The art of astrronmical navigation

The Art of
Astronomical Navigation
BY

S. M. BURTON
Fellow of the Institute of Navigation
Master Mariner

Revised by

CHARLES H. COTTER
Fellote of the Royal Institute af Navigation
Master Mariner

BROWN, SON & FERGUSON,
52 DARNLEY STREET

LTD.

•


CopYl'ight in all countries signatory to the Berne Convention
All rights reserved

PREFACE

First Edition 1955
Second Edition 1962
Third Edition 1975

©

1975 BROWN, SON & FERGUSON, LTD, GLASGOW G41 2SG

Printed and Made in Great Britain

TO THE FIRST

EDITION

THIS book first appeared in 1935 under the title of A Manual of
Modern"Navigation: its purpose then being to hasten what seemed
the overslow conversion of Merchant Navy navigators from older
methods to position.line methods. With or without the assistance
of the Manual that conversion may now be thought of as complete. In the meantime, however, the sun and the moon and the
stars have not changed in their courses, and Sumner's discoveryand it was a discovery, be it noted, not an invention-remains
forever with us. Consequently the principles enunciated in the
original book have not changed either. As to the practices: they
are still taught and used today, and will, no doubt, long continue
to be taught, although the normal method of applying them may
(or may not) come to be through the medium of precomputed
tables instead of by logarithmic calculation. Parts I and II of the
book have therefore been carefully combed through and brought
up to date where necessary (with a chapter on the 'Short Methods'
added) and Part III entirely re·edited: the effect of which, it is
hoped, will be to revive the book's former popularity as a guide
and friend to the practising navigator, making it one which he
will always like to possess, and keep within easy reach.
For the benefit of those who may not have been acquainted
with the original book it might be eXplained that its main purpose,

as developed in Parts I and II, might be described as being to
give a good general overall view of the astro-navigational wood
for the benefit of those who have at least some nodding acquaintance with its trees-a view such that the navigator is enabled
first to choose his own route through the wood, and then to keep
his chosen course so as to reach his objective on the other side.
Part III is an effort to collect and compress within a relatively
small space all the formulae and other procedures used in or in
connection with navigation, many of which, although of casionally
needed, are not sufficiently used to keep them always fresh in
the memory.
v


PREFACE

TO THE SECOND

EDITION

RECENT changes in The Nautical Almanac have made it desirable
that a new edition of this book should be brought into being. The
chief alterations will be found on pp. 107-9, necessitated by the
discontinuance
of Right Ascension as a navigational
concept.
Other minor corrections
and adjustments
have been made as
necessary.
S. M. BURTON

March 1962

PREFACE TO THE THIRD EDITION
THE late Stephen M. Burton was well-known over many decades
for his deep interest in the practical needs of navigators.
To this
end his Nautical Tables, first published in 1936, and now well
established
as a thoroughly
practical tool of navigation,
was
produced (as those who knew him will testify) with the fervour
of a crusader.
His little manual on the Art of Astronomical Navigation, like
his Nautical Tables, is standing up well to the test of time.
In
this new edition the only major change from the Second edition
of 1962 has been in Chapter XVIII, which now draws attention
to the existence of several navigation tables which may still fall
into the hands of navigators, but which are rapidly being made
obsolescent by the splendid inspection tables now to be found
on almost every sea-going ship. The revised chapter concentrates,
therefore, on modern inspection tables and their use.
CHARLESH. COTTER

CONTENTS

PART
CHAP.

I
II
III
IV
V
VI
VII
VIII

THE

I

PRELIMINARY REMARKS
THE TRAVERSETABLE
POSITION LINES
THE TRANSFERREDPOSITION LINE
THE AsTRONOMICALPOSITION LINE
THE AsTRONOMICAL TRIANGLE
THE FOUR STANDARDMETHODS
THE POSITION LINE AT CLOSE QUARTERS

THE

XI
XII
XIII
XIV
XV
XVI

XVII
XVIII

V-VI

PRINCIPLES

PART

IX
X

PAGE

.

PREFACES

3

5
7
10

17
21
2434

II

PRACTICE

SUCCESSIVEOBSERVATIONS
REMARKS UPON SUCCESSIVE OBSERVATIONS IN
GENERALANDTHENOON POSITIONIN PARTICULAR
SIMULTANEOUSOBSERVATIONS
STARS
THE ANGLE OF INTERSECTION
HEIGHT OF ALTITUDES
THE PARTICULARCASE OF VERYHIGH ALTITUDES
THE USES OF A SINGLE POSITION LINE.
POSITION LINE BY D.F. BEARINGS
NAVIGATION TABLES

·
·
·
·

.

January 1975
PART
MISCELLANEOUS

NOTES

VB

56

60
6470
72
75
79
83
91

III
AND

FORMULAE

MISCELLANEOUSNOTES AND FORMULAE
VI

43

.

•

106


PART I
THE PRINCIPLES

CHAPTER

II

THE TRAVERSE TABLE
only in the sailings (D.R. navigation), but in every method
of fixing a ship's position, either celestial or terrestrial, the rightangled triangle is continually turning up and demanding quick
and easy solution. Now, although it is quite generally known that
inspection of the Traverse Table is the quickest and easiest way
of approximately solving any right-angled triangle, it is a curious
fact that the method is by no means quite generally used. N evertheless, the navigator who will take the trouble to acquire the
Traverse-Table-habit early in life will find it a valuable convenience for the rest of his days.
Being set down in terms of the Plane Sailing triangle, it is
necessary to have that figure always present in the mind when
referring any other triangle to the Traverse
Table. Under the circumstances it is
hardly possible for any person having to do
with the navigation of ships to forget the
Plane Sailing figure. With a little trouble,
however, the mind can be so trained that,
so far from any effort being required to
recall it, the merest suggestion of D'lat. or
Dep. will have the effect of instantaneously conjuring it up into
the mind's eye..
With the Plane Sailing triangle so fixed in the subconscious
mind it is only necessary, when referring any other triangle to the
Traverse Table, * to bear in mind the following facts:
NOT

(i) The angle considered being always represented by the

Course, consequently,
(ii) the side adjacent to this angle is represented by the D'lat.,

•

* Most Traverse Tables now have 'Dep.' and 'D'long.' included in their
headings, and some go even further and include 'Hyp.', 'Adj.' and 'Opp.'
5


6

THE

(iii)
and
(iv)
The
reader
of it':

ART

OF ASTRONOMICAL

NAVIGATION

the side opposite to this angle is represented by the Dep.,
the hypotenuse is always represented by the Dist.
following few examples may be of some use in helping any

not already conversant with the process to 'get the hang

Angle 25°, opposite side 75, required hypotenuse. (177'5)
Angle 76°, adjacent side 143, required opposite side. (573'4)
Hypotenuse 297, adjacent side 72, required opposite side. (288'4)
Hypotenuse 426, angle 31°, required adjacent side. (365'2)
Opposite side 79, adjacent side 342, required angle. (13°)
Adjacent side 23, hypotenuse 165, required angle. (82°)

The commonest use to which the Traverse Table is put, other
than that for which it was originally intended, is to inter-convert
Dep. and D'long. This is done in accordance with the proportions

shown in the adjacent (Parallel Sailing) triangle: that is to say,
with the latitude of the required conversion as 'Course' the
corresponding Deps. and D'longs. will be found respectively in
the 'D'lat.' and 'Dist.' columns. There should not arise any
confusion as to which is which if it is borne in mind that the 'Dist.'
column is the hypotenuse column and D'long. always exceeds
departure. This same triangle also applies for Mid. lat. Sailing,
the angle used being Mid. lat. instead of Lat.

CHAPTER

III

POSITION LINES
IF the mind is brought to consider the matter for a moment it
will be realised that the word 'position' when used in the sense
of geographical locality, must mean relative position.

Again, by further consideration of the point it will also be
realised that relative position is composed of two elements, namely
direction and distance, and that either one of these elements considered separately resolves itself into a line. Therefore it will be
evident that an unknown position can only be fixed by means of
lines.
In navigation these two kinds of lines when so used are given
the common name of position lines. *
Position lines defined by direction (bearings) are, of course,
derived from light waves, detected by the eye, or sound or radar
waves, and are therefore always of a true or great circle nature.
This means that the principle of cross-bearings, which is that the
mutual bearing of any two places on the earth's surface is reciprocal, is fundamentally untrue. In practice, however, this seldom
makes any appreciable difference for ordinary compass bearings
by eye, as the following table will show. In the case of radio
position lines (and possibly radar ranges) the matter can be of
more importance, owing to these bearings being sometimes used
across relatively long distances, and we shall therefore touch upon
this matter in another place (Chapter XVII) ..
These maximum errors arise, of course, on east and west
bearings.
The table on page 8, like other small tables in this book, is not
intended for 'use' in the ordinary sense of that word. It is only
* Position lines can, in point of fact, be:derived from other things than simple
lines of direction or distance. There are what might be called 'conditional
position lines'. The arcs derived from horizontal sextant apgles or t.hf paraboli~
curves from certain types of radio beacons are instances. See also p. 85 for the
'curve of constant bearing',
7

8

THE

ART

OF

ASTRONOMICAL

NAVIGATION

POSITION

inserted here to assist the reader to form an opinion for himself.
Position lines defined by distance are necessarily of a circular,
or 'arciform' shape. As they are less easily obtainable under all
circumstances, we do not use them so frequently in coastal naviTABLE

SHOWING

MAXIMUM

NEGLECT

ERROR IN BEARING

OWING TO

LINES

9

dence of tangent and circumference, the only way to find the
exact point of contact is to find that perpendicular to the tangent
which passes through the centre of the circle.
It is important to notice that the larger the radius of the circle
the longer the length of apparent coincidence of tangent and
circumference.

OF GREAT CIRCLE
--.-

Distance

(M.)

Lat.

30°
40°
50°
60°
70°

20

30

40

50

.1°
·1
'2
,3
,5

.p
'2
,3
'4
,7

'2°
,3
,4
'5

'2°
'4
,5
'7
1·1

,9

gation as those derived from bearings; but in astronomical positionfinding they are, always have been, and probably always will be
the only kind that can be obtained, and the following points concerning them are therefore worth particular notice.

A circle is a line drawn at an equal distance round a fixed point
called its centre; the obvious but important corollary of which is,
that every point on this line (the circumference of the circle) is at
an equal distance from its centre.
Some people may become indignant upon the discovery that
they have bought this kind of information, but it is curious how
often the point has been missed in circumstances in which its
recollection might have been most enlightening. There are, however, two other attributes of the circle which it is extremely important that the reader should recall to his memory. They are the
tangent and the chord.
A tangent is a straight line drawn so as to touch, without
cutting, the circumference of a circle.
Two points of navigational interest to notice about the tangent
are: (i) that if a line (radius) be drawn from the centre of the
circle to the point of contact it will always be perpendicular to the
tangent, and (ii) that a tangent always appears to coincide with the
circumference of the circle for a short distance on each side of
the point of contact. Incidentally, these t~o facts are very closely
associated with each other, because, owing to this apparent coinci-

A chord is a straight line joining the ends of an arc, X Y in
Fig. 1.
If a chord be bisected by a line at right angles to itself, this line
will pass through the centre of the circle to which the chord
pertains.
It should be noticed that when a chord subtends a very small
arc of a circle it amounts to much the same thing as a tangent.

AAN

B

THE TRANSFERREDPOSITION LINE

CHAPTER

IV

THE TRANSFERRED POSITION LINE
ALTHOUGHit will be generally conceded that the most accurate
astronomical positiom that can be obtained at sea are derived
from stellar observations, the fact remains that the sun is by far
the most accessible of the heavenly bodies, and consequently the
most used and the most useful. Since, however, all astronomical
positions are obtained from distances, it follows that the sun can
only give one position line at a time, and therefore every actual
position, or fix, by the sun must involve a 'transferred position
line'. This means that the principle of, and all principles associated with, the transferred position line are of vital importance to
astronomical position-fixing, and should be examined with according due care. Whether position lines are derived from terrestrial
or celestial sources does not in any way affect the manner of their
use, and for certain reasons it will be convenient to demonstrate

upon the latter. Let A B (Fig. 2) be a position line (P.L.). Then
at the time this P.L. was obtained it is known that the ship was
somewhere on it. Let it be given that from the time this P.L.
was obtained the ship steamed the course and distance represented
by the line c c'. Then if she had been at c she would have arrived
at c'. Now let it be assumed that she had been at d. In this case
she would have steamed an equal distance on a parallel course and
10

II

have arrived at d'. Draw the line A' B' through c'd'. Then
because c c' and d d' are equal and parallel it follows from the law
of parallelograms that A' B' is parallel to A B.
This means that wherever the ship may have been on A B,
she will, after making good the given course and distance, be on
the line A' B'. A' B' is therefore a transferred position line. To
transfer a position line, therefore, we simply move any point in it
the course and distance made good in the interval, and through
the point so found draw a line parall~l to the original position line.
We may call this a 'hardy' principle, since it holds good even
though the ship's course be altered, or, which is the same thing,
it is necessary to allow for a current in the interval. Figs. 2a and
2b are intended to demonstrate this. It will be seen that the
reason why the principle holds good despite an alteration of course
is because two (or more) separate and distinct parallelograms are
formed.
N ow consider an everyday instance of the principle applied in
practice.
A ship steaming on the course 025° takes a bearing of the
lighthouse L (Fig. 3) and finds it to be 050°, which bearing she
draws on the chart (050° + 180° = 230° from L). After continuing
on this course for 9 miles she takes another bearing which she
finds to be 085°, which she also draws on the chart (265° from L).
Required the position.
Draw in a line 025° to represent the course line and mark off
along it 9 miles from the first bearing (A B in Fig.). Through
the point so found draw a line parallel to the first bearing, and the

ship must be at C, the intersection with the second bearing.
(Note that had any other line, such as the dotted line, been
assumed as the course line, the same position must necessarily
have resulted.)
.
The case of two distances is of special interest as being the only
means by which a position can be fixed by two altitudes of the
same or different heavenly bodies successively obtained. In
practice, of course, we generally use approximate position lines
in the shape of tangents, or, possibly, chords; but in addition to
the fact that the following method can, under certain circumstances
(radar ranges, for instance), be actually used in practice, it is also
illustrative of an important principle.
•
First, it must be understood that the only practicable way to


THE TRANSFERREDPOSITION LINE

13

C' as centre and the same radius as the first position line, describe
the arc A' B'. Then A' B' is the transferred position line. (e'
represents a small portion of the position line transferred as a
tangent. )
Now an example in practice.
A ship steaming 010 takes a sextant angle or radar range of
the lighthouse L (Fig. 5), which gives her distance from same as
0

7 miles. After continuing her course for 6 miles another sight
places her 3 miles off. Required the position.
From L layoff 010 6 miles (run) to L'. Then with L' as centre
and 7 miles as radius, describe an arc. This will be the first
position lines transfured. With L as centre and 3 miles as radius
describe the second position line. The intersection then shows the
position.
THE ISOSCELES
TRIANGLE.Whenever two bearings are SQ taken
that the angle between the course line and the second bearing is
double that between the COurse line and the first, an isosceles
0


14

THE

ART

OF

ASTRONOMICAL

NAVIGATION

triangle is formed in which the course line intercepted between
the two bearings and the second bearing line are the two equal
sides. Thus, taking the first triangle A B L
(Fig. 6) the angle inside the triangle at

B = 180°- 40° = 140°, and consequently the
angle at L must be 180°- (140° + 20°) = 20°.
That is to say, the angles at A and L are
equal, being each 20°, and therefore, by the
rule of isosceles triangles, the sides opposite
those angles must also be equal. Thus
A B=B L, or 'distance off equals distance run'.
Similarly it will be seen that B C = C L,
and C D = D L. It is this isosceles triangle
problem which is familiarly known at sea
as 'doubling the angle on the bow', a
method of which the four-point bearing is
a much used example. It should always be
borne in mind, however, that a fix by two
observations not simultaneously obtained
can only be had through the principle of
the transferred position line, and that
doubling the angle on the bow is merely a
convenient method of obtaining the position
without the necessity of actually drawing
the transferred position line on the chart.
Also, it cannot be too frequently pointed
out that the four-point bearing is useless
as a means of safely passing a danger. It is simply a convenient
method of obtaining a departure.
When doubling the angle on the bow the angle of intersection
will always be the same as the first bow angle.
Having crawled under the transferred position line and had a
look at its works, let us now proceed to take notice of a point
which always arises when fixing position by successive observations. We say 'always' because the point arises in one form or

another irrespective of whether the observations are of terrestrial
or celestial objects.
We know that the reliability of any fix depends on a good angle
of intersection of the position lines. In the case of a 'running fix'

THE

TRANSFERRED

POSITION

LINE

15

it will also depend on the course and distance in the interval being
made good. In practice it is never exactly known, and the error
may generally be considered as a percentage of the run.
Our difficulty, however, in the case of terrestrial position lines,
is that the angle of intersection (when the same object is used)
always depends on the length of run, and we have, as it were, to
split the difference. In doing so we make use of the valuable fact
that although the change of bearing is dependent on the length of

run, it is never proportional to it. Take the case of a light picked
up 10° on the bow and 30 miles distant. The angle between the
first bearing and the beam bearing would be 80°, yet we generally
prefer to use the four-point bearing, because 45° is a very fair
angle and the length of run a great deal shorter: the Traverse
Table tells us 5·2 miles as against 29·5. This, of course, is because

the rate of change of the bearing increases until the object is
abeam. It will be evident, then, that two bearings of an object


16

THE ART OF ASTRONOMICALNAVIGATION

when it is at an equal angle on either side of the beam bearing
will intercept a shorter length of course line than two bearings
the same angle apart on one side of it. Thus in Fig. 7 if a ship is
passing 10 miles off the lighthouse L and it is on the four-points
at A the distance A B will be 10 miles and the angle of intersection of position lines at B 45°. If, however, the two bearings
had been taken at two points before and abaft the beam as at
X and Y, then the distance X Y would have been only 8·3 miles,
while the angle of intersection at Y would still have been 45°.
We can state, then, as a definite rule, that a more reliable fix can
always be had from a bearing on either side of the beam bearing
than can be had from two bearings on one side of it. Consider
the beam bearing as the meridian and the whole argument applies
equally to astronomical fixes.
Now it is quite obvious that to be able to fix our position from
a single object, other than by simultaneous bearing and distance,
it is necessary that the ship be moving at a known course and
speed. If, however, we could imagine a lofty beacon in actual
motion across the earth's surface, and that the direction and rate
of its motion were so known and tabulated that we could plot its
position at any moment on the chart, it will be easily seen that we
could fix our position from it even were the ship stationary. By
means of the transferred position line we could also fix our position

by it while the ship were in motion, provided the rate and direction
of her motion were not such as to neutralise the relative motion
of the beacon.
Such beacons are the sun, moon, planets and stars, and their
rate of geographical motion being roughly 900 knots there is no
great danger of any ship in navigable latitudes neutralising their
bearing-change in the very near future.

CHAPTER

V

THE ASTRONOMICAL POSITION LINE
BEFOREproceeding with this subject it may be as well to recall
the following definitions to the reader's memory:
A great circle is a circle on a sphere whose plane passes through
the centre of that sphere; that is, it always divides the sphere into
two equal parts, or hemispheres.
A spherical triangle is formed by the intersection of three great
circles, and if any three of its six parts be known the remaining
three can be found by calculation. It thus differs from the plane
triangle in that it can be solved through the knowledge of its three
angles alone: which three angles do not add up to 180°, or to any
other fixed sum.
The nautical mile, for the purposes of navigation, is the length
of an arc on the earth's surface which subtends an angle of one
minute at its centre.
An observer on the earth's surface, being unable to judge the
distances of the heavenly bodies, sees them apparently on the
interior surface of an infinitely large sphere.

The rational horizon is the circle where the plane, horizontal
to the observer and passing through the centre of the earth, cuts
the celestial sphere. It thus divides this sphere into two halves,
the upper (i.e. observer's) half of which is called the celestial
concave..
The true altitude of a heavenly body, that is-the altitude
which is obtained by correcting the observed altitude, is the angle
at the centre of the earth between the body and the rational
horizon.
The geographical position of a heavenly body is the point on
the earth's surface vertically below the body. The latitude of this
position is equal to the body's declination, and its longitude is
equal to the Greenwich hour angle of the body, subtracting from
360° to get east longitude if the G.H.A. exceeds 180°. (There is
17


THE

be an observer situated somewhere on the earth's surface. Then
Z is his zenith, H H' his rational horizon, and Z H = 90°.
Suppose, now, he sees a celestial body at X (whose geographical
position is therefore at X') and takes an altitude of it. By correcting
this altitude he gets the arc H X, and by subtracting from 90° he
gets the arc Z X, his zenith distance. But the arc Z X in degrees
and minutes is the same as the arc 0 X' in degrees and minutes;
and if he knows the length of the arc 0 X' in degrees and minutes
he knows its actual lineal length in nautical miles.
This means that for the purposes of navigation zenith distance
is the same as geographical distance; and herein lies the whole

principle of nautical astronomy, and of every method of fixing a

ASTRONOMICAL

POSITION

LINE

19

ship's position by that science which is in common use today.
Containing, as it does, the whole principle of astronomical
position-finding in a nutshell, the importance of understanding this
point can hardly be overestimated. Pursuing the argument with
this principle as primary foundation, we proceed as follows:
The navigator in his ship at sea keeps a careful account of the
courses and distances he makes in order that he may be able to
estimate his position at any required moment. For various wellknown reasons, however, this estimate is liable to be dangerously
in error at times, and he therefore has to find some means of
checking it. To do so, it is obvious that he must obtain some kind
of measurements from objects whose true positions are accurately
known. Such objects are the heavenly bodies.
Now, the navigator's instruments are his sextant, his chronometer, and his nautical almanac. With his sextant he takes an
altitude of a heavenly body, noting simultaneously the chronometer time. This altitude gives him one element of his position
relative to the body, namely, his distance from it. But since the
body itself is always in rapid motion, it is necessary for him to
know its exact position at the instant of observation. This information is supplied by his chronometer and almanac, the declination being the body's latitude, and its G.H.A. (west or east) its
longitude.
Thus any single sight simply gives a circle of position, and
nothing else but a circle of position. Since, however, the position

of the ship is always known within certain limits, he knows that
the ship is on a certain limited arc of the circle of position. And
if, as is generally the case in practice, this arc is very small in
comparison with the radius, it may be considered a straight line
running at right-angles to the bearing of the body of which the
sight was taken ..
It is interesting, and perhaps not without value, to note the
difference between this modem conception of sextant and chronometer position-finding as presented above, and the older attitude
to the subject which persisted for so many generations-and,
indeed, still persists in some places. Under this older attitude
latitude and longitude were thought of as the means by which a
ship's position was found, and hence the value of any single sight
was appraised in accordance with its suitability for determining
either the latitude or the longitude. In the light of the preceding


20

THE ART OF ASTRONOMICALNAVIGATION

arguments, however, it will be seen that just as one does not
endeavour to obtain terrestrial bearings which coincide with the
latitude or the meridian, so also is it pointless to try to obtain
astronomical position lines which do so. The value of a plurality
of position lines depends on their angle relative to each other
(angle of cut), and that of a single one upon its angle relative to
the land or other dangers, or the course. As one recedes from the
land a position line's bearing relative thereto becomes of less importance, but as latitude and longitude are intangibilities with
which there is no danger of collision, it does not, as one recedes
from the land, become more desirable that position lines should

coincide with them. This point may become clearer if we consider the following definitions:
Position lines are the means by which an unknown position is
determined from actual measurements.
Latitude and longitude are (i) a means by which a known or
given position may be expressed, and (ii) the names of certain
angular values used in the mathematical calculations connected
with navigation.
Another point which it might be worth clearing up at this
juncture is that relating to the true function of the chronometer
in nautical astronomy. Under the older system time was thought
of as necessary to the finding of longitude. But let it be considered
how things would be if the earth were brought to a standstill on
its axis. Although the heavenly bodies would then become
stationary (or nearly so) their value to the navigator would become,
if anything, slightly enhanced, for their altitudes could be measured
with more care and convenience. They would be, as it were,
eternally on the meridian in all directions. The question of time,
it will be seen, would not arise; and if there were thought to be
any point in ascertaining the longitude in particular it would be
done by simply taking an altitude of an east- or west-bearing body.
The earth's habit, therefore, of continually revolving on its axis
is merely a navigational nuisance which has to be countered by
means of the chronometer, the function of which instrument is
to reveal the position of the earth on its axis at the instant a sight
is taken.

CHAPTER

THE ASTRONOMICAL

VI

TRIANGLE

FROM the preceding chapter it would seem that to draw the
position line on the chart we have only to 'prick off' the position
of the body, and then, with the zenith distance as radius, sweep
an arc in the direction of the D.R. position.
Theoretically this is perfectly true; but as this zenith distance
is liable to be anything up to 5000 miles or so, various difficulties
make this sort of straightforward geometry impracticable. Besides
the smallness of the chart scale that might have to be used, the
matter of chart projection obtrudes itself in an acute form. It
could, of course be done on a globe, but not accurately-or very
conveniently. For a satisfactory solution we find ourselves forced
into the field of mathematics.
Now the mathematical part of nautical astronomy simply consists of working out certain trigonometrical formulae connected
with a particular figure known as the 'astronomical triangle'.
There is no practical necessity for the navigator to make a
detailed investigation of the derivation of these formulae, since
they are acknowledged mathematical facts not open to question.
All that he requires is a sufficient understanding of the principles
involved to be able to turn the formulae with which the mathematicians supply him to useful account, and the first step to this
end is to understand the astronomical triangle ..
Fig. 10 represents the two concentric spheres consisting of the
earth and the celestial sphere in which C is the common centre,
P S the common polar axis, and the arc through R Q the celestial
equator. The arc P G S has also been inserted to represent the
Greenwich celestial meridian.
Let 0 be an observer on the earth's surface, then Z is his zenith.

Let X be a celestial body of which a sight has been taken, then
R X is its declination and X' is its geographical position. • P Z X
is the astronomical triangle. The angle within the triangle at Z
21


22

THE

ART

OF ASTRONOMICAL

NAVIGATION

is, of course, the azimuth, and it might be mentioned that the
angle at X is called the 'parallactic angle' and the celestial point
opposite the zenith is called the 'nadir'.
Now, in this triangle both the side P X (90° - dec.) and the side
Z X (90° - alt.) are known, and we thus have two parts. To solve
it, however, we require a third part, which in practice we never
have. We would not, of course, expect to have a third part;
because if we had we could solve the triangle and deduce the
actual position of Z, whereas from the preceding chapter we
already know that our single sight has only given us a position
line. What we can do is to assume such other parts of the triangle

THE

ASTRONOMICAL

TRIANGLE

23

The angle at P in the astronomical triangle here shown is the
hour angle of X westward from Z. It is called the local hour
angle (of X), and written L.H.A. The angle G P X, on the other
hand, is the Greenwich hour angle of X, which is tabulated in the
almanac under G.H.A. of the body, if the body be sun, moon or
planet. G.H.A.'s so tabulated are given in the westerly sense from
Greenwich, from 0° to 360°. Lastly, there is the angle G P Z
which is the longitude of Z (the observer).
It can be seen at a glance that in the case of the relative positions
of G, Z, and X in this figure:
L.H.A. = G.H.A. - Long., and
Longitude=G.H.A. - L.H.A.
but in as many cases as not the relative positions of G, Z, and X
will be quite otherwise. It will be found, however, that the
following equations always hold good for the sun, moon and
planets, whatever the relative positions of G, Z, and X.

+ East
L.H.A. =G.H.A .
.
_
- ':,est Long. rejecting 360° when necessary.
}
Longltude-G.H.A.

L.H.A.
The longitude is applied in accordance with the rule:
Longitude East, Greenwich time least.
Longitude West, Greenwich time best.
The same equations also hold good for the stars; but individual
star G.H.A.'s are not given in the almanacs. Star G.H.A.'s have
to be formed by adding together the tabulated values G.H.A. 'I'
and S.H.A. (sidereal hour angle), as eXplained in due course on
p.64.
as will enable us to calculate a position through which the position
line may be drawn on a large scale chart or diagram. Each method
of working up an astronomical sight is therefore based on certain
assumptions, which assumptions and methods are explained and
demonstrated in the next chapter.
Before passing from this chapter it may be as well to make use
of Fig. 10 to refresh the memory on the subject of the standard
H.A. equations which come into all problems in nautical
astronomy.


THE FOUR STANDARDMETHODS

CHAPTER

VII

THE FOUR STANDARD METHODS
TURNINGfor a moment to Fig. 10, it will be seen that the two
shaded triangles (celestial and terrestrial) are exactly similar, each
being composed of three arcs which subtend the same angles at

the common centre. In expounding nautical astronomy it is usual
to refer the student with his earth, its poles, meridians, etc., to

25

Also, since the observer is always situated at the centre of his
horizon, it will be convenient to place him there in future figures.
Thus, if we slew Fig. 10 so as to place the observer 0 in the
centre of the figure (covering C), the pole p' will be brought
within the figure, and the observer's meridian P'O S' will become
a vertical circle represented by a straight line. Lastly, to comply
with the usual lettering employed with the astronomical triangle
we will call the observer's position Z.
Fig. 11 shows things portrayed in this manner. Such a figure
is described as being 'on the plane of the observer's horizon', and
it will be found the most convenient type of figure for depicting
most problems in nautical astronomy. There have been added,
in Fig. 11, the points N.S.E.W., which are the north, south, east
and west points of the horizon, Q, to show E Q W, the equator,
and H, to show H X, the altitude, (complement of Z X). This
might also be a suitable occasion to remind the reader that P is
described as the 'elevated pole', and that the elevation NP is
always equal to the latitude, Q Z.
It will be remembered that:
The complement of an angle is the amount by which it differs
from 90°. Thus the complement of 75° = 15°, the complement
of 110°=20°.
The sine of an angle is the cosine of its complement, etc.
The reciprocals are: sin and cosec, tan and cot, sec and coso
Thus sin 8 C 1 8'

osec

Cos 8= S 1 8 ' etc.
ec

The three sides of the astronomical triangle are:
Polar distance (90° ±Dec), usually denoted by p.
Zenith distance (90° -Alt.), usually denoted by z ..
Co-latitude (90° - Lat.), usually denoted by 1'.
And their complements are, respectively,
the celestial concave. Instead, however, of recklessly projecting
the whole system into the infinity of the heavenly regions; by
simply bringing down the single point X to the position X', we
can stay at home and deal with the terrestrial triangle with the
comfortable knowledge that the length of its sides in minutes of
arc are also their actual lineal length in homely nautical miles.
24

Declination, in this book denoted by d.
Altitude, in this book denoted by a.
Latitude, in this book denoted by 1.
Therefore in all formulae connected with the astronomical
triangle the values d, a, and 1 may be used in place of p, ~ and l'
AANC


THE

In the rare cases where it is necessary to work out the azimuth
this formula will be found the most convenient: in fact, with threefigure logarithms, which are sufficient for practical purposes

within 40° of the meridian, it is scarcely longer than by ABC
table. On the other hand, the angle produced being the angle
(0° to 90°) from the meridian, the result is ambiguous when the
bearing is near 90° (or 270°).
Many books purporting to be treatises on practical navigation
used to contain examples of navigation problems worked out with
the co-latitude and polar distance. Now the use of these values
always necessitates a subtraction step which, though simple in
itself, is unnecessary. It would not be too much to say-that in
all problems connected with the navigation of a real ship, any
unnecessary step, however simple, increases the chances of
making a mistake, and is therefore a breach of good seamanship.
The four standard methods of locating the position line are:
(i) Marcq St. Hilaire's (to find z).
(ii) Longitude (to find L.H.A.).
(iii) Ex-meridian (to find mer. z dist., i.e. I ;:t;d).
(iv) Meridian altitude.
They are placed here in order of their availability. That is
to say, while the St. Hilaire method is always 100 per cent
available; the longitude method ranges from about 90 per cent
to 60 per cent available; the ex-meridian from about 25 per cent
to 1 per cent; and the meridian altitude about 1 per cent. Many
different formulae have been devised and used, or advocated, in
the past for working the three out-of-meridian problems. The

FOUR

STANDARD

METHODS

27

'reigning favourites' are all taken from the same equation, thus:
(i) Hav z=hav (I ;:t;d)+cos I cos d hav L.H.A.
(ii) Hav L.H.A. = [hav z - hav (I ;:t;d)] sec I sec d.
(iii) Hav (I ;:t;d)= hav z - cos I cos d hav L.H.A.
The quick eye may detect something suspicious in (iii), since
the latitude (which is the value sought) appears on both sides of
the equation. In actual fact all ex-meridian tables and formulae
in general use are based on certain assumptions which only hold
good when the body is at moderate altitude and near the meridian.
The procedure for the use of the ex-meridian method in practice
is explained and demonstrated in due course on pp. 51 and 52. A
'working formula' for each of the above is given below and on
pp. 37 and 38.
In the St. Hilaire method we assume a position (such as the
D.R.) in order to get a Lat. and Long. for Z. This gives us the
three parts of the astronomical triangle I', p and P (two sides and
the included angle) and with these three parts we proceed to
calculate the zenith distance corresponding to the assumed position. Now, since the true zenith distance (z) is our true distance
from the body, it follows that the difference between the calculated
and true zenith distances ('intercept') must represent the amount
that we are either nearer to or further away from the body than
we assumed ourselves to be. The rest is easy! The position line
runs at right-angles to the body's bearing. Through the assumed
position draw a line to represent this bearing and mark off along
it, either towards or away from the body as the case may be, the
intercept. Through the point so found draw the position line at
right-angles to the bearing line.

The working formula for z is:
Log hav 8= log hav L.H.A. + log cos I + log cos d.
Nat hav z=nat hav 8+nat hav (I ;:t;d).
In all navigation formulae where lat. and dec. are combined the
rule is, like names subtract, unlike names add. If it is borne in
mind that what is always wanted is the d'iat. (i.e. the distance
apart in latitude) between lat. and dec., no mistake should arise.
The St. Hilaire process by description is as follows:
With the date and G .M.T. enter the almanac and tak60out the
G.H.A., and to it apply the longitude. Result-L.H.A.
Under
the L.H.A. put down the lat. and dee. and form their d'lat. Take



30

THE ART OF ASTRONOMICALNAVIGATION

centre and X Z as radius an arc has been swept in to represent
the resulting position line. Call the lat. of Z Lat. 1. It can be
seen that the longitude resulting from the solution of this trianb~ will obviously be that of Z-i.e. the longitude where the position
line cuts Lat. 1.
Now assume some other latitude such as Lat. 2 to be used.
In this case the triangle necessarily becomes P Z' X, and the
longitude resulting from its solution will be the longitude of Z'
-i.e. the longitude of the point where the position line cuts
Lat. 2.
So that whatever latitude we work with we shall always get
the longitude of the point where the position line cuts the assumed

latitude.
It will be remembered that when locating the position line by
this method the body's bearing is always assumed to be the same,
whatever the latitude used, whereas in our present figure there is
a considerable difference. This is because the lats. here shown
are several hundred miles apart, whereas in practice they would
seldom exceed twenty. *
THE Ex-MERIDIAN
ANDMERIDIAN
ALTITUDE.By the nature of
things these two should be considered together.
To take the meridian altitude first-when an astronomical body
is actually on (or, to speak technically, 'in transit of') the meridian
the position line derived from its zenith distance is a tangent to
the parallel of latitude which the observer happens to be on at
the moment, and it therefore determines the latitude. Not a particularly profound remark, this, perhaps: but it is intended to
make the point that the fact that the meridian altitude determines
the latitude is merely coincidental.
The latitude from a meridian altitude is deduced from the
equation-Lat. =z ;td. Some momentary confusion sometimes
arises as to how z and d should be combined in particular instances.
The simplest way to settle this point is to draw a vertical line
to represent the meridian and then fill in N, S, Z, Q and X,
in that order, taking care to place X the right side of Q, and
of Z.
• Jt may be of interest to recall that the old 'Sumner' method consisted of
working out both the positions of Z and Z' and obtaining the position line by
joining them. In this case the position line is not a tangent to the position
circle, but a chord. Hence the method was sometimes known as the 'method
of chords'.

that the ex-meridian method was devised: its original purpose
being to adjust an intended meridian altitude for a 'near miss'.
At one time the ex-meridian process was the only known method
by which a sight taken very near the meridian could
made use
of. The St. Hilaire method has robbed it of this distinction, but
by reason of the ease and expedition with which it can be solved
by means of specially prepared tables the ex-meridian still holds
its own.
The principle of the method being to treat the sight as a
meridian altitude, it is necessary to find the amount by which the
measured altitude must be increased (or the zenith distance
reduced) to make it what it would have been had the boJy actually
been on the meridian at the moment of observation.

be


THE

FOUR

STANDARD

METHODS

33

L.H.A. By adding this amount to the altitude (or deducting it

from the zenith distance) the meridional zenith distance is obtained
and thence the latitude of the point Z. This is the usual explanation
of the ex-meridian problem from the mathematical point of view,
but from the navigator's point of view we must proceed a little
further. Thus, since the longitude used is generally only an
assumed longitude, it is necessary to understand what happens if
this assumption is much out.
Suppose, then, that the observer had assumed some other
meridian, such as the (dotted) meridian of Z'. In this case the
astronomical triangle becomes P Z' X, and the reduction which
the ex-meridian tables would give for this-triangle would be M'R',
yielding the zenith distance M' Z', and thence the latitude of the
point Z'.
So for every different longitude (or L.H.A.) we use, we get a
different reduction and, consequently, latitude, this latitude in
each case being the latitude in which the position line cuts the
assumed longitude.
There is yet one other method of obtaining a position line which
should undoubtedly be rated as 'standard', and that is by an
altitude of the pole star. It is simply an ex-meridian, to which the
above remarks therefore apply. That is to say, it determines a
position line at right-angles to the bearing of the star at the
moment of observation.
The procedure for deducing a position line from a pole star
sight is explained and demonstrated in the pole star tables in the
almanac.


CHAPTER

VIII

THE POSITION LINE AT CLOSE QUARTERS
SO far we have been dealing orrty with the astronomical side of our
subject. It is one thing, however, to be able to work out a problem,
but quite another thing (0 understand how to make the best use
of it: and to this end it is necessary to understand certain local
triangles which establish connection between the D.R. or assumed
position and the astronomical position line.
Summarising the preceding chapter we have the following facts:
(i) The St. Hilaire method determines the bearing and distance
of the point on the position line nearest to the D.R.
(ii) The Longitude method determines the longitude in which
the position line cuts the assumed latitude.
(iii) The Ex-meridian method determines the latitude in which
the position line cuts the assumed longitude.
It will be highly convenient to have some general term for these
points, and in default of any existing British expression we shall
borrow from the U.S.A. the description 'COMPUTEDPOINT',
which term we shall abbreviate in future into 'C.P.'
Then:
(i) In the St. Hilaire method the C.P. is found by applying the
intercept to the D.R. position as a course and distance.
(ii) In the Longitude method the c.P. =lat. D.R. and long. obs.
(iii) In the Ex-meridian method the C.P. = long. D.R. and lat.
obs.
And to plot the position line on the chart in each case we simply
have to mark off the C.P. and through it draw a line at right-angles
to the body's bearing. The C.P. is defined by three terms, of
course, viz. lat., long., and bearing. Without the bearing the thing

is useless.
Let us suppose, now, that a ship in a given D.R. position takes
an altitude of a heavenly body. Then this altitude determines
her distance from the body, and consequently gives her a position
34

position J E X the body's bearing, and the distance E J 10 miles.
Through J draw the line M L at right-angles to X E J. Then
M L is the position line.
Now, if this ship works up her sight as a St. Hilaire and ~pplies
the intercept (320° 10 miles) to her D.R. position she will get the
lat. and long. of the C.P. J; if she works it as a Longitude she will
get the lat. and long. ofthe C.P. L; and if she works it as an exmeridian sh~ will get the lat. and long. of the C.P. M (i.e. she
would if the body were within ex-meridian limits). For convenience we might call E L the 'longitude intercept' and E M the
'ex-meridian intercept'.
a
From the figure it will be seen that there are three right-angled
triangles which connect the D.R. position with the astronomical


36

THE

ART

OF

ASTRONOMICAL

NAVIGATIO

N

position line. Thus there is the larger triangle L E M in which E
is the right-angle, and also the two smaller triangles E] L and
E] M, in both of which the right-angle is at].
Also, since in
this system of right-angled triangles all the angles are known, it
follows that if anyone of the sides be known, the position of
L, ], M or E with relation to any other of those positions may be
calculated. In practice this would be done by means of the
Traverse Table.
Example I: Knowing E ] = 10 miles, to find M.
In triangle E] M, with 40° as course, and E] 10 miles as
D'lat., EM = Dist. = 13 miles.
Therefore M lies 13 miles North of the D.R. position.
Example II: Knowing E M = 13 miles, to find L.
In triangle M E L, with 50° as course and E M 13 miles as
D'lat., E L=Dep.=15·5
M. Turn 15·5 into D'long. in the usual
manner and apply to the D.R. long.
As we have said, to understand these local triangles is to understand 'applied nautical astronomy', and for this reason it cannot
be deprecated as waste of time to consider them from every
possible angle. There is a certain experiment which most of us
have either tried or seen tried in our time in which a sight taken
near ex-meridian limits is worked up both as an ex-meridian and
as a longitude. From Fig. 16 we know, however, that if we work
up the sight in question as an ex-meridian we shall obtain the
latitude of the point M. And if we had any doubt on the matter

before, it should now also be clear that by reworking the sight by
the longitude method (using the ex-meridian lat.) the resulting
longitude must also be the longitude of M-that is to say, the
D.R. long. Similarly, by working the sight first as a longitude we
shall get the longitude of the point L, and if with this longitude
we rework it as an ex-meridian, we shall get the latitude of Lthat is to say, the D.R. lat.
Since any single sight simply determines a position line, and
nothing else but a position line, it follows that whatever method
we employ to work it up we can only get the same useful result,
providing always that the method is appropriate. The following
experiment in which the same sight is worked by the three standard
out-of-meridian methods and the results reduced to a common
C.P. for comparison is worthy of attention.


THE

POSITION

LINE

AT CLOSE

QUARTERS

39

its place. It will be obvious, then, that the most likely of these
new possible positions will be that which lies nearest to the
original estimate of the ship's position.

The point may be illustrated as follows:
Let E, Fig. 18, be the D.R. or estimated position at the moment
when a position line (X Y in figure) was obtained; and let the
circle ABC bound the extent of possible error in the D.R.
position. Then it is evident that the line A B contains all the
possible positions which the ship might occupy, and since] is
the mean of all these positions it is therefore the most probable.

It will be found that the sight here worked is considerably outside the limits of any ex-meridian tables and could not, therefore,
have been worked by means of such tables. Although the formula
employed in this example will give satisfactory results up to considerable angles from the meridian, it should be clearly understood
that the only object in working an ex-meridian is in order to take
advantage of the handiness of specially designed ex-meridian
tables. When this object is not available the St. Hilaire method
is to be preferred as being more safe.
Presuming it to be generally known that the longitude method
should not be employed on a bearing of less than (about) 20° from
the meridian while the St. Hilaire method will give an equally
accurate result whatever the body's bearing, it is desired hereby
to draw attention to yet another property of the St. Hilaire method
which is worthy of serious attention, namely, that it at once
determines the bearing and distance of the most likely position of
the ship resulting from any single sight. Assuming the D.R.
position to be the most probable position of the ship at the
moment a sight is taken, the resulting position line, if it does not
pass through or near to the D.R. position, renders that position
an impossible one, and creates a number of new possible ones in

Now, it is extremely important to understand that the ship iE
under no obligation whatever to be at the most probable position.

Indeed, since the line A B contains an infinite number of points,
the chances of her being right on the point ] are very nearly
infinitely small. Under no possible circumstances can J be regarded
as a substitute for a fix.

This may read rather like a negation of the matter contained in
the preceding paragraph. There is, however, a more practical
way of looking at the thing than as a question of probability;
namely, that subject, of course, to the sight being a good observation, correctly worked, the point ] must always, under every
combination of circumstances, actually lie nearer to the ship's
true position than the D.R.; whereas any other position, such
as that derived from the longitude or ex-meridian metQod,
may very easily lie further away. And since for the purposes
of navigation it is necessary to assume some position from which


40

THE

ART

OF ASTRONOMICAL

NAVIGATION

to set the course, ] is the most logical position to assume.
It would be not inappropriate to call this] position the 'sightamended D.R.'
The above train of argument is intended to apply more particularly to deep-sea navigation. When a landfall has to be made
from a single sight, however, and it is not possible to use it in the

manner explained on pp. 79 and 82, it is the general practice of
prudent navigators to assume.their ship to be in the most dangerous position. In this case mark the position line and the D.R.
on the chart. Then, taking on the dividers a distance equal to the
amount of possible error in the D.R., place one leg on that position
and with the other note the point on the position line on the side
towards the danger (A or B, Fig. 17). From this place set the
course.

PART II
THE PRACTICE


CHAPTER

IX

SUCCESSIVE OBSERVATIONS
THE chief desideratum of the navigator is that of being able to
fix his position whenever his discretion suggests the necessity.
This being impossible, those methods which will most nearly
supply the want should claim his attention first, and on the
assumption that two observations of the sun are the most frequently
available means under all circumstances, two such observations
with the necessary change-of-bearing interval between are treated
first. Four examples of the same pair of sights worked by different
methods are given as under:
Method I.
" II.
" III.
" IV.

Two St. Hilaires plotted.
Two St. Hilaires and Traverse Table.
Two Longitudes plotted.
Longitude and St. Hilaire with Traverse Table.
METHODI
Two ST. HILAIRESPLOTTED

In this case the two intercepts are calculated from their respective D.R. positions and the results are then plotted in the following
manner:
Assume any convenient point as D.R. and through it erect a
perpendicular to represent the meridian. From this point layoff
the two intercepts on any convenient scale, such as the lines in
the work-book, and thence the two position lines. The D'lat.
and Dep. to be applied to the D.R. position to obtain the observed
position will then be apparent, the Dep. of course, being referred
to the Traverse Table for D'long.
Note. It will be seen that no special charts, diagrams, squared
paper, or such-like paraphernalia are required, the only instrument necessary being an ordinary protract:or such as is frequently
43

•