Mathematics

(Navigation). Educational manuscript titled "The Mariner's Compass"

  • An elegant and substantial 114-page “schatkamer” educational manuscript produced by a student of navigation in Georgian England.

    As historian Margarette E. Schotte explains in Sailing School: Navigating Science and Skill 1550-1800 (2019), between the 16th and 18th centuries northern European navigation developed from a craft learned by practice aboard ship to a profession taught in specialised schools. Central to this pedagogy was the creation of “manuscript ‘treasure chests’ (schatkamers) – in which members of the maritime community puzzled through lessons related to their profession” (Schotte, p. 9).

    As is typical for schatkamers, the present manuscript comprises both notes and practice problems, and combines traditional nautical knowledge with trigonometry. It opens with the compass rose, “perhaps the most fundamental of all navigational devices. One early seventeenth-century English maritime writer likened the ability to recite the thirty-two points to mastering the rudiments of alphabet... Mariners used this framework not only to orient themselves geographically – with the help of a simple magnetic compass and the stars – but also as a type of notional clock, to keep track of tides in various ports” (Schotte, p. 31). Another set of traditional skills was determining tides. The required pieces of information were the Golden Number, the epact, the Sunday letter, and the Moon’s age in the current month, which could be determined by calculations on the knuckles and then combined with times from local tide tables for the final result. Explanations of the method and calculations for determining all these numbers, as well as the Moon’s “southing” or angle above the horizon, are recorded here.

    Much of the manuscript addresses different techniques for reckoning, including notes, practice problems, and diagrams. Eleven pages are on plane sailing, a simplified form of trigonometric reckoning used over small ranges of latitude and longitude; fourteen cover middle latitude sailing, a method of determining departure by assuming that a course is steered at the middle latitude; and eleven are on Mercator sailing, which utilises a Mercator projection to solve navigational problems. Oblique sailing, in which a course oblique to the meridian is charted, and current sailing, the technique of predicting the current’s effect on a vessel’s course, also make appearances.

    Of particular interest are the five pages on traverse sailing, including two completed traverse course diagrams with charming illustrations of ships. As Schotte explains, these diagrams were not generally used aboard ship, but in the classroom they helped students develop the skill of precisely tracking a ship’s location. “Once a day, at least, the navigator added up the numerous legs from the preceding twenty-four hours to determine a vessel’s current position. Navigators were accustomed to tracking their progress on traverse boards, but these pegboards could document only the barest essentials of the course. As expectations increased about the accuracy of the day’s records, sailors were taught how to record their motion in geometric terms and then to analyze the resultant shapes with trigonometry. Textbook authors devoted whole chapters to the ‘traverse course’, explaining the fairly straightforward process of converting a day’s travel into a form compatible with a tabular logbook. Instructors walked students through the construction of a right-angled triangle that represented the ‘difference of latitude’ and ‘course’ for a single trajectory and then demonstrated how to add together multiple triangles’... Teachers believed in the pedagogical value of drawing such courses and included set examples in their teaching. Although publishers chose to conserve paper rather than print these sprawling diagrams, students frequently produced them in class” and “regularly pushed beyond the printed text. When it came to traverse courses, teachers took opportunities to make exercises more interesting and memorable” by using both real and fanciful place names and designing courses in “witty shapes” like hearts, anchors, spirals, and fortifications (Schotte, pp. 127-129). In this manuscript, one of the traverse charts includes a delightful drawing of a three-masted sailing ship, remarkably detailed for its diminutive size.

    As important as these mathematical techniques was the navigator’s ability to keep accurate records and to correct dead reckoning while at sea, to which twenty pages of the manuscript are devoted. These lessons culminate in a remarkable 17-page exercise charting in detail a ship’s progress from London to Madeira, based on a sailing of the Nancy, commanded by “A. B.” with the log kept by the mate T. Weir. The exercise includes elaborate calculations as well as notes on the weather and astronomical information. A second exercise, based on the sailing of the Frances of London, commanded by William Johnson, from London to Madeira and Tenerife was begun but seems to have been left incomplete. There are also sixteen pages on surveying coast and harbours, including two carefully drawn charts of fictional coasts. Finally, the manuscript ends with a short chapter on the curvature of the earth and its effect on one’s view of objects at a distance.

    Schatkamers like this one could have a long life, and might be returned to for reference, or plumbed for practice questions when the mariner needed to take tests to advance in their career. Today they are an important and under-utilised historical source on the ways that individuals learned their craft, as well as how the teaching and practice of navigation adapted to the new realities of long-distance voyages (Schotte). This is a superb example, comprehensive, complete, and neatly written, with great care taken in preparing the charts and drawings.

  • England, late-18th or early-19th century.

    Folio, 114-page manuscript. Early 20th-century marbled boards, black morocco label to the spine, endpapers contemporary with binding. Spine panel detached and loosely inserted, edges of boards rubbed, marbling faded, heavy spotting to title and early leaves, particularly in the lower corners of the fifth and sixth leaves, small torn and creased area affecting the gutter and limited portions of the contents in the final half of the text.