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Nautical Astronomy : From the Sailings to Lunar Distances

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From the mid sixteenth century through the mid eighteenth century advances in science, mathematics, and technology enabled the navigator, cartographer, or surveyor to determine both his latitude and longitude from celestial observations. This paper explores the history of the development of those techniques. The period during which these ideas grew from theoretical speculations to practical tools, a period of two and one-half centuries, was spurred on by the �nancial encouragement of governments and commercial interests and the contributions of major �gures such as Kepler, Newton, Briggs, Napier, Vernier, Harrison, Mayer, Flamsteed, Maskelyne, Bowditch, and others. ====================================== �A partir du mi seizi�eme si�ecle par le mi dix-huiti�eme si�ecle, les avances en science, les math�ematiques, et la technologie ont permis au navigateur, au cartographe, ou �a l'arpenteur de d�eterminer sa latitude et longitude des observations c�elestes. Cet article explore l'histoire du d�eveloppement de ces techniques. La p�eriode l'o�u ces id�ees se sont d�evelopp�ees des sp�eculations th�eoriques aux outils pratiques, une p�eriode d'environ deux cents et cinquante ans, a �et�e incit�ee par l'encouragement �nancier des gouvernements et des interests commerciaux et les contributions des �gures principales telles que Kepler, Newton, Briggs, Napier, Vernier, Harrison, Mayer, Flamsteed, Maskelyne, Bowditch, et d'autres.
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Nautical Astronomy : From the Sailings to Lunar
Distances
Joel Silverberg
Dept of Mathematics
Roger Williams University
Bristol, Rhode Island
USA
Presented at York University, Toronto, Ontario, May 26, 2006
Canadian Society for the History and Philosophy of Mathematics
Abstract
From the mid sixteenth century through the mid eighteenth century advances
in science, mathematics, and technology enabled the navigator, cartographer,
or surveyor to determine both his latitude and longitude from celestial observa-
tions. This paper explores the history of the development of those techniques.
The period during which these ideas grew from theoretical speculations to prac-
tical tools, a period of two and one-half centuries, was spurred on by the financial
encouragement of governments and commercial interests and the contributions
of major figures such as Kepler, Newton, Briggs, Napier, Vernier, Harrison,
Mayer, Flamsteed, Maskelyne, Bowditch, and others.
`
A partir du mi seizi`eme si`ecle par le mi dix-huiti`eme si`ecle, les avances en science,
les math´ematiques, et la technologie ont permis au navigateur, au cartographe,
ou `a l’arpenteur de d´eterminer sa latitude et longitude des observations c´elestes.
Cet article explore l’histoire du d´eveloppement de ces techniques. La p´eriode
l’o`u ces id´ees se sont d´evelopp´ees des sp´eculations th´eoriques aux outils pra-
tiques, une p´eriode d’environ deux cents et cinquante ans, a ´et´e incit´ee par
l’encouragement financier des gouvernements et des interests commerciaux et les
contributions des figures principales telles que Kepler, Newton, Briggs, Napier,
Vernier, Harrison, Mayer, Flamsteed, Maskelyne, Bowditch, et d’autres.
1
Introduction
The growing importance of the exploration of the New World spurred an urgent
need for enhanced navigational methods beyond dead reckoning and observa-
tions of the highest elevation of the sun above the horizon. Although theoretical
speculations on how longitude might be determined from celestial observations
can be found as early as 1514, it was in the early 1600’s that the search began in
earnest. The hundred and fifty years between the early seventeenth and the mid
eighteenth centuries witnessed the invention of accurate quadrants and sextants;
the first published tables of lunar positions with respect to the sun, planets, and
fixed stars; the invention of logarithms and logarithmic scales and tables; and
the invention of accurate pendulum clocks for land use and chronometers for
marine use. This paper will outline these advances and will explore the origins
and development of the mathematics that made these methods practical.
Early Navigational Techniques
The earliest navigational techniques were based on sailing within sight of land,
hugging the coast from one port to another. As commerce took sailors further
afield, the need arose for navigation techniques which could be used out of
sight of land. The discovery and use of the magnetic compass became the
foundation for these early methods. The earliest European guides to navigation
were called portolans and gave compass headings and distances between various
ports of call. Some gave detailed descriptions of harbors including descriptions
of prominent landmarks and warning of hazards to navigation. More elaborate
portolans included sketches of the landscape in various ports to aid the sailor.
Some very early examples of portolans [circa 1400] can be viewed at the Dibner
Institute’s web site devoted to the manuscript of Michael of Rhodes. 1
Parallel Sailing
When sailing long distances out of sight of land, as was necessarily the case for
transatlantic travel (and for some Mediterranean routes of travel), the practice
was at first to sail as closely as possible along a parallel of latitude. This proce-
dure was used because it is possible to verify the predicted latitude. At dawn,
noon, and dusk, the latitude provided by the dead reckoning was compared with
the observed latitude and changed if necessary. But no procedure was available
to validate the longitude of the dead reckoning. Sailors developed the strategy
of sailing northward or southward until they reached the desired latitude, vali-
dated by measuring the altitude of the sun at noon or the altitude of the pole
star at dawn or dusk, then attempting to sail eastward or westward along that
1See http://dibinst.mit.edu/michaelofrhodes/manuscript/.
2
parallel until the ship approach a coastline. If the ship missed the desired port,
another noon observation could tell them whether to travel northward along the
coast or southward until they reached their destination.
Other Sailings
But the helmsman can rarely choose his direction of sail. The direction in which
a ship can travel is greatly constrained by the direction of the wind. The path
of ships is also effected by unseen currents and the tendency of the vessel to
slip to leeward (downwind). The method of parallel sailing was a hazardous
one as well, and many a ship was lost on a reef or shoal before it reached the
coast if it sailed into waters other than those the navigators thought they were
approaching.
Methods which allowed ships to take their natural course with respect to the
wind, rather than attempting to sail along a parallel of latitude were used in
some parts of the Mediterranean as early as 1400 [?] and were refined in the
16th and 17th century following the insights of Gerard Mercator (ca. 1556) and
Edward Wright (ca. 1596). [?,?,?,?] With the development of nautical charts
uthe Mercator projection, and the development of tables of “meridional parts,”
sailors adapted these techniques to take into account that ships sailed, not upon
a plane surface, but upon the surface of a globe, and that they sailed neither
upon straight lines nor upon great circle routes, but upon loxodromic spirals.
The complex set of methods that produced a correct dead reckoning for a ship
sailing upon a series of loxdromes was known as “the sailings.” The methods
of the various sailings, used for transatlantic navigation until well into the 19th
century, are extensively discussed in a previous article prepared for this journal.
[?].
The Determination of Latitude: The Noon Sight
Perhaps the most fundamental of celestial observations was the noon sight.
The sun is at its highest elevation above the horizon at local apparent noon.
The sextant (and before that the backstaff, cross staff, or astrolabe) was used to
measure the altitude of the sun above the horizon for a period of time bracketing
local noon. The maximum altitude could be used to find the latitude of the
observer and the moment when that maximum occurred was used to mark the
moment of local noon. The angular distance of the sun above or below the
celestial equator varies with the seasons throughout the year and was available
in tables of “declinations” for each day of the year from classical times.
At local noon, the sun, the zenith (the point in the sky directly overhead of the
observer), and the celestial poles all lie on the same meridian. The latitude is the
arc length from the celestial equator to the zenith. If, on a given day, the position
3
of the sun is north of the celestial equator, as illustrated in the diagram below,
the latitude of the observer (arc length of the observer’s meridian, between the
equator and the observer) can be found by either the arc length of the meridian
between the celestial pole and the observer’s horizon or the arc length of the
meridian between the zenith and the celestial equator, each of the arcs being
complementary to the same arc.
A twilight observation of the altitude of the pole star above the horizon uses
the first possibility, while the zenith distance of the sun (the complement of the
altitude above the horizon), corrected for the declination of the sun above or
below the celestial equator uses the second possibility. Thus the sailor measured
the altitude of the sun above the horizon at its highest point, then subtracted
that value from 90to find the zenith distance of the sun. Then the sailor
consulted an almanac to determine the sun’s declination above or below the
celestial equator for that date. If the sun was above the equator, the declination
was added to the zenith distance to determine the observer’s latitude, else it
was subtracted.
The Determination of Longitude
The early years of the 16th century saw the first theoretical speculations on how
one might determine the longitude of a particular location. Johann Werner, a
follower of Regiomontanus from Nuremburg produced a work on geography enti-
tled In Hoc Opere Haec Continentur Moua Translatio Primi Libri Geographicae
Cl’Ptolomaei, published in 1514, in which he describes a method of lunar dis-
tances, mapping the positions of fixed stars and calculating the longitude from
the passage of the moon across each one. He also describes an instrument with
an angular scale on a staff from which degrees could be read off. He makes a
4
study of map projections and gives a method to determine longitude based on
eclipses of the moon. In 1531, Gemma Frisius, a Flemish cartographer and as-
tronomer, published Principiis astronomiae in which he describes how a clock
could be set on departure, keeping absolutely accurate time, and compared
with local time on arrival (as determined by solar observation). Both men were
important influences on Gerard Mercator who first developed and published
maps and charts uwhat has become known as the Mercator projection. Neither
Werner’s nor Frisius’ proposed methods were practical for centuries to come.
The difficulties to be faced and overcome were considerable and spread over a
wide range of disciplines.
This illustration shows a surveyor ua cross staff to determine the height of a
tower, a pair of surveyors ucross staffs to determine how far away a fortification is
from their location, and an astronomer (note the globe, quadrant, and dividers)
measuring the distance between the moon and either a planet or a bright star
near the ecliptic in order to determine his longitude. The illustration appears
on the fronticepiece of Introductio geographica by Petrus Apianus, published in
Ingolstadt in 1533.
In this paper we will concentrate on describing the evolution of the method of
lunar distances, as proposed by Johann Werner, although we will also comment
from time to time about the similarities and differences between this method
and the use of time pieces to determine longitude. We will describe both the
observational and computational methods that were developed to determine
longitude from lunar distances along with the challenges and difficulties that
presented themselves along the way.
5
Developing Tables of Lunar Positions : Problems in celes-
tial mechanics
The essence of the lunar distance method is that a table has been prepared that
records the position of the moon, sun, and planets against the fixed stars at
frequent intervals throughout the day. The sun moves through the fixed stars
at a rate of approximately one degree per day. The planets generally move even
more slowly, but the moon moves through half a degree of arc in a hour and
thus provides a time keeper useful for measuring hours and minutes, rather than
days. A detailed and accurate prediction of the moon’s orbit is needed to pro-
vide such a table. As the contributions of Galileo, Copernicus, and Kepler were
applied to this problem, predictions were made, but proved inaccurate when
compared to direct observations of the moon’s position. Newton also tackled
this problem, but in the end accurate predictions of lunar position evaded the-
oretical solution due to the fact that the moon is strongly influenced by both
earth and the sun, a classic “three body problem.” An empirical approach was
eventually successful. The founding of Greenwich Observatory (and others) and
the collection of nearly a century of data culminated in tables of accurate lu-
nar positions. The observatory was established in 1675 and Flamsteed began
to collect his lunar observations, a project that was to continue for nearly 40
years.
The first accurate tables of lunar positions were prepared by Tobias Mayer, a
professor of economics and mathematics at G¨ottingen, in 1753 who sent them
to the British government in 1755. These tables were good enough to determine
longitude at sea with an accuracy of half a degree. It was Mayer who discorvered
the libration of the moon and the first edition of Maskelyne’s Nautical Almanac
(1767) included Mayer’s tables, with a statement that they were sufficiently
accurate to determine one’s longitude to within a degree. Comment is also
made that the difficulty and length of the necessary calculations seemed the only
obstacles to hinder them from becoming of general use, a consideration that we
will return to at a later point. Mayer’s method of determining longitude by lunar
distances and a formula for correcting errors in longitude due to atmospheric
refraction were published in 1770 after his death. [?]
Accurate tables are a necessary tool for determining longitude, but not sufficient.
One also needs to be able to measure both angles and time with great accuracy.
The first issue is a technological one, but issues surrounding the measurement
of time has, in addition to technical issues, both theoretical and mathematical
components.
The problem of angles
Werner envisioned ua cross staff or an angular scale on a staff to measure the
angles required, but it’s accuracy fell far short of what was needed. The backstaff
6
was invented by John Davis, an English captain in 1607. The accuracy and
usefulness of the medieval quadrant was also improved and we have the first
report of a trial of the method of lunar distances at sea in 1615. The sextant
was developed shortly afterwards, and soon supplemented by Vernier scales by
1638. By the 1730’s the design of sextants and quadrants were refined to their
modern form and were capable of measuring angles large and small, and with a
precision of a tenth of a degree.
The Celestial Sphere and Spherical Trigonometry
The earth and the observer are imagined to sit at the center of a large sphere,
upon which are projected the positions of the sun, moon, planets, and stars.
Also projected upon this sphere are the equator and poles of the earth, the lines
of latitude and longitude on the earth’s surface, and the horizon and zenith as
seen by the observer.
Spherical triangles are formed on the surface of a sphere by the intersection
of three great circles. In spherical trigonometry both the sides and vertices of
a spherical triangle are measured in degrees of arc. The length of a side is
measured by the central angle defining the portion of the great circle that is
the side of the triangle. In the figure left hand below, the central angle c is the
measure of side c on the surface of the sphere.
7
The vertex angles are measured by the dihedral angle of the planes defining the
great circles that form the two sides meeting at that vertex. In the right hand
figure above, the vertex labeled A is formed by the intersection of two planes
pasthrough the center of the sphere. The angle between these two planes at the
center of the sphere, labeled αis the measure of the vertex angle at A.
Each spherical triangle has three sides and three vertex angles, each measured in
degrees or minutes of arc. In the figure shown below, the measure of the three
sides are labeled a, b, and , c and the measure of the three angles are labeled
α, β, and γ. Given any three of these six quantities, it is possible to determine
the remaining three quantities. There is a spherical law of , which looks much
like the law of in plane trigonometry, but it is the laws of cothat are most
useful for navigational problems. There are two forms of these laws, one which
was called the law of cofor sides and one which was called the law of cofor
angles.
Lunar Distances: the simple version
The basic concept of determining lunar distances is to measure the angular
separation between the moon and another object (usually the sun) and to find
from tables in the Nautical Almanac the moment (Greenwich apparent time)
when that angular separation was predicted to occur. Then one calculates the
difference between the local apparent time and Greenwich apparent time, and
converts the time difference to a difference in longitude (15 degrees of longitude
= 1 hour time difference). The implementation is far more complex. The first
issue is to consider how to determine the local apparent time. The second issue
is that the observed distance between the moon and the sun requires a host of
corrections. We consider each in turn.
The problem of local time
It is a misconception that the invention of the marine chronometer removed
the need for celestial observations for determining longitude. The chronometer,
when working correctly, reports the time at the reference meridian. But the
ship’s clock can not report the local time – no clockwork, however accurate can
determine local time, that time depends upon the observer’s position, which
on a ship is constantly changing. Only locations on the same meridian share
the same local time. Our modern “standard time zones” are more than 1000
miles wide, and all who live within a time zone share the same time, but two
locations 50 or 100 miles apart have very different local apparent times. ships
are constantly in motion, they are constantly shifting their local time. The noon
sight, discussed earlier, provides the sailor with the ship’s latitude and the time
of noon, and would appear to solve the problem of local time, provided any
reasonable timepiece is aboard ship. However the moment at which the sun is
8
at its highest is impossible to measure with accuracy, the sun is at the peak of
its arc and its altitude changes very slowly at that point. The sun seems to hang
at the same altitude for fifteen to thirty minutes, the uncertainty in the moment
when it was exactly noon is considerable. It is when the sun is in the East or in
the West that it’s altitude above the horizon changes most rapidly, and thus it
is in the early morning and late afternoon that the sun’s altitude provides the
most accurate information about local time, rather than at mid-day when the
sun’s altitude changes most slowly. The “time sight” was developed to provide
an accurate measure of the local time, that is the number of hours before or
after local apparent noon.
The basis of the time sight is the spherical triangle formed by the sun, the
zenith, and the celestial pole. The three sides of this spherical triangle are
the complementary angles of the sun’s altitude above the horizon, the solar
declination , and the observer’s latitude.
9
In the figure below, Pmarks the celestial north pole, Zthe zenith (the point
directly above the observer) , and Mthe position of the sun. The arc NW S
marks the observer’s horizon, and the arc QW Q0marks the celestial equator.
The altitude of the sun above the horizon is labeled h, the declination of the
sun above the celestial equator is labeled d, and the latitude is shown as the
altitude of the celestial pole above the horizon (rather than the distance from
the celestial equator to the zenith). With respect to the spherical triangle P Z M,
side P Z is a portion of the meridian of the observer, while side P M is a portion
of the meridian of the sun. The angle at P, labled tin the figure is therefore a
measure of the time that has passed local noon.
The solar altitude is measured ua sextant or similar instrument, the solar dec-
lination is determined from the almanac entry for the date of the observation,
and the latitude is estimated from a deduced reckoning based on the most recent
noon sight and the information on course and speed in the ship’s log (as used in
the sailings) [?]. The ship’s latitude can not be determined from direct observa-
tion at the time of the “time sight” uthese methods. three sides of a spherical
triangle are now known, the three vertex angles can be determined. The angle
at the celestial pole is called the local hour angle and is directly proportional
to the time , or before, noon. The angle at the zenith is called the azimuth
and the angle at the solar position is called the parallactic angle or the angle of
position.
Clearing the lunar distance
The time sight provided the sailor with a knowledge of local apparent time.
The time sight required for its calculation a previous noon sight (for finding
the ship’s latitude) and a dead reckoning of the ship’s movements the noon
sight was performed (so that the ship’s latitude could be updated for the posi-
10
tion of the ship when the time sight was performed.) The lunar distance was
designed to provide the apparent time at the reference meridian, so that the
difference in longitude between that meridian and the ship’s position could be
determined.
Calculating the distance from center to center
The distance (in degrees of arc) between the moon and the sun was measured,
but that distance was measured from limb to limb, that is from the edges of
each disk, not from their centers, while the entries in the Nautical Almanac
were calculated from the center of the sun’s disk to the center of the moon’s
disk. Thus the first corrections in the measured values were to look up the
semi-diameters of the sun and the moon. These vary with the distance of the
earth from the sun and of the earth from the moon, so they were tabulated by
date.
Augmentation
When the moon is high in the sky, the observer is closer to the moon than he
would be if the moon were on the horizon. If the moon were directly overhead,
it would be closer by the radius of the earth (about 4,000 miles closer). Thus
the higher in the sky the moon is, the larger it’s semidiameter. Tables in the
almanac provide a correction, called augmentation, for this effect.
Allowance for dip
The height of the eye above the surface of the water effects the location of the
sea horizon. Due to the curvature of the earth, any additional height allows
the observer to see further over the curve of the ocean surface. Tables in the
almanac provide a correction in altitude to provide the angle that would be
observed from the surface of the sea.
11
Allowance for parallax
The positions of the sun and moon in the Nautical Almanac are calculated on
a geocentric basis . . . as seen from the center of the earth. The observer on
the surface sees the moon in a different position due to the parallax of the two
different locations. the moon is relatively close to the earth, the parallax can be
large (nearly one degree at times) and requires correction. In the figure below,
the the altitude of the moon, as seen from position A is angle H’ A S, while the
altitude as “seen” from the center of the earth (as reported in the Almanac)
would be angle HCS. A celestial object at point S would appear higher in the
sky from the center of the earth at C than it would appear to the observer on
the surface at point A.
Consider the diagram below. The moon is centered at M, the earth at C, and
an observer at point O views the moon at high in the sky. The zenith distance
of the moon, as observed from point O is given by angle DOM. An observer
at point O1, however observes the moon to be on the horizon (with a zenith
distance of 90 degrees). The angle O1M C is called the horizontal parallax and
is tabulated in the Nautical Almanac. The angle OM C is the parallax for the
observer at point Oand needs to be computed.
Distance MC is the distance between the earth and the moon, and distances OC
and O1Care equal to each other and are equal to the radius of the earth. Con-
sidering triangle O1MC we see that sin(HP ) = r
MC where H P is the horizontal
parallax and ris the radius of the earth. Considering triangle OCM and apply-
12
ing the law of to that triangle, we see that sin(p)
r= sin(180zdm)M C where p
is the local parallax (angle OM C ), ris the radius of the earth, and zdm is the
zenith distance of the moon. the of the supplement of an angle is equal the the of
the angle, we can replace sin(180zdm) by sin(zdm). Thus sin(p)
r=sin(zdm)
MC .
Solving for sin(p) we get sin(p) = sin(zdm)r
MC = sin(z dm) sin(H P ). both p
and HP are small (1 degree or less), a small angle approximation sin(x)x
can be used to obtain the relationship p= sin(zdm)H P .
Allowance for refraction
Rays of light traveling through the atmosphere at an angle will be bent through
the effects of refraction, in accordance with Snell’s Law, so that they appear
higher in the sky than they really are. This effect is proportional to the tangent
of the zenith angle. Objects at the zenith are not effected, objects near the
horizon experience the greatest effects. Tables for the effects of refraction appear
in the nautical almanacs. Nineteenth and twentieth century almanacs may
also have corrections for temperature and barometric pressure, which effect the
density of the atmospheric gases.
Computational difficulties in clearing the lunar distance
The effects of dip, parallax, augmentation, and refraction all effect the altitude
at which the moon, sun, and stars are observed, but not their horizontal posi-
tion in the sky. With the aid of modern computers and calculators, a straight
forward way of determining their effect on the lunar distance can be formulated.
[?]
13
First, measure the lunar distance, correcting for semidiameter and augmenta-
tion. Next, measure the altitude of sun and moon and calculate their com-
plements. Thirdly, use the spherical law of cofor sides to calculate the vertex
angle at the zenith. Adjust the altitudes for sun and moon for the effects of
dip, parallax, and refraction. Recalculate complements of the adjusted alti-
tudes, and use the zenith vertex angle found with the uncorrected sides. Note
that the zenith vertex angle is not effected by changes in altitudes. Uthe two
sides (complements of corrected altitudes) and the vertex angle, use the spher-
ical law of cofor angles to calculate the length of the side connecting sun and
moon. That is your corrected lunar distance. Consult the tables in the Nautical
Almanac to find out what time it was in Greenwich when the bodies were so
separated. That is apparent time in Greenwich. Find the difference between
that time your local apparent time as determined by your time sight, and con-
vert the time difference to a longitudinal difference. This is an elegant solution,
but the computational burdens of carrying this out in the absence of electronic
calculators and computers were insurmountable. Another method needed to be
found. Two major mathematical contributions enabled the lunar distances to
be computer without the aid of electronic computers (although many human
computers were in fact used in preparing the tables for these almanacs). The
first of these contributions were the invention of logarithms by Napier and their
perfection by Briggs. The second of these resulted from increasophisticated sets
of trigonometric manipulations, culminating, in clodecades of the eighteenth
century, in scores of competing methods, direct and approximate, for clearing
the lunar distance.
The problem of computational complexity
The central computation for performing a time sight is to solve the equa-
tion
cos(a) = cos(b) cos(c) + sin(b) sin(c) cos(α) (1)
14
for the unknown hour angle α, given the sides a, b, and c. Thus
cos(α) = cos(a)cos(b) cos(c)
sin(b) sin(c)(2)
or
α=arc cos( cos(a)cos(b) cos(c)
sin(b) sin(c).(3)
Tables of and coof sufficient accuracy were available from the mid sixteenth cen-
tury, but the multiplication of two 6 or 7 digit numbers was both tedious and
error prone, the division of two such numbers even more so. The discovery of
logarithmic numbers and the prompt development of tables of logarithms and
logarithms of trigonometric functions at about the same time, eased such prob-
lems of multiplication and division, but the form of equation (??) or equation
(??) does not lend itself to logarithmic computation due to the subtraction in
the numerator.
Adaptations for Logarithmic Computation
many of the trigonometric equations that navigation required involved finding
sums and differences, as well as products and quotients, it was often the case that
logarithmic computations were not as useful as one would wish. Considerable
ingenuity was exercised to find trigonometric identities and manipulations that
would change the needed calculations into the form of products and quotients.
Often this entailed employing trigonometric functions less common than chords,
, tangents, and secants. Occasionally it entailed inventing new trigonometric
functions, and of course computing and making available trigonometric and log-
trigonometric tables for these unusual or new functions. Tables of the versed ,
haver, suver, and log squares, were a part of many sailor’s toolkit. [?]
These functions often have interesting geometric interpretations in terms of
the chords, tangents, and secants of the circles through which such functions
were understood before the late eighteenth century, and in addition they often
simplified the equation into a form which could be more easily memorized and
remembered. The versed was understood as that part of the diameter between
the and the arc, the haverwas defined as one-half of the versed , the coverwas
the versed of the complement of the arc, and the suveras the difference between
the diameter and the versed . 2
2Algebraically, vers(x) = 1 cos(x), hav(x) = 1
2vers(x), suvers(x) = 1 +
cos(x),and covers(x) = 1 sin(x).The functions are also closely linked with the half-angle
identities. sin2(x
2) = hav(x) and cos2(x
2) = suvers(x).
15
The Versed
The versed was used occasionally in navigational computations throughout the
18th century, but became more widely appreciated in the 19th century as use of
lunar distances to determine longitude required more intricate calculations. The
use of the versed avoided the potential errors caused by the algebraic sign of the
co, tangent, or secant functions of second quadrant angles. The function results
in ambiguous cases where two different directions have the same . The cofunc-
tion requires the consistent and accurate use of signed arithmetic or algebraic
sums. The versed however has neither problem, but increases monotonically
from 0 to 2 as the angle passes through the first two quadrants.
The Haver
The haveris defined as one-half of the versed , and arises naturally as a part of
the half-angle identities. Today we teach students that sin( x
2) = ±q1cos(x)
2,
but until the nineteenth century this would have been understood as sin2(x
2) =
hav(x). This formula is much easier to remember and is very suitable for loga-
rithmic calculation.
Recall the equation for finding the local hour angle in a time sight:
cos(α) = cos(a)cos(b) cos(c)
sin(b) sin(c).(4)
A common eighteenth century manipulation was to introduce the variable s=
a+b+c
2, called the half-sum. Uequation (??),
vers(α)=1cos(α)=1cos(a)cos(b) cos(c)
sin(b) sin(c)
=sin(b) sin(c) + cos(b) cos(c)cos(a)
sin(b) sin(c)
=cos(bc)cos(a)
sin(b) sin(c)(5)
Adding the expansions cos(AB) and cos(A+B)leads to the identity cos(A
B)cos(A+B) = 2 sin(A) sin(B) thus replacing a difference by a product. The
offending difference in our problem is cos(a)cos(bc) Setting A+B=aand
AB=bcwe find that A=a+bc
2= (sc) and B=ab+c
2= (sb).Thus
the expression cos(bc)cos(a) can thus be replaced by 2 sin( ab+c
2) sin( a+bc
2)
and equation (??) becomes
vers(α) = 2 sin(sb) sin(sc)
sin(b) sin(c)(6)
16
. In terms of haveor square ,
hav(α) = sin2(α
2) = sin(sb) sin(sc)
sin(b) sin(c)(7)
or even more concisely,
hav(α) = csc(b)csc(c) sin(sb) sin(sc) (8)
For a time sight, a, b, and care the complement of the altitude, 90h, the
polar distance, p, and the complement of the latitude, 90φ, respectively.
In terms of these variables cos(sb) = cos(90h+p+φ
2) and cos(sc) =
cos(p+φ+h
2h).
If αrepresents the local hour angle, the equation becomes
hav(α) = csc(p)sec(φ) cos h+p+φ
2cos h+p+φ
2h(9)
Note how perfectly suited for logarithmic calculations this formulation of the
problem is. The items involved in addition and subtraction are direct measure-
ment, not the results of intermediate calculations. Furthermore, whereas the
intermediate calculations (based on looking values up in logarithmic or trigono-
metric tables) required 6 to 8 significant figures of accuracy, the direct mea-
surements required many fewer significant figures. Once the table lookups are
performed, determing the value of loghaver(α) required only sums and differ-
ences, along with one more table lookup to obtain the inverse log haver. Many
other such rearrangements appear in the manuals for seamen and the textbooks
for navigation. Hardly ever are explanations included for where these remark-
able formulas came from.
The expression h+p+φ
2was called the half-sum, and the expression h+p+φ
2hwas
called the remainder. Such forms populate many pages of Bowditch’s manual
for seamen and are reflected in the training of seamen up into the twentieth
century. An oral history of one such sailor recorded this conversation how he
was taught to find local time.
“Add the secant of latitude, the of polar distance, the coof the half sum, and
the of the remainder . . . the logs of course. . . and you gotter remember what it
gives yer . . . it gives yer the log haverof the hour angle” [?]
This semi-poem exactly reflects equation (??), and although it is doubtful that
this elderly seaman knew where such a rule came from, he knew exactly what
he was doing when he used it.
17
From Mendoza y Rios to Bowditch
Don Jos´e de Mendoza y Rios was a retired Spanish admiral, an expatriate
during the wars between England and Spain, and a member of the Royal Society
of London. He presented a monumental overview of nautical astronomy and
navigational science to the Society in 1796. As published in the Philosophical
Transactions of the society it ran to seventy-nine pages of detailed explanations
and complex trigonometric calculations. Written in the French language, it was
“read” to the Society by Sir Joseph Banks, and detailed no fewer than forty
different methods for clearing the lunar distance. I have selected one to present
here both because the method is interesting and because it was very widely used
by navigators in the late 18th and early 19th century. With some adjustments it
becomes incorporated in Nathaniel Bowditch’s ground breaking work, the New
American Practical Navigator (1802).
The goal of clearing a lunar distance was to correct all apparent measurements
to “true” ones, that is, measurements as they would be seen by a hypothetical
observer at the center of the earth without the distorting effects of the earth’s
atmosphere. The corrected value of the lunar distance is then compared with
the table of lunar distances in the Nautical Almanac, and interpolating between
the table entries (recorded at three hour intervals), determining the apparent
time at the reference meridian.
Let Mand Sbe the apparent lunar and solar altitudes respectively, and let
mand sbe the true lunar and solar altitudes. Let Dbe the apparent lunar
distance and dbe the true lunar distance and let Zbe the zenith angle.
Then the spherical triangle MZS leads to the relationship
cos(D) = cos(Z) cos(M) cos(S) + sin(M) sin(S) (10)
via the spherical law of cofor sides, while the triangle mZs leads to the equa-
tion
cos(d) = cos(Z) cos(m) cos(s) + sin(m) sin(s).(11)
Solving equation(??) for cos(Z) we get
cos(Z) = cos(d)sin(m) sin(s)
cos(m) cos(s).(12)
Substituting this value for cos(Z) into equation(??) we get
cos(D) = cos(d)sin(m) sin(s)
cos(m) cos(s)cos(M) cos(S) + sin(M) sin(S)
cos(D) = (cos(d)sin(m) sin(s))cos(M) cos(S)
cos(m) cos(s)+ sin(M) sin(S)
18
cos(D) = cos(d) cos(M) cos(S)
cos(m) cos(s)tan(m)tan(s) cos(M) cos(S) + sin(M) sin(S)
(13)
The moon appears lower in the sky than its geocentric position would indicate,
M=m+uwhere uis positive. The effect of refraction on the moon is much
less than the effect of parallax. The sun appears higher in the sky that its true
postion (due to refraction) S=sv, where vis positive. The parallax of the
sun is negligible in comparison with the effect of refraction the sun is so far
away from the earth.
Expanding the and coof the sum and differences of angles ubasic trigonometric
identies we see that
sin(M) = sin(m+u) = sin(m) cos(u) + cos(m) sin(u) (14)
cos(M) = cos(m+u) = cos(m) cos(u)sin(m) sin(u) (15)
sin(S) = sin(sv) = sin(s) cos(v)cos(s) sin(v) (16)
cos(S) = cos(sv) = cos(s) cos(v) + sin(s) sin(v).(17)
Uequations (??) through (??) to calculate sin(M) sin(S) and cos(M) cos(S),
which we will need for equation ( ?? ) we get
sin(M) sin(S) = sin(m) sin(s) cos(u) cos(v) + cos(m) sin(s) sin(u) cos(v)
sin(m) cos(s) cos(u) sin(v)cos(m) cos(s) sin(u) sin(v) (18)
and
cos(M) cos(S) = cos(m) cos(s) cos(u) cos(v)sin(m) cos(s) sin(u) cos(v)
+ cos(m) sin(s) cos(u) sin(v)sin(m) sin(s) sin(u) sin(v) (19)
Medoza y Rios expands sin(u),sin(v),cos(u), and cos(v)intoaT aylor seriesexpansion, keepingal ltermsof ordertwoor less.T hussin(u)
u, sin(v)v, cos(u)1u2
2,and cos(v)1v2
2.Substituting the values uand
vfor sin(u) and sin(v) and substituting 1 u2
2for cos(u) and 1 v2
2for cos(v)
in equations (??) and (??) we can express sin(M) sin(S) and cos(M) cos(S) in
terms of m, s, u,and v.
sin(M) sin(S) = sin(m) sin(s) + ucos(m) sin(s)vsin(m) cos(s)
u v cos(m) cos(S)1
2u2sin(m) sin(s)1
2v2sin(m) sin(s) (20)
cos(M) cos(S) = cos(m) cos(s)usin(m) sin(s) + vcos(m) sin(s)
u v sin(m) sin(s)1
2u2cos(m) cos(s)1
2v2cos(m) cos(s) (21)
19
Substituting the values from equations (??) and (??) into the expression for
cos(D) in equation (??) we get
cos(D) = cos(d) + usin(s)
cos(m)ucos(d)tan(m)vsin(m)
cos(s)+vcos(d)tan(s)
+u v (sin2(m)cos2(s)cos(d) sin(m) sin(s))
cos(m) cos(s)1
2u2cos(d)1
2v2cos(d).
(22)
Taking D=d+δone has cos(D) = cos(d+δ) = cos(d) cos(δ)sin(d) sin(δ) and
uthe second order approximations for the error term, δ, we can replace sin(δ)
by δand cos(δ) by 1 1
2δ2to get
cos(D) = cos(d)δsin(d)1
2δ2cos(d) (23)
which when substituted into equation(??) gives the correction needed for the
lunar distance in terms of the apparent lunar distance and the apparent lunar
and solar altitudes, along with the corrections to be made in the lunar and solar
altitudes:
δ=usin(s)
cos(m) sin(d)+u cot(d)tan(m) + vsin(m)
cos(s) sin(d)v cot(d)tan(s)
+u v (sin2(m)cos2(s) + cos(d) sin(m) sin(s))
sin(d) cos(m) cos(s)+1
2u2cot(d) + 1
2v2cot(d)
1
2δ2cot(d).(24)
Note, however, that the desired lunar distance correction, δappears on both
sides of this equation. Mendoza takes an additional step, regrouping the terms
of the above equation in terms of powers of u,v, and δ.
δ=usin(s)cos(d) sin(m)
sin(d) cos(m)+vsin(m)cos(d) sin(s)
sin(d) cos(s)
+u v cos(d) sin(m) sin(s)sin2(m) + cos2(s)
sin(d) cos(m) cos(s)+1
2u2cot(d)
+1
2v2cot(d)1
2δ2cot(d) (25)
and then procedes to square that above value for δ, eliminating any products
of u, v, or δthat are higher than second order. He gets a second order approx-
imation for δ2
20
δ2=u2(sin(s)cos(d) sin(m)
sin(d) cos(m))2+v2(sin(m)cos(d) sin(s)
sin(d) cos(s))2
2u v (sin(s)cos(d) sin(m)
sin(d) cos(m))(sin(m)cos(d) sin(s)
sin(d) cos(s)),(26)
which he subsitutes for δ2in the right hand side of equation(??).
δ=u(sin(s)cos(d) sin(m)
sin(d) cos(m)) + v(sin(m)cos(d) sin(s)
sin(d) cos(s))
+u v 2 cos(d) sin(m) sin(s) + sin2dsin2(m)sin2(s)
sin2(d) cos(m) cos(s)
+1
2u2cot(d) 1sin(s)cos(d) sin(m)
sin(d) cos(m)2!
+1
2v2cot(d) 1sin(m)cos(d) sin(s)
sin(d) cos(s)2!.(27)
Medoza concudes, “Voil`a la formule qui exprime g´en´eralement les corrections
qu’on doit appliquer `a la distance apparente d, pour avoir la distance vraie D,
ayant ´egard `a toutes les ´equations qui d´erivent de u,v, et des produits du second
ordre de ces ´el´ements.” [?]3
Of course, no one aboard ship would be expected to carry out such calculations.
As much as possible would be calculated ahead of time, on land, and presented
to the sailor as a series of tables, which he would employ to find his various
corrections, and these tables were detailed and numerous, forming the vast
majority of the bulk of any nautical almanac or manual for seamen.
3[And here we have the formula which generally expresses the corrections that one must
apply to the apparent [lunar] distance d, to determine the true [lunar] distance D, including
in all equations the effects of any terms which involve uand v[the corrections for for lunar
and solar altitude] and any second order products of those quantities.]
21
On the left is a page from the first edition of the Nautical Almanac, prepared
by Nevil Maskelyne. On the right is a set of instructions for uthe tables for
clearing the lunar distance, from an 1801 American edition of John Hamilton
Moore’s New Practical Navigator.
The mathematical creativity, ingenuity, and sheer tenacity that underlies the
calculations that were behind the tables and methods made available in such
works is impressive. That being said, it was not expected that the navigator
or sailor understood or cared where such methods came from, only that they
worked and that they were within his abilities to carry out with accuracy in
the difficult envirnoment of a ship at sea. “Seamen of all times have been
content to work according to the rule, caring little for the derivation of the
rule. . . . Moreover, once a specified method . . . had been accepted, mastered
and committed to memory, a seaman tended to use it throughout his sea-going
career.” [?, p. 246] The same comment would not hold true for the inventors of
these methods, their advocates in the admiralty, and the teachers of navigation,
who often came to their positions with considerable mathematical talent and
continually strove to improve the accuracy and efficacy of these methods.
22
References
[1] Bok, B.J. and Wright, F.W., Basic Marine Navigation,, Houghton
Mifflin, Boston, New York, 1944.
[2] Bowditch, Nathaniel, The New American Practical Navigator, F.
& G. W. Blunt, New York, 1833.
[3] Brink, R.W., Spherical Trigonometry, D. Appleton-Century Com-
pany, Inc., NY, London, 1942.
[4] Calahan, H. A., The sky and the sailor; a history of celestial nav-
igation (1st ed. ed.). Harper, New York, 1952.
[5] Cotter, Charles H., A History of Nautical Astronomy, Hollis &
Carter, London, 1968.
[6] Ifland, P. Taking the Stars: Celestial Navigation from Argonauts to
Astronauts, Mariners’ Museum, Krieger Pub. C., Newport News,
VA, 1998.
[7] O’Connor, J. J. and Robertson, E. F., Tobias Mayer,
The MacTutor History of Mathematics Archive, Uni-
versity of St. Andrews, Scotland, www-gap.dcs.st-
and.ac.uk/˜ history/Biographies/Mayer Tobias.html.
[8] Michael of Rhodes: A Medieval Mariner and His Manuscript.
Dibner Institute for the History of Science and Technology,
http://dibinst.mit.edu/DIBNER/Rhodes/manuscript, 2005.
[9] Petze, C.L., The Evolution of Celestial Navigation, Motor Boating,
New York, 1948.
[10] Rios, Joseph de Mendoza y , Recherches sur les principaux Prob-
lems de l’Astronomie Nautique. Par Don Jesef de Mendoza y Rios,
F.R.S. Communicated by Sir Joseph Banks, Bart. K.B.P.R.S ., De-
cember 22, 1796. Philisophical Transactions of the Royal Society
of London, Series I, Vol. 87, pp 43 – 122, 1797.
[11] Silverberg, Joel, The Sailings: The Mathematics of Eighteenth
Century Navigation in the American Colonies, Proceedings of the
Canadian Society for the History and Philosophy of Mathematics,
Volume 18, 20005, pp 173–199.
[12] Vincent, Fiona, Positional Astronomy, University of
St.Andrews, Scotland, 1998 and 2003. http://star-www.st-
and.ac.uk/˜ fv/webnotes/index.html
[13] Wroth, L.C., Some American Contributions to the Art of Naviga-
tion, 1519–1802,, Associates of the John Carter Brown Library,
Providence, 1947.
23
Appendix A: List of Manuscripts Consulted
Rhode Island Historical Society Manuscripts
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
24
... During the later half of the 18th century, two competing methods were developed to determine the apparent time at the prime meridian the measurement of lunar distances, and the development of marine chronometers. [9,1,2,3] Since 1767, the Nautical Almanac has listed the distance between the moon and the sun, planets, and selected stars at intervals throughout each day. Comparing the observed lunar distance with those tables tells us the time in Greenwich at the moment the observation was made. ...
... A considerable amount of mathematics was needed to correct refraction, parallax, and a variety of other distortions in the measurements. [9,1,7] Determining Local Apparent Time ...
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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 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.
Basic Marine Navigation
  • B J Bok
  • F W Wright
Bok, B.J. and Wright, F.W., Basic Marine Navigation,, Houghton Mifflin, Boston, New York, 1944.
The sky and the sailor; a history of celestial navigation
  • H A Calahan
Calahan, H. A., The sky and the sailor; a history of celestial navigation (1st ed. ed.). Harper, New York, 1952.
A Medieval Mariner and His Manuscript. Dibner Institute for the History of Science and Technology
  • Rhodes Michael
Michael of Rhodes: A Medieval Mariner and His Manuscript. Dibner Institute for the History of Science and Technology, http://dibinst.mit.edu/DIBNER/Rhodes/manuscript, 2005.
Some American Contributions to the Art of Navigation
  • L C Wroth
Wroth, L.C., Some American Contributions to the Art of Navigation, 1519-1802,, Associates of the John Carter Brown Library, Providence, 1947.