see differences of opinion between scientists as indicative of personal dislike, but scientists on opposing sides of scientific issues are frequently close friends. Scientists Virginia Trimble and Markus Aschwanden concluded that “Simon Newcomb has, in recent years, been something of a victim of bad publicity, being cited for his reluctance to include spectroscopy … in our discipline and societies. … In his own later writing, however, he comes across as the sort of person you might well want to go on an observing run or a country ramble with.” 3 3. V. Trimble, M. J. Aschwanden, Astron. Soc. Pac. 114, 475 (2002). https://doi.org/10.1086/341673 It is about time that many of the unfounded myths about Newcomb be exposed and expunged. He was certainly one of America’s greatest astronomers, a respected colleague, and a devoted husband and father, fully worthy of the respect and honors accorded him in his own time.REFERENCESSection:ChooseTop of pageREFERENCES <<1. B. E. Schaefer, J. Br. Astron. Assoc. 103(1), 30 (1993). Google Scholar2. A. C. Doyle, The Valley of Fear, originally published by A. L. Burt, New York, 1914; available at http://www.gutenberg.org/etext/3776. Google Scholar3. V. Trimble, M. J. Aschwanden, Astron. Soc. Pac. 114, 475 (2002). https://doi.org/10.1086/341673 , Google ScholarCrossref© 2008 American Institute of Physics.

Implementation of the new nomenclature in The Astronomical Almanac

AbstractThis talk charts the implementation of the resolutions of the IAU XXIV General Assembly in 2000 (IAU 2000) in The Astronomical Almanac (AsA).

A History of Mathematical Tables: From Sumer to Spreadsheets (review)

Reviewed by: A History of Mathematical Tables: From Sumer to Spreadsheets Peggy Aldrich Kidwell (bio) A History of Mathematical Tables: From Sumer to Spreadsheets. Edited by Martin Campbell-Kelly et al.New York: Oxford University Press, 2003. Pp. viii+361. $89.50. In 1823 the English-born machinist and instrument maker Timothy Claxton sailed from Russia to the United States. Claxton spent much of the nine weeks of his voyage doing calculations on a slide rule. He later published the results in a small book titled Concise Decimal Tables for Facilitating [End Page 662] Arithmetical Calculations . . .(1830). Few twenty-first-century travelers spend odd moments doing calculations by hand, and even fewer publish the results in printed tables. As the volume under review demonstrates, however, the humble Mr. Claxton was part of a rich tradition of table making that stretches back to ancient Sumer. Moreover, though table making is now done largely by electronic computers, the tabular presentation of data lives on in software such as spreadsheets. In September 2001, the British Society for the History of Mathematics brought together a remarkable group of historians for a conference on the history of mathematical tables. The volume under review is an outgrowth of the papers presented at that conference, providing readers with an excellent introduction to a growing body of literature in the history of mathematics, astronomy, and computing. The opening chapter describes the oldest surviving tables, which are from ancient Mesopotamia. These range from a simple table for land measurement from Sumer (circa 2600 B.C.E.) to a tabular account of sheep and goats from the archives of Ur (circa 2028 B.C.E.) to more standardized administrative tables of the eighteenth century B.C.E.Later, Babylonian astronomers regularly produced ephemerides listing the times of astronomical events; the last of these appeared after the birth of Christ. From this wide-ranging opening the book moves on to innovations in printed tables, particularly the early logarithmic and actuarial tables prepared in seventeenth-century Britain. Subsequent chapters discuss groups of people organized to produce tables, from the unemployed hairdressers of postrevolutionary France to the table-making enthusiasts of the late-nineteenth- and early-twentieth-century British Association for the Advancement of Science to those employed by the Mathematical Tables Project of the Works Projects Administration in the United States. While some organized people to compute tables, others sought to automate the process. Two chapters are devoted to machines known as difference engines, which were designed in the nineteenth and early twentieth centuries both to compute tables and to print the results of calculation. Entries in a table, however they are calculated, depend on underlying assumptions about the interpretation and classification of data. Detailed accounts of the theory underlying the preparation of planetary tables, of the selection of categories used to present census data, and of table making at Britain's Nautical Almanac Office (NAO) suggest some of the complexity that underlies the ordered columns of tables. Many of the tables of the NAO are now presented exclusively in electronic form. A concluding chapter examines another electronic version of the mathematical table, the spreadsheet, and focuses in particular on the history of Visicalc, Lotus 1-2-3, and Microsoft Excel. One characteristic of ardent table makers that emerges from this book is a meticulous—some might say obsessive—attention to detail. Happily, [End Page 663]this spirit seems to have worn off on the authors, editor, and publisher. Each section includes not only notes and suggestions for further reading but also well-chosen illustrations of tables, people, and places. Relevant historical, biographical, and technical details are presented in separate text boxes. Such careful design, combined with thorough planning, a thoughtful introduction, cross-referencing, and an index, does much to unify the disparate contributions. A book that surveys more than four thousand years of accomplishment is, perforce, selective. This reviewer would have been happy to learn more about the hundreds of inexpensive books of tables like Claxton's, as these played a much larger role in nineteenth-century engineering and commerce than the handful of difference engines discussed at length. The book also says very little about tables published in Germany, important...

A Benevolent Astronomer: Further notes on L. J. Comrie

Britain, and particularly London, was neither a pleasant nor a safe place to be in 1940. Nor were the prospects of the city all that welcoming. In May Winston Churchill became Prime Minister – he offered ‘blood, toil, tears, and sweat.’ By early June the ‘miracle’ of Dunkirk was offering some encouragement, but this was offset by the evacuation of British troops from Norway with several thousand casualties. July to August saw the Battle of Britain with the Blitz on London, and throughout this period the loss of Allied shipping caused continuing apprehension. By the end of the year Britain stood alone against Germany and Italy, with Russia uncommitted in the wings. It was against this background that Leslie John Comrie made the decision to ship a large quantity of books and serial and periodical publications to the Carter Observatory in Wellington. Comrie was an expatriate New Zealander who had been Superintendent of the British Nautical Almanac Office, had established Scientific Computing Services Ltd in London and was an acknowledged leader in numerical computation before the advent of electronic computers. He was also regarded as the leading table maker and table editor of his day. 1

The History of mathematical tables: from Sumer to spreadsheets

Introduction Table and tabular formatting in Sumer, Babylonia and Assyria, 2500 BCE - 50 CE The making of logarithm tables The computation factory: de Prony's project for making tables in the 1790's Difference engines: from Muller to Comrie The 'unerring certainty of mechanical agency': machines and table making in the nineteenth century Table making in astronomy The General Registry Office and the tabulation of data, 1837 - 1939 Table making by committee British table maker 1871 - 1965 Table making for the relief of labour The making of astronomical tables in H.M. Nautical Almanac Office The rise and rise of the spreadsheet Biographical Notes