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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 90◦to 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(180◦−zdm)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(180◦−zdm) 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) = ±q1−cos(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)=1−cos(α)=1−cos(a)−cos(b) cos(c)

sin(b) sin(c)

=sin(b) sin(c) + cos(b) cos(c)−cos(a)

sin(b) sin(c)

=cos(b−c)−cos(a)

sin(b) sin(c)(5)

Adding the expansions cos(A−B) 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(b−c) Setting A+B=aand

A−B=b−cwe ﬁnd that A=a+b−c

2= (s−c) and B=a−b+c

2= (s−b).Thus

the expression cos(b−c)−cos(a) can thus be replaced by 2 sin( a−b+c

2) sin( a+b−c

2)

and equation (??) becomes

vers(α) = 2 sin(s−b) sin(s−c)

sin(b) sin(c)(6)

16

. In terms of haveor square ,

hav(α) = sin2(α

2) = sin(s−b) sin(s−c)

sin(b) sin(c)(7)

or even more concisely,

hav(α) = csc(b)csc(c) sin(s−b) sin(s−c) (8)

For a time sight, a, b, and care the complement of the altitude, 90◦−h, the

polar distance, p, and the complement of the latitude, 90◦−φ, respectively.

In terms of these variables cos(s−b) = cos(90◦−h+p+φ

2) and cos(s−c) =

cos(p+φ+h

2−h).

If αrepresents the local hour angle, the equation becomes

hav(α) = csc(p)sec(φ) cos h+p+φ

2cos h+p+φ

2−h(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

2−hwas

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=s−v, 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(s−v) = sin(s) cos(v)−cos(s) sin(v) (16)

cos(S) = cos(s−v) = 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)≈1−u2

2,and cos(v)≈1−v2

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) + sin2d−sin2(m)−sin2(s)

sin2(d) cos(m) cos(s)

+1

2u2cot(d) 1−sin(s)−cos(d) sin(m)

sin(d) cos(m)2!

+1

2v2cot(d) 1−sin(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