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The Sailings: The Mathematics of Eighteenth
Century Navigation in the American Colonies
Joel Silverberg
Dept of Mathematics
Roger Williams University
Bristol, Rhode Island
USA
Presented at the University of Waterloo, Ontario, Canada, June 5,
2005
Abstract
Throughout the 17th and 18th centuries, sailing ships regularly plied the At-
lantic between the old world and the new. The survival of the newly planted
colonies, and all hopes for their growth and prosperity, depended upon safe and
reliable maritime transport. To determine one’s position at sea, one needs the
ability to measure both time and angles with considerable accuracy. At the
start of the ages of exploration and colonization, sailors could do neither.
Pendulum clocks were first developed by Galileo in 1602 and Huygens in 1640,
but these clocks could not function on board a moving ship. The most accurate
time piece aboard ship, until the development of the chronometer by Harrison in
the 1760’s, was the sand filled “hour glass,” which was employed in sizes which
measured one-half minute and one-half hour. Quadrants and Sextants precise
enough to determine latitude with the needed accuracy were not developed
until the 1670’s. The concept of the logarithm introduced by Napier in 1614,
shortly supported by accurate tables by Briggs, substantially eased the burden
of calculation required to solve trigonometric problems with great precision.
The concepts of calculus developed in the 1670’s played a role in advancing
navigational science. The first tables of lunar distances appeared in the Nautical
Almanac in 1767, which allowed sailors to use the position of the moon to
accurately determine the local time.
This paper will describe the navigational techniques recorded in 18th century
manuscript copy books written by young men in Rhode Island between the dates
of 1712 and 1840. This paper describes those techniques, known collectively as
“the sailings.” The sailings begin to appear in British sources dating from
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the late 17th century, were fully developed by the middle of the 18th century,
and were used until the middle to late 19th century. They allow the sailor to
approximate the position of his vessel without recourse to either chronometer or
celestial observations. The practice of this period among merchant vessels was
to combine these sophisticated methods of dead reckoning with a noon-time sun
sighting, which provided a measure of the ship’s latitude once a day, weather
permitting.
Sources and Context
The 17th and 18th centuries saw many societal changes as well as technological
advances. The turn of the 18th century saw the disintegration of the Puritan
theocracies established at Boston, Plymouth, Salem, and Hartford, while the
Glorious Revolution of 1688 in England – replacing Catholic James II with
Protestant William of Orange created a climate in which more autonomy was
granted the New England colonies. A secular enlightenment inspired society
with a rising merchant class evolved whose future prosperity depended on safe
and reliable maritime commerce. Many of the best minds of the age, including
mathematicians, craftsmen, inventors, and sailors made important contribution
to improving navigational methods, but it took more than two centuries from
the landing at Plymouth Rock to develop the set of 19th century methods
that would safely guide ships to ports around the globe before the advent of
electronics and radio.
The manuscript collection of the Rhode Island Historical Society, which started
collecting historical documents and artifacts in the 1820’s, contains about 18
manuscripts written by teenage boys as they were introduced to the mathematics
needed for navigation. Most of these manuscripts date between 1710 and 1830,
and detail a collection of techniques, known as “ the sailings ”, which teach how
the information recorded in the ship’s log can be used to provide a calculated
estimate of the ship’s latitude and longitude at the end of the day. Here the
curriculum ends for most students, whether they be ordinary sailors or students
studying the material as a classroom exercise, as was done in both Academies
and Colleges through the late 18th and early 19th centuries. But for those who
were trained for positions of more responsibility such as ship’s Master, navigator,
supercargo, or ship’s officers of naval vessels, there remains more work to be
done and more mathematics to be learned, and a few of the manuscripts reflect
the more challenging aspects of determining the ship’s position from celestial
observations, rather than by deducing the position from the ship’s log.
2
Small Circles, Great Circles, and Loxodromes
European and American sailors and navigators were familiar with both plane
and spherical geometry. That ships sail not on a plane, but upon the surface
of a sphere is clear. What is not as apparent to the modern reader is that
spherical trigonometry does not address the problem. All sides of spherical
triangles are arcs of great circles, but a ship can not sail along a great circle
unless it is constantly changing its heading and direction of travel. Vessels
under sail, left to their own devices, travel at a constant angle to the wind.
The helmsman makes small corrections via his control of the rudder to sail at
a constant compass heading – which approximates a constant direction. The
bearing of a vessel is defined by the angle the vessel’s centerline makes with a
meridian (a great circle which passes through the North and South poles.) If
the vessel is heading due North or due South, she will travel along a meridian
of longitude (a great circle.) If the vessel is heading due East or due West, she
will travel along a parallel of latitude (a small circle.) But a vessel on any other
point of sail will travel along a “loxodromic spiral,” which may circle the earth
many times, coming ever closer to one of the poles. All such paths approach
either the North or the South pole.
Figure 1: The Loxodromic Spiral
Since lives, ships and quite literally treasure were at risk with every voyage,
and since men lacked the ability to measure either angles or time with sufficient
accuracy to determine their position, how did they find their way back and forth
without meeting disaster even more frequently than they did? How do you keep
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track of where you are and where you are going if your maps are flat and the sea
is curved, and the route on which your path takes you traces a spiral upon the
surface of the sea? The techniques of plain, parallel, mid-latitude, Mercator,
current, oblique, and other sailings were developed precisely to answer these
questions.
Figure 2: John Brown: 1750 copy book. Replica of his 1767 Sloop Katy. Later
recommissioned as the Sloop Providence, first ship of the Continental Navy,
1775. First command of John Paul Jones.
The Ship’s Log and Deduced Reckoning
Direction
Every point on the globe lies on a unique great circle that passes through that
point and through the North and South poles. That circle is the meridian on
which the ship finds itself, and the direction of that arc points directly North and
South. All directions are measured with respect to that arc. The direction may
be measured in degrees, minutes, and seconds clockwise from due North, but
more commonly that circle of 360 degrees is divided into four, eight, sixteen, or
thirty-two equal parts, depending on the precision required. One thirty-second
of a circle is called a point of the compass. If further precision was desired, each
point was divided into quarters. The compass card pictured in Figure 3 has
been divided into 32 points, and a course might be referred to as being 2 points
West of South, each point being equivalent to 11◦150.
4
Figure 3: The Points of the Compass
Distance, Time, and Speed
In order to measure the speed of a ship, the sailor measured the distance the
ship had moved in a fixed amount of time. The tools used to do this were the
log, the log-line, and the half-minute glass. The log-line is a cord, one end of
which is attached to and wrapped around a reel. The other end is attached to
a quadrant shaped piece of wood, called a log. The log is thrown overboard at
the stern of the ship. The arc of the log is weighted with a lead so as to keep
it in a perpendicular position. The positioning of the log in the water makes
it much more difficult to pull the log through the water than it would be to
pull the log-line off it’s reel. Consequently the log acts as a “sea anchor” and
does not move through the water, while the ship and the reel pull away from
the log. The log-line has a knot tied in it every 120th part of a nautical mile.
The sand-glass measures the 120th part of an hour (one-half minute). This
creates a proportional situation . . . the number of knots that the ship sails in
half a minute is the same as the number of miles she would sail in one hour,
since an hour is 120 times as long as the glass measures and a mile is 120 times
as long as the distance between the knots. That is, the number of knots that
pass through the sailors hands in half a minute is numerically equivalent to the
number of nautical miles the ship will move in one hour. The instruments used
are pictured in Figure 4.
Figure 4: The Log, Log-line, and Half-minute glass
5
The speed measured by these instruments is not the speed of the ship with
respect to objects on land, or progress as measured on the nautical chart. It is
the speed with respect to the log, which is a sea anchor. The log is not attached
to the sea floor or to any land. By design, it will not move through the water,
but it will move with the water. If there were no wind to move the ship, but
the ship were in an ocean current, the log and the ship would both move with
the current, and the relative velocity of the ship with respect to the log would
be zero. In the presence of wind, the ship would pull away from the log, but
both the log and the ship are still being moved by the current. The technique
tells us the speed with which the ship moves through the water (which itself is
moving.) A ship moving 3 knots through the water, when the water itself is
moving at 3 knots, may be stationary with respect to the ground, or may be
moving 6 knots with respect to the ground, or some intermediate figure.
The Ship’s Log
The sailor at the helm maintains a detailed record of every change in direction
and every change of speed throughout his watch. The logline and glass are
used to measure speed, and the mariner’s compass with appropriate corrections
measures the direction. This record, called the ship’s log, is used to calculate
the paths along which the ship is moving and to estimate the position of the
ship at the end of his watch. This deduced (or calculated) reckoning of the
ship’s position over time has come to be called a “dead reckoning,” which is
believed to be a corruption of the term deduced reckoning. Once within sight of
an object or location of known position, the sailor replaces his deduced position
with an observed position, and begins a sequence of deduced reckonings over
again, starting from the observed position, until such time as another observed
position can be obtained.
Eighteenth Century Trigonometry
Many navigational cyphering books begin with a study of geometry and trigonom-
etry. The 18th century understanding of trigonometry was quite different than
our understanding today. The concept of sines,cosines, tangents, etc as functions
defined upon the real numbers simply did not exist. Angles were described and
defined by the length of the arc of a circle, rather than the other way around.
The sine or tangent or secant of an angle (or sine of an arc) was understood as
a physical length, and therefore varied with the size of the circle under consid-
eration. The manuscripts speak of the sine of an arc of 30 degrees of a circle
of radius 25, not of the sine of 30 degrees. Tables of sines, tangents, etc., were
constructed based upon circles of convenient radii. Tables exist for a variety of
radii, as small as 1/2 and as large as 10000000000. When using a new table,
one examined the sine of 90 degrees, which would be the length of the radius
of the circle used to construct the table. Further the radius was not always the
6
hypotenuse of a right triangle, but sometimes one of the legs. Their understand-
ing of sine, tangent, and secant is perhaps most easily described in terms of two
diagrams.
Figure 5: Definitions of sine, tangent, and secant
In the left hand diagram in Figure 5, the choice of radius has placed the side
opposite the central angle inside the circle, and therefore the side opposite is half
of the chord of twice the angle. In the right hand diagram, however, a different
choice of radius has placed the side opposite the central angle outside the circle,
in fact, tangent to the circle, and it is this length that is the tangent of the
central angle. The hypotenuse of this triangle is the secant of the central angle.
This is reflected in the origins of the words tangent and secant, which come
from the Latin words for lines that touch and lines that cut (the circle).
Consider the complex of circles superimposed on the triangle in Figure 6. These
four circles define the six trigonometric functions of the acute angle that appears
in the lower left corner of the triangle. Let us call that angle x.
7
The lower two circles are centered on x, the upper two circles are centered on
the complement of x. The larger of the lower circles defines sin(x), the smaller
of the lower circles defines tan(x) and sec(x). The larger of the upper circles
defines the sine of the complement of x(that is, the length of the side opposite
x’s complement), which came to be called the cosine(x), while the smaller of
the upper circles defines the tangent of the complement of xand the secant of
the complement of x, that is x0s co-tangent and co-secant.
Figure 6: An Eighteenth Century View of Trigonometry
The lengths of the appropriate sides of these triangles are the trigonometric
values, rather than the ratio of sides as we define them today. To take into
account the differences between the size of the circle of our navigational problem
and the size of the circle used in our tables, these students used the fact that the
corresponding sides of similar triangles are in the same ratio. Since specifying
a single acute angle of a right triangle determines all three angles, the triangle
in the problem at hand and the triangle used to construct the table of sines
or tangents will always be similar. All problems are problems in proportions,
and algebraic solution of equations is not needed. A proportion is described,
the product of the means is set equal to the product of the extremes, and
the unknown quantity is determined with a single multiplication and a single
division (possibly with the aid of tables of logarithms).
8
Plain Sailing
The foundation of all of the methods for dead reckoning is the technique called
plain (or plane) sailing. In the navigational manuscripts, a path of constant
direction is most often called a rhumb line, rather than a loxodrome. If a
ship sails on a constant course, cutting each meridian at a constant angle she
traverses a rhumb line, and that angle is her course. Knowing how far she
traveled along that course allows us to find the lengths of the two legs of a
right triangle, knowing the angles and the hypotenuse. But the manuscripts
universally explore every possible combination of what could be known or given,
and what could be desired.
The essence of plane sailing is an application of plane right angle trigonometry,
in which one of the acute angles is the course of the ship. What is remarkable
is that although distance, departure, and difference of latitude are each curved
lines upon the surface of a sphere, their lengths are correctly and exactly given
by an application of plane trigonometry, provided that the path of the ship is
along a rhumb line or loxodrome !
If the path of the ship cuts each meridian at a constant angle, a simple calculus
analysis of the incremental northerly (or easterly) progress along the loxodrome
is identical to the northerly (or easterly) progress predicted by the plane right
triangle used in plane sailing. If the course is variable, or if the path is a series of
different courses (course used here in the technical sense of the angle the path
makes with the meridian), then this equivalence between travel on the plane
triangle and travel along a rhumb line does not hold. It is the special quality
of the rhumb line that gives plain sailing its power. The plane triangles that
are used are not mediocre approximations to travel on a sphere, but rather a
full and accurate description of travel on a sphere for paths that are rhumb
lines.
However, this victory is only part of the story. Although plane sailing accurately
tells us how far East or West we have travelled along our rhumb line, what
we really need to know is how many degrees of longitude East or West we
have gone during that passage, and it does not tell us that. At the equator,
traveling 60 miles due East would change the longitude by one degree. We would
have travelled exactly 1/360th the way around the world. Yet at a latitude of
60 degrees, travelling 60 miles due East would change the longitude by two
degrees, travelling 1/180th the way around the world. If we were close enough
to the pole, traveling 60 miles due East could result in a circumnavigation !
Such considerations are beyond the scope of plane sailing, but it is precisely to
account for such situations that the other sailings were developed.
9
Figure 7: Plane Sailing – Sailing on a Rhumb Line
The distance (run along the rhumb line) and the departure (from the meridian)
were always measured or reported in nautical miles. The difference in latitude
was measured or reported in either minutes of arc, or in nautical miles, the
two values being numerically identical, since differences in latitude were always
measured along a meridian (and therefore a great circle.) The manuscripts
are uniform in separately presenting each of the possible cases. Of the four
quantities of interest (the course, the distance, the departure, and the difference
in latitude), the navigator is given the value of two of them and is required to
calculate each of the other two.
If the course (the angle the ship’s path makes with any meridian) is known,
there are three possible cases.
•If the course and the distance are known, then the distance becomes the
radius, and the departure and difference of latitude are the sine and cosine
of the course, respectively, for circles of appropriate radius.
•If the course and difference of latitude are known, then the difference
of latitude becomes the radius and the departure and distance are the
tangent and secant of the course, respectively, for circles of appropriate
radius.
•If the course and departure are known, then the departure becomes the
radius, and the difference of latitude and distance run are given by the
cotangent and cosecant of the course, respectively, for circles of appropri-
ate radius.
10
The eighteenth century understanding of trigonometry makes all of the
problems very easy ... one need only choose the appropriate radius, and
use rules of proportionality to scale upwards to the size of the radius.
If the course is not known, then we will be given two of the sides and asked to
find the course and the missing side. Again, we have three possible cases. We
first list the methods used for determining the course.
•If the distance and departure are known, then the distance is to the de-
parture as the radius of our table is the the tabular sine of the course.
Since the distance, departure, and tabular radius are known, we can solve
for the sine of the course. Then the table of sines is used to find the angle
with the calculated sine. That angle is the desired course.
•If the distance and difference of latitude are known, then the distance is
to the difference of latitude as the radius of our table is to the tabular
cosine of the course. The known values allow us to compute the cosine
of the course, and the tables are used to find the angle with the desired
cosine.
•If the difference of latitude and the departure are known, then the dif-
ference of latitude is to the departure as the radius of the table is to the
tangent of the course. The value of the tangent is found and a table of
tangents used to find the course.
Now, in order to determine the length of the third side, some manuscripts
teach the student to use the Pythagorean theorem, but many manuscripts
instead teach the student to first find the course, and then to use the
course and one of the sides to find the missing side. For example, if the
departure and difference of latitude is known, and the distance sailed is
desired, rather than taking the square root of the sum of the squares of
the sides to find the distance run, instead proceed as follows.
First observe that, since the difference of latitude is a side adjacent to the
course, treat the difference of latitude as a radius. Then, as the difference
of latitude is to the departure, so is the radius of the table to the tangent
of the course. Thus tangent(course) = departure/difference of latitude.
Use a table of tangents to find the course if we are given the tangent
of the course. Then, continuing to view the difference of latitude as the
radius, observe that as the radius of the table is to the tabular secant,
so is the difference of latitude to the distance run. Thus the distance
run is the difference of latitude times the secant of the course (divided
by the radius of the table, which is most likely equal to 1.) This is an
elegant way to view the problem and avoids the tedious and error prone
procedures for extracting square roots. Further, use of logarithms can
reduce the needed multiplications or divisions to additions or subtractions.
(Of course logarithmic tables could be used to assist in the taking of roots,
but I found no evidence in these manuscripts that they were used to do
so.)
11
Parallel Sailing
A single course can be resolved into a departure (from the initial meridian) and
a difference in latitude (from the initial parallel of latitude.) What is desired for
locating the ship, however, is not how far East or West the ship has travelled
but the difference in longitude. If a vessel is sailing due East or due West, the
departure can be converted to a difference in longitude. The course when sailing
due East or due West is a path along a small circle, which on the globe is a
parallel of latitude. The radius of this small circle varies as the cosine of the
latitude. Along a great circle, one minute of arc traces a path whose length is
by definition one nautical mile. Along a parallel of latitude the length of a path
corresponding to one minute of arc on the small circle (measured in nautical
miles) is equal to the cosine of the latitude. Therefore, in the special case when
a ship is sailing along a parallel of latitude, the difference in longitude can be
calculated by dividing the departure by the cosine of the latitude, or equivalently
by multiplying the departure by the secant of the latitude. Such a course, winds
permitting, can be sailed by maintaining a compass heading of East or West
and verifying that the latitude remains constant by measuring the altitude of
Polaris above the horizon or the altitude of the sun above the horizon at local
noon.
Mid-Latitude Sailing
When sailing on a course other than 90 ◦or 270 ◦, mid-latitude or middle lati-
tude sailing was often used to convert a departure into a difference in longitude.
Since the ship is not sailing at a constant latitude, the average of the initial and
final latitude for the course was defined as the mid-latitude. The departure was
divided by the cosine of the mid-latitude to provide an estimate of the difference
in longitude. This technique is reasonably accurate for short distances, partic-
ularly near the equator where the meridians do not converge quickly as one
travels North or South. For distance greater than a few hundred miles or when
sailing in northern or southern waters, this method is not sufficiently accurate,
and the student was advised to use the technique of Mercator Sailing.
Mercator (or Wright’s) Sailing
When sailing at high latitudes (North or South), or when sailing for a distance
of more than a few hundred miles, the approximations of middle-latitude sailing
(even if corrected via a table of corrections) lead to errors large enough to
endanger the safety of the ship.
If the course is not due East or West, the path of a ship will cross every latitude
between the initial and final latitude, thus requiring a different factor to convert
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miles of departure to difference of longitude for each of the infinite number of
parallels crossed.
Methods based upon the charts of Gerard Mercator (ca. 1556) and upon the
treatises of Edward Wright (ca. 1596) provided an accurate method for calculat-
ing the difference in longitude when a course was sailed under these conditions.
Mercator’s chart was developed so that any rhumb line (or path of constant
direction) appeared as a straight line upon the chart. Since all meridians are
rhumb lines (direction North or South), the meridians appear as straight lines.
Since any loxodrome will, by definition, cross each meridian at a constant angle
(the very definition of constant direction), the meridians, as drawn on the chart,
must not only be straight, but parallel to each other.
Figure 8: Design of Mercator’s Chart
Since the distance between any two meridians on the surface of the globe, as
measured along a parallel of latitude, will diminish as the cosine of that lati-
tude, the distance between two meridians on a Mercator chart at any particular
latitude will not be the true distance, but the distance scaled by the reciprocal
of the cosine of the latitude (that is, scaled by the secant of the latitude). In
order to preserve rhumb lines as straight lines, the same scaling must be applied
to distances between parallels.
If we imagine travelling along a meridian, starting at one parallel and ending at
another. The distance between the initial and final parallels, as they appear on
a Mercator chart, will vary with the latitude of those parallels. If the path is
divided into a large number of very small steps, then each increment in latitude
13
will appear on the chart multiplied by the secant of the latitude at that point
along the way. The distance between the initial and final parallels on the chart
will be the sum of these products. The tables of “meridional parts” that appear
in the manuscripts and texts of the time give the results of calculating such
sums for paths divided up in segments of one minute of arc. The tables list,
for every degree of latitude, the sum of the secants of each latitude from the
equator to the tabular entry where the secant is calculated for each minute of
arc along the way. By subtracting cumulative sum of secants for the latitude
of the point of departure from the cumulative sum of secants for the latitude
of the point of arrival, a “meridional difference of latitude” is determined. This
value approximates the integral Rθ=final latitude
θ=initial latitude sec(θ∗π/180)dθ where θis
measured in degrees of arc. This meridional difference of latitude is used to
determine the difference of longitude which corresponds to the known departure
from the meridian.
Mercator sailing is founded on the similarity of two right-triangles, in one of
which the perpendicular sides are the proper difference of latitude and the de-
parture, and in the other , the meridional difference of latitude and the difference
of longitude. The hypotenuse of the first triangle is the distance on the rhumb,
while the hypotenuse of the second triangle is unrelated to either latitude or lon-
gitude; and since it had no navigational significance, it was given no name. Since
corresponding sides of similar triangles are proportional, as the true difference
of latitude is to the departure from the meridian, so is the meridional difference
of latitude to the difference in longitude. Thus the difference of longitude is
equal to the meridional difference in latitude times the departure, divided by
the proper difference in latitude. Equivalently, the difference in longitude is the
meridional difference of latitude times the tangent of the course.
Figure 9: Mercator Calculation of Difference in Longitude
14
Navigators were aware that this method, although precise in theory for any
situation, suffered from the discrete approximations of the table of meridional
parts, and students were advised that unless they obtained a more finely divided
table, the method of middle–latitude sailing would give more accurate results
than the method of Mercator sailing if distances were small or if the course
were near the equator. Mercator sailing was also avoided for courses near 90 ◦
or 270 ◦, where the small errors in the value of the course lead to large errors
in the difference of longitude, since the tangent of an angle varies extremely
rapidly for angles near 90 ◦or 270 ◦.
Traverse Sailing
The preceding techniques allow the sailor to estimate the distance travelled, the
departure, etc. for a single course. Yet ship’s are rarely able to complete a
voyage by traveling along a single course or direction of travel. The journey
instead proceeds along a series of many courses due to the need to travel in an
upwind direction or due to the nature of shifting and changeable winds. Such
a change of course is likely to occur every few hours, or in channels, bays, etc.,
several times per hour. The irregular path of a ship sailing in this manner is
called a traverse. Every day at noon, the practice was to reduce the previous
24 hours of courses to a single equivalent course. The process of reducing a
compound course to a single course was called resolving a traverse. Several
successive days of sailing could also be resolved to give an indication of the
current position of the ship.
Two methods were used. The first was appropriate only for small distances
sailed in the vicinity of the equator. A more accurate method was used for
longer distances and when far from the equator. Considering the earth to be
a plane surface, the distance traveled on each course can be resolved into a
departure from the initial meridian and a difference in latitude, each measured
in nautical miles. The algebraic sum of the departures (E vs. W) and the
algebraic sum of the differences in latitude (N vs. S) give a resolved departure
and difference of latitude, which can be expressed in terms of a resolved course
and distance run. The method of Mercator sailing can then be used to calculate
the difference in longitude, thus predicting the latitude and longitude at the
end of the series of courses. The method assumes that the departure of the ship
traveling along several courses is identical to the departure of a ship sailing along
a single course to the same place. When the direction of the ship on each leg
is different, so are the two departures. However, both difference of latitude and
difference of longitude do remain the same, whether the path is along a single
course or several. The more accurate method calculates not the difference of
latitude and departure for each course, but rather the difference of latitude and
difference of longitude for each course, then finds the algebraic sums of difference
of latitude and difference of longitude to resolve the traverse.
15
Current Sailing
If there is a tide or current and a log is thrown upon the water and left at liberty,
then the log will move along with the water in the direction of the current and at
the same speed as the current. The motion of a ship at rest would be the same.
For this reason, the rate of sailing as measured by the log-line is the motion of
the ship through the water rather than the motion of the ship compared with
an object on land or objects such as navigational aids attached to the sea floor.
If a ship is steered in the direction of the current, its actual rate of progress
will be the sum of the speed of the current and the speed of the vessel through
the water as measured by the log-line. If a ship is steered against the current,
its rate of progress will be the difference between the motion through the water
and the speed of the current. If the ship steers at an angle to the current,
the movement of the ship is the vector sum of the ship’s movement through
the water and the movement of the water over the ground. The method of
current sailing advises the sailor to treat the motion of the water, if known, as
an additional and separate course and distance to be added to the courses and
distances recorded in the ship’s log.
Suppose a ship sails NE for 2 hours at a rate of 8 kts and then sails SE for 4 hours
at a rate of 7 kts in an area where the current is known to be 30 degrees South
of West at a rate of 3 kts. The first course travels 16 miles in a NE direction.
The northing will be 11.3 miles and the easting 11.3 miles. The second course
has a distance run of 28 miles in a southeasterly direction, with a departure of
19.8 miles to the eastward and a similar movement southward. In the absence of
any current, the vessel has moved 8.5 miles southward and 31.1 miles eastward.
The current during those 6 hours has been moving at a rate of 3 kts, so the
water has moved 18 miles in a direction 30 degrees South of West. The water
has moved 9 miles South and 15.6 miles westward. Adding this motion to the
previously calculated movement in the absence of current gives us a combined
movement of 17.5 miles South and 25.5 miles east of the original position. The
combined effect of current and the two courses is to have moved a distance of
30.9 miles on a course of 34◦27 0South of East.
Oblique Sailing
Oblique sailing was used to determine the latitude and longitude of a ship as
she left a harbor or bay at the start of a journey. It is crucial to start a de-
duced reckoning from a known latitude and longitude rather than an estimated
one.
16
Figure 10: Obtaining an Initial Position
As a vessel left a bay or harbor, a bearing was taken on a prominent object
on land whose latitude and longitude were known from the chart. In Figure
10 above, the castle on land (whose latitude and longitude is known) is labeled
A, and the ship starts its voyage from the point labeled B. A bearing measures
the angle between the North-South line and the object observed. The bearing
between the ship at point B and the castle is marked b1, and its equal opposite
interior angle is marked (b1). If the distance between points A and B were
known, the methods of plane sailing and middle latitude sailing could determine
the position of the ship, but the distance from B to A is not known. The
ship proceeds along any convenient course from point B to point C. The only
requirement is that the distance traveled from B to C be measured (marked d1
on in the diagram,) and that the bearing to the castle change significantly from
the original bearing. A new bearing on the land object is taken at C. The second
bearing is marked b2, and its equal opposite interior angle is marked (b2). All
three angles of the oblique triangle ABC can are now determined. The angle at
B is the sum of b1 and c1, the angle at C is the difference of b2 and c1), and
the angle at A is 180◦−b1−b2. Since all three angles and one side of triangle
ABC are known, the lengths of the unknown sides can be determined by use of
the Law of Sines. The distance we are interested in is the distance from A to
C, marked d2 in the diagram. d2 = d1 sin(B)
sin(A). If the sailor imagines, or accepts
17
the fiction, that the vessel departed not from point B, but from point A, along
a course given by (b2) for a distance run given by d2, the methods of plain
and mid-latitude sailing will provide the latitude and longitude of the ship at
point C. The sailor now has a very accurate starting point for the latitude and
longitude of the ship as it heads out to sea, and the keeping of a dead reckoning
may begin upon a solid foundation.
The method of oblique sailing was also used to plan tacks of the ship when
sailing to windward. Ships can not sail directly into the wind, but can progress
in that direction by a zig-zag path, first sailing as close to the wind as possible
but to one side of the wind, then sailing at the same angle from the wind, but
on the opposite side of the wind. To determine how far to sail in one direction
before changing to the other direction requires solving for an oblique isosceles
triangle, also using the Law of Sines.
Great Circle Sailing
Circles upon a sphere
The curve of intersection of a sphere made by a plane is a circle. If the plane
passes through the center of the sphere, the section is a great circle, while if
the plane does not pass through the center of the sphere, the section is a small
circle. The radius of a great circle is the same as the radius of the sphere.
Distances between points on a sphere
Just one plane can be passed through the center of a sphere and any two points
on the surface, unless the two points are the ends of a diameter of the sphere.
This will define a unique great circle passing through the two points, and the
two points divide that great circle into two portions. The length of the smaller
portion is the distance between the points. Since all distances on a sphere are
measured by arcs of the same radius and since the length of an arc of a circle
is measured by the angle it subtends at the center of the circle, distances on a
sphere are best measured in degrees or radians.
Spherical Angles
The intersection of two arcs of great circles, defines a spherical angle. The angle
is measured by the dihedral angle formed by the two planes that define the great
circles involved, and subsequently to the angle between the tangents to the two
circles at their point of intersection.
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Spherical Triangles
A spherical triangle is formed by the arcs of three great circles that do not all
pass through the same point. The three sides are measured in degrees of arc
(each measured by the central angle of the arc of the great circle that forms a
particular side.) The three vertices are each spherical angles, also measured in
degrees (each determined by the dihedral angle of the planes defining the two
sides that intersect at a particular vertex.) There are many relationships among
the six parts that define the triangle, but the ones needed for great circle sailing
are the Spherical Law of Sines and the Spherical Law of Cosines, as detailed in
the paragraphs that follow.
The Methods of Great Circle Sailing
Two points on the surface of the globe, along with the North pole, determine a
spherical triangle for sailors in the northern hemisphere. (Sailors in the southern
hemisphere used the South pole as their third vertex.) Two of the sides of this
triangle are arcs of meridians which extend from each point to the pole, and
thus the length of each of those sides (measured in degrees as appropriate for
distances on the surface of a sphere) is the complement of the latitude of the
respective points. The third side is the great circle route between the two points,
and it is the length of this side which we wish to find. The difference in longitude
between the two points is a dihedral angle which measures the spherical angle
at the North pole. Since we know the lengths of two sides and the measure of
the included angle, the Law of Cosines for spherical triangles can be used to
find the length of the side opposite the included angle. The value of the two
other angles in the triangle can be found using the Law of Sines.
Figure 11: The Navigational Triangle
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In the triangle drawn in Figure 11, sides b and c are the co-latitudes of the
points of departure and arrival, respectively, and the angle αis the difference
in longitude between those points (all measured in minutes of arc).
From the Law of Cosines it follows that cos(a) = cos(b)cos(c)+sin(b)sin(c)cos(α).
Since all three sides and one angle are now known, the Law of Sines for spherical
triangles, sin(a)
sin(α)=sin(b)
sin(β)=sin(c)
sin(γ)), can be used to find angles βand γ. These
values can be used to determine the initial course when departing and the final
course when arriving for a ship sailing a great circle course, but does not tell
the sailor what his course should be at any other time on the journey. Thus this
sailing is more of academic interest, rather than of practical use, and is absent
in many manuscripts.
Celestial Observations
The various sailings allowed the sailor, starting from a known position on the
globe, to use the information recorded in the ship’s log to calculate or deduce
the latitude and longitude of the ship at every step of its journey, in ways which
accounted for the curvature of the globe, the nature of the loxodromic paths
the ship followed, the effects of currents, and the convergence of meridians with
increasing distance from the equator. Regardless of the Sailing used, however,
eventually it is necessary to determine how close the predicted position is to the
ship’s true position. If in mid-ocean, no land references are available, and only
visible reference objects are in the heavens.
Observations Which Determine the Latitude
The Polaris Sight
By observing the North star (Polaris) at a time when the horizon is visible, one
may determine the latitude of the ship (the altitude of Polaris above the horizon
is the latitude of the observer). If the sky is too bright, the Pole star can not be
seen; if the sky is too dark, the horizon can not be seen. These polar sightings
can thus only be performed at dusk or at dawn, or when the moon brightens
the sky – but not so much so as to prevent the sighting of Polaris, which is but
a second magnitude star.
20
Figure 12: The Noon Sun Sight
The Noon Sun Sight
Every noontime, when the sun was not obscured by clouds, a Noon Sun Sight
was taken to determine the ship’s latitude. A sextant or similar instrument was
used to measure the altitude of the sun above the horizon at its highest point.
Almanacs provided tables of the sun’s declination (the altitude of the sun above
(or below) the celestial equator) for each day of the year. The measured altitude
of the sun at its highest point, adjusted for the declination, provides the latitude
of the observer.
Correcting the Dead Reckoning Position
If the dead reckoning position has a latitude other than the latitude observed
through a polar or noon-sighting, then the latitude of the deduced position
must certainly be altered, but it is unclear in what way we should alter the
predicted longitude. The error in predicted position could be due to an error in
estimating the distance run. In that case the direction of the rhumb line should
be accepted as correct, and the line extended (or contracted) to intersect the
21
observed parallel of latitude. On the other hand, the error could be due to an
error in estimating the course, while the estimated distance run is correct. In
that case the corrected position will be at the intersection of the observed parallel
of latitude with a circle, centered at the beginning of the course segment, with
the distance run as its radius. A ship under the influence of a strong current will
likely show errors in both the course and the distance run. In practice, sailors
picked what seemed to them a reasonable point in the vicinity of the deduced
position, but with the observed latitude, and hoped for the best.
The latitude can be directly determined at noon and, with lesser precision, at
dawn and dusk. Yet ships do not stay stationary, and so their latitude will
change throughout the day and must be estimated from the last Polaris or noon
sun sight. In nautical terminology, this is called advancing or retarding the
previous position, and it is the methods of the sailings and of dead reckoning
that are used to update the last known latitude.
Only the most advanced of the manuscripts examined considered techniques
beyond those already described. However, more powerful techniques, developed
in England and in France, and promoted by forward looking practitioners such
as Nathaniel Bowditch. However, such methods were seldom used in the Amer-
ican colonies or upon merchant ships until the early decades of the nineteenth
century. Only two manuscripts examined contained considerations of celestial
observations for the purpose of determining longitude (those of Page and Peck)
. Those manuscripts, in addition to a detailed study of the sailings, contain
examples of calculations needed for the use of lunar distances and time sights
which, in combination, provide an indication of longitude.
Observations Which Determine the Longitude
This inability verify or validate the longitude of the dead reckoning position
spurred efforts to use celestial observations, sometimes together with mechan-
ical devices, to determine the ship’s true longitude. In order to determine the
longitude of a vessel, one must know the local apparent time at the ship’s merid-
ian together with the local apparent time at the same instant at a location of
known longitude (a reference meridian.) Each hour of time difference corre-
sponded to a 15◦difference in longitude.
The reference meridian for British and American shipping was the meridian at
the Royal Observatory at Greenwich. The two most practical ways of determin-
ing the time at Greenwich were to use the tables of ephemera published in the
Nautical Almanac or to use the newly invented chronometer.
The Time Sight
The determination of local time required the taking of a “time sight.” At first
glance, the sun noon sight should provide an indication of both latitude and
22
local time, but since the sun appears to hang stationary in the sky for up to
several minutes as it approaches its highest point, it can be difficult to determine
the exact moment of local apparent noon. Any error in the determination of
local time, leads to an error in longitude. The estimate of both local apparent
noon and of longitude can result in considerable errors in the position of the ship
– as much as 100 miles or more (certainly enough to put the ship in danger.)
The time sight was a technique used to more accurately find the local time, and
therefore the longitude.
The sun moves more rapidly through the sky when it is not at the peak of its
arc. A time sight was taken in the morning or in the afternoon, when the sun
was in the East or in the West, by measuring the altitude of the sun above
the horizon. A spherical triangle was visualized with the location of the sun
against the celestial sphere as one vertex of the triangle, and the zenith (the
point in the heavens directly overhead) and the celestial pole as the other two
vertices.
Figure 13: The Navigational Triangle
Measuring the sides of this spherical triangles in angular measure, the distance
from the sun to the pole is the complement of the sun’s declination (obtainable
from the almanac entry for that day), the distance from the sun to the zenith
is the complement of the sun’s altitude (measured with the sextant), and the
distance from the pole to the zenith is the complement of the ship’s latitude
(which was measured some hours earlier at the sun noon sight, and advanced
to the current time through dead reckoning and use of the sailings.) Since we
know three sides of a spherical triangle , we can calculate any of the angles.
The angle of interest is that between the meridian passing through the sun
and the meridian passing through the zenith. This angle, called the local hour
angle, tells the difference between the ship’s meridian and the sun’s meridian.
Our knowledge of Greenwich time tells the difference between the Greenwich
meridian and the sun’s meridian (called the Greenwich hour angle). A simple
addition or subtraction tells us the difference between the ship’s meridian and
that of Greenwich, i.e. the longitude. If we are interested in time rather than
longitude, the local hour angle tells us how much time separates us from local
apparent noon, and does so with much more precision that the noon sight.
Note, however, that the method requires a value for the latitude of the ship in
the early morning or in the late afternoon. That information will have been
collected at the previous Polaris or noon sun sight and advanced to our current
time through dead reckoning and the sailings.
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The Day’s Work
The determination of the ship’s position at day’s end was referred to as “the
day’s work,” and the gathering of the data needed to determine that position
was referred to as “working the day’s work.” A full understanding of the ability
to determine latitude and longitude at one point in time with a few celestial
observations took most of the nineteenth century to develop. Until the middle of
the nineteenth century it was necessary to gather bits of information throughout
the day: a polar sighting at sunset and another at sunrise, a morning time sight,
a noon sun sight, and perhaps an afternoon time sight, together with information
from the ship’s log and use of the sailings to bring the latitude measured at noon
backwards to the earlier sightings, and forward to the afternoon sightings, to
determine the latitude and longitude at day’s end. A dead reckoning would also
be kept and the two would be reconciled. Doing the “day’s work” seems an
appropriate description of the task.
Continued Use of the Sailings in the Nineteenth Century
At first, chronometers were scarce, extraordinarily expensive, and of variable ac-
curacy; and for many decades they were available only to naval vessels of impor-
tance. Most navigation continued to be carried out through deduced reckoning
supported by the mathematics of the various sailings, with periodic checks on
the accuracy of the reckoning through determination of latitude via a noon sun
sight. Even when chronometers became more affordable and readily available in
the 1820’s and 30’s, it was necessary to continue the use of the sailings, both to
advance the noon latitude position for the determination of local time, and to
maintain knowledge of the ship’s position when clouds prevented any sightings
of sun, moon, or stars.
For these reasons, the methods of the sailings continued to be a central feature in
the education of those who aspired to go to sea throughout the nineteenth cen-
tury, despite the advent of plentiful and affordable chronometers and advances
in the navigational sciences. In the end, new insights into celestial observations,
the demands of pilots, not of ships, but of aeroplanes, and the development
new technologies such as radio beacons, radiotelegraph time signals, and most
recently satellite and GPS navigation equipment have replaced these centuries
old navigational techniques with other methods. Yet, for over two hundred and
fifty years the contributions of those who developed and taught the sailings re-
mained vital for keeping precious lives and cargo safe, and for maintaining the
flow of maritime commerce around the globe.
Appendix A: List of manuscripts consulted
Rhode Island Historical Society Manuscripts
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•1712 Jahleel Brenton, age 22, aristocrat, merchant, captain
•1719 James Browne, age 18, ship owner, merchant
•1726 Edouard LeGros, age unknown, Newport seaman and merchant
•1750 Moses Brown, age 12, merchant, industrialist, educator, Quaker,
abolitionist
•1753 John Brown, age 17, merchant, China Trade
•1763 & 1770 George Arnold, age 16 & 23, captain of both fishing and
trading vessels
•1792 Eliab Wilkinson, age 19, schoolteacher, almanac writer, surveyor,
banker
•1792 George Utter Arnold, age 16, mill owner, store owner, justice of the
peace
•1805–1818 Martin Page, age 15, Seaman, Captain, Ship’s Master and
supercargo for Brown and Ives, merchants in the West Indies and China
Trades
•1829, 1835, 1840 Viets Peck, age 15 – 26, Merchant, Captain, father in-
volved in slaving and smuggling at Port Royal (Jamaica) & Havana
25