University of California.
PRACTICE OF NAVIGATION
PEAGTICE OF NAVIGATION
HENRY RAPER, Lieut. R.N., F.R.A.S., F.R.G.S
REVISED AND ENLARGED.
PUBLISHED BY J. D. POTTER,
Admiralty Agent far Cliarls
145 MINORIES, E.
REAR-ADMIRAL SIR FRANCIS BEAUFORT,
HYDROGRAPHER TO THE ADMIRALTY
The eminent station which you occupy in
tlie naval scientific world renders it highly gratifying
to me to dedicate the following Work to you as
a testimony of my regard and esteem ; while the
general accordance of my views on the subject with
those of your more experienced judgment, gives me
the greater confidence in laying my labours before
1 have the honour to be,
Your obedient Servant,
Tdis Wcrk is intended for the use of all persons concerned
either with the navigation of ships or with the determination of
latitude and longitude on shore.
The present volume, which is devoted exclusively to tlie prac-
tice, contains all the rules and tables necessary in navigation, and
for the determination of latitude and longitude by means of the
sextant or reflecting circle. The study of its contents demands no
|>revious attainments beyond the knowledge of the elements of
arithmetic. Every endeavour has been made to render the whole
easy of reference, and to adapt it to the use of those who may
desire to instruct themselves. Rules which admit of more cases
than one, as, for example, that for applying the equation of equal
altitudes, are given in the form o( tables ; so that the several con-
ditions involved, and their mutual connexion, being exhibited to the
eye, the computer is relieved from the sense of complication, and
the cliance of a mistake is materially diminished. An ample alpha-
betical index is annexed, by whicli tlie reader is at once referred to
all the inforiuation which the volimie can afford him.
Those who have been brought u]) to the sea, and who have
experienced the distaste for long calculations which that kind of
life inspires, will not hesitate to admit that the only means of
inducing seamen generally to profit by the numerous occasions
â€¢A-hich offer themselves for finding the place of the ship is extreme
VI PREFACE TO THE FIRST EDITION.
breviiy of solution. It is not, however, merely as a concession to
indolence, that rules should be made as easy and simple as possible;
the nature of a sea life demands that every exertion should be made
to ? bridge computation, which has often to be conducted in circum-
stances of danger, anxiety, or fatigue, and so to separate the several
points, that the seaman may be referred directly to what concerns
his case, to the exclusion of all other matter. These considerations
have lieen carefully kept in view in the rules, in the examples, and
in the form and order of the tables.
Two kinds of solutions are employed, and, in general, two only;
naicely, an approximate method, and a complete, or, as it is called,
rigorous, method. The former may often serve in cases of haste,
or when precision is not necessary, and will also afford a conve-
nient check against the effects of a mistake in the more elaborate
All the computations are effected by the well-known methods of
inspection and logarithms; and as the former, it is presumed, leave
but little to be desired in point of expedition, Gunter's scale, or other
mechanical methods, are not employed.
Sailing on a Great Circle is, in this work, reduced, like Plane
Sailing, to Inspection, by means of the Spherical Travebsk
Convenient rules are given for finding the distance of the land
by its change of bearing, and by its altitude observed above the
The seaman will find every necessary information on the subject
of local magnetic deviation.
The highly useful problem of determining the latitude at sea, by
the reduction of an altitude to the meridian, will be found greatly
abridged ; and a table is added for the purpose of shewing the
limits within which the result may be depended upon when the
time at shi|) is in error. This table will bo found, it is presumed,
of considerable utility, as it is perhaps from the want of some
specific information as to the degree of confidence which it is safe
to place in the result, no less than of a short and easy rule, that this
excellent observation is almost entirely neglected ; and, in conse-
tjiienee, the latitude, when the meridian altitude is not exactly
obtained, is too often lost for the day.
The approximate solution of the double altitude, as a questioL
of Time, will be found, it is hoped, well adapted to general use:
since unless tLe latitude by account is very much in error this
PREFACE TO THE FIRST EDITION. VU
aietnud determines both the true latitiiJe anil tlie time at ship;
and the computation of the time is one with which seamen are
familiar in the next degree to tnat of the latitude by meridian
altitude. The principle is not new, but rules have not hitherto
been given for computing directly the error of the latitude by
The first approximate method of clearing the lunar distance ia
new, being effected, like many other problems, by the Siiherical
Traverse Table. The rigorous method is a modification of Borda's,
and employs five logarithms, of which two only are taken out to
In a work in which many of the methods are new, I have felt it
would be more satisfactory to the professional reader to find them
illustrated by observations actually taken at sea. The examples
are accordingly selected from the journals of Captain W. F. W.
Owen, who kindly lent them to me for the purpose; though, neces-
sarily, in proceeding by fixed rules, I could not introduce the
solutions employed by that distinguished navigator. The remaining
observations have been furnished to me by the Rev. G. Fisher,
astronomer to Sir Edward Parry's expedition to the Polar Seas.
In order to enable the computer to judge of the degree of pre-
cision to which he attains, the degree of dependance to be placed on
tiie result, or the limit of probable error, is indicated. This is the
more important, as very indistinct and erroneous notions prevail
among practical persons on the subject of accuracy of com])utation ;
and much time is, in consequence, often lost in computing to a
degree of precision wholly inconsistent with that of the elements
themselves. The mere habit of working invariably to a useless
precision, while it can never advance the computer's knowledge of
the subject, has the unfavourable tendency of deceiving those who
are not aware of the true nature of such questions into the per-
suasion that a result is always as correct as the cop puter chooses
to make it; and tlius leads them to place the same confidence in
all observations, provided only they arc ivorhtd to the same degree
of accuracy. By habitually following the short precepts laid down
on this point, tlie computer will learn insensibly to estimate the
value of liis results ; of which, since the limit of error is the sole
ciiterion of the accuracy of any determination, he cannot otherwise
be a judge. The degree of precision to which it is proper to carry
the work in any case is observed, in general, in the examples.
In the Tables every endeavour has been made to repde"- the
Vm PREFACE TO THE FIRST EDITION.
collection complete for the purposes required, and to compress the
whole into small compass. For the sake of clearness, a different
figure has beei) adopted for the argument and for the numbers in
the body of each table. In the logarithms six places of figures
only are employed, because a single result in which six places are
necessary cannot be depended upon to the degree of precision
obtained. On the same principle, some of the logaritlims are given
to three places only.
The log. sine square of half the arc, Table 61, universally
familiar to seamen in finding the time, is given, for the convenience
of this constant computation, to every second of the 12 hours. By
means of this term tables of versed sines are dispensed with, all our
solutions being either numeral or purely logarithmic.
I have not, either in the Rules or the Tables, aimed to make
that additive which is in the nature of things subtractive. The
precept subtract is as easy as the precept add; and when the
student has the natural process before him he may be led to dis-
cover the reason of it; and must thus, by attention, always advance
in knowledge of the subject. But an artificial process obstructs the
exercise of the faculties, or leads the student, who reflects on what
he does, to false conclusions.
The composition of the Table of Maritime Positions has been
a very laborious task, and has caused great delay in the appearance
of the Work. The numerous chronometric measures furnished of
late years liave rendered it necessary to deduce longitudes in a more
systematic and accurate manner than that hitherto followed, which
has cliiefly consisted in modifying former determinations by means
of those succeeding them. Absolute, or astronomical positions, and
relative positions, being distinct things, and the latter being by far of
the greater consequence to navigation, it is necessary, preparatory
to a complete and final arrangement, to separate these two kinds of
determinations. Accoi'dingly, in a series of papers, some of which
have been already published in the Nautical Magazine,* I have
endeavoured to arrange the chronometric differences of longitude
with reference to certain fixed points, convenient for the purpose,
which it is proposed to call Secondary Meridians. These standard
â€¢ The data or evidence for the several positions being given in these papers, the
ralue of each determination is easily appreciated ; and accordingly, individuals in pos-
session of one or more good watches may, by correcting defective measures, or by
establisliing new linlcs of connexion, render material service to maritime geography.
See Nautical Maijazine, 1839, and followin;^ years.
rUEFACE TO THE FIRST EDITION. IX
Iiositioiis, of which the number assumed is eighteen, being con-
siderably distant from each other, are determined nearly enough
for present purposes, and would, according to the system proposed,
be finally settled by long series of astronomical observations.
An account of the principles adopted in this arrangement, and
of the several voyages and surveys from which the materials have
been taken, will be found, together with some suggestions for the
advancement of the subject, in the Nautical Magazine. But it is
necessary to state here, that the late determinations of the longitude
of Madras have, from the importance of that position, occasioned a
long and intricate discussion. Mr. Riddle and Mr. Maclear have
compared observations of moon culminating stars made at Madias,
with like observations made in Great Britain and at the Cape of
Good Hope respectively. According to their computations, which
agree very nearly, the received longitude, 80" 17' 21", is about 3' "21"
too great. The number and superior character of these observa-
tions, and the agreement of the results, liave led me to adopt, with-
out hesitation, 80Â° 14' 0" ; while the magnitude of the correction has
rendered it indispensable to trace its effects on the longitudes of the
Precision in the Maritime Positions, especially in the longitudes,
becomes, as navigation advances to perfection, a matter of increasing
importance; because, where longitudes are well determined, the
error of a chronometer may be ascertained on every occasion of
making the land.
It will not be out of place to remark here that it is high time the
chronometer should be found, like the compass, among the stores of
every vessel beyond a mere coaster. It would be superfluous to
attempt to prove that the hardships and privations consequent on
missing a port, the losses of ships from being out in their reckonings,
and the evils incident to navigation generally from the want of a
ready means of checking the enormous errors to which the dead
reckoning is liable, would, in numy cases, have been prevented by
In urging tiiis recommendation, it is, of course, taken for granted
tliat they to whose hands the chronometer is entrusted are qualified
to make a ])roper use of it. Employed merely as a check, a single
chronometer cannot fail to prove of great service ; but too firm a
reliance on such an instrument would lead to the dangerous error
â€¢ The accepted LnngituJe of Madras, India TrigonoiDetiical Survey, 1878 (sea
pnge ?94), is 80Â° 14' SÂ« ' E
K PREFACE TO THE FIRST EDHION.
iÂ»f relaxing that vigilance which tlie known uncertaintj of the dead
re-jkonuig keeps perpetuallv alive.
A list of times of high water, or, as they are now called, Esta-
blishments of Ports, is not given. The researches on tlie tides
made of late years by Mr. Lubbock and the Rev. W. Wheweil,
iiave proved that the establishment cannot be truly deduced but
from numerous observations, and consequently that a sim])le recorded
time of high water is altogether insufficient. Moreover, if the esta-
blishment were correctly known, the time of high water, as also the
height of the tide, cannot be determined without other elements,
which, except in comparatively few places, are not afforded. But
in navigation it is not the true instant of high or low water that is
required so much as the time at which the flood or ebb stream turns,
oecause tliis last affects every vessel when near the shore ; and the
proper place for information of this kind is, obviously, the Sailing
Although some results of the kind might be advantageously
placed in a general work on navigation, yet the uncertainty of
almost all that has been published, and the difficulty of collecting
better materials, will, it is hoped, excuse the omission, at least for
It may, however, be remarked, that under whatever form it maj
hereafter be found advisable to publish particulars of the tides, the
observations required are so numerous, the discussions so tedious,
and the whole subject so complicated, that no individual could
undertake successfully to treat this branch of navigation unless ia
a work devoted exclusively to its consideration.
The subject of Maritime Surveying, usually treated in works of
this kind, has been omitted. Surveying is no part of the navigation
of a ship, and a survey having any pretensions to authority can
scarcely be made by a person whose qualifications for the task are
confined to the slender information contained in a few pages. A
survey is a matter of too great consequence to the security of navi-
gation to be received from incompetent hands ; and the seaman who
desires to acquire a knowledge of surveying should study works
treating expressly of this branch of science.
The customary chapter on the Winds has likewise been omitted.
The subject, generally, does not belong to tfie navigation of a ship,*
and, even if it did, the general information contained in a few
pages, thougli interesting as a branch of natural philosophy, is
PH.KTACE TO THE FIRST EDITION. xl
neoessarilv too vague to be effective in shaping tlie course. The
same applies to Currents, and also to the Marine Baronietc r ;
which, though matters of important consideration in sea-voyages,
are not concerned in the practice of navigation, since this term, in
btrictness, comprehends only tlie consideration of the place of tlic
ship when her circumstances and destination are given.
Tlie space gained by the omission of these collateral subjects,
and other matters sometimes introduced, is appropriated to the
numerous practical details of the proper subjects of such a treatise.
The Work will be completed by another volume, wliicli will be
entitled the Theory of Navigation, and will contain the construc-
tion of the rules and tables, for the advantage of those who desiie
to confirm their practical knowledge by matliematical investigation.
It will contain, likewise, those methods in which the transit and
azimuth instruments are employed. The present volume being thus,
in tlie ordinary practice of navigation, indepentlent of the second, no
notice of anotlier volume appears in the title-page.
By the term Theory is commonly understood, in this particular
:nibject at least, the scientific principles on which the rules are
formed. Considerations of this kind are thus altogether excluded
from the present volume; but, on tiie other hand, that rationale, or
process of reasoning, which, in considering the nature of the case, is
obvious to common sense or apprehension, is, in raost cases, intro-
duced, as necessary to a clear understanding of important points.
The theory and the practice are thus kept purposely distinct.
The former is not always necessary to successful practice; and rules
constructed for ready and general application approacli to perfec-
tion in proportion as they leave less to individual judgment or skill.
It is the custom, generally, to teach the theory first ; the impression
forced upon me is, on the contrary, that the practice is itself the
best foundation for sound and rapid advancement in the theory.
For he who has acquired the practice knows the nature and extent
of the subject; and in proceeding to the theory he has a distinct
pev^cption of the object to be attained. This is not the place for a
discussion on these points; but it was incumbent on me to state, in
a few words, the grounds of the arrangement adopted.
It is manifestly the duty of a writer, who undertakes to treat a
subject in a thoroughly practical manner, not only to discuss every
point which ])resents itself, but also to pronounce a decided (>)iiiiii)n
in every case. It is proper to bring this point under tlic notiru nl
Xli PREFACE TO THE 1-IRST EDITION
the reader, who, especially if he has more experience in these inattprs
than myself, might otherwise be disposed to consider many things
In this volume as laid down too positively.
I cannot close the preface to a work which has been some years
in preparation, and in which I have endeavoured to reduce to a
practical form every useful consideration which has been suggested
by my own experience or by intercourse with eminent officers and
men of science, without soliciting the indulgence of the reader to
errors and to deficiencies. Absolute correctness, especially in tables,
is scarcely attainable ; and in a treatise which contains much that
has not appeared before, I cannot reasonably fiatter myself that,
notwithstanding every care and attention, some small inaccuracies
may not be found.
Saptemhkr 1 6'}0
In tlie Advertisement to tlie Second Edition Iliad the satisfaction
of being able to state that the Eoyal Geographical Society had
conferred the flattering distinction of their gold medal on the
first edition, and that the Lords Commissioners of the Admiralty
had honoured my work by ordering it to be supplied to Her
Majesty's Navy as ship's stores.
The present edition has been greatly augmented. Much of
the -work has been rewritten. Two approximate methods of
determining the time, though of inferior value, are introduced,
since a work aiming to be complete for practice should contain
provision for extreme cases. Nos. 789, 791.
The introductory portion, it had often been suggested, was in-
sufficient for the purposes of elementary instruction. It is easier
to allege this, than to lay down a condition which is to determine
the extent of such preliminary matter. An attempt, however,
has been made to fix a limit, on the following grounds : â€”
The most general defect, perhaps, in the education of seamen,
as regards the present subject, is an insufficient knowledge of
iiiitlimetic; by which I mean, not of the more advanced rules,
but of the elements, and especially of proportion. Now all ques-
tions to which arithmetical processes are applied involve some
proportion, which the operation is to bring out, or distinctly
assign ; and it appears, accordingly, a great omission in our
education that we are not more exercised on this point, which is
tbe sole object or end of the processes which we learn to practise
KIV ADVKRTl&EMKiNT TO THR THIRD KUITION.
Again, in geometry, it is not tlie variety of problems wliicb
benefits the practical man, but a v^ell-grouiided and familiar know-
ledge of a few comprehciibive propositions, which he applies readily,
and with confidence; and the geometrical knowledge which appears
to me to suffice to our present purpose is comprised in, â€” 1, the
property of the square of the hypothenuse; 2, the measure of an
nngle at the cir^'iuiference ; 3, the similarity of pl-ine triangles The
first is of general importance; the second includea the problem of
fixing a station by means of two angles subtended by three objects;
and the third is the basis of trigonometry.
In this edition, therefore, proportions and fractions are treated
at some length, and illustrated by numerous examples which afford
tlie student abundant exercise ; and a short course of geometry
is given, after the manner of Euclid, sufficient to establish the
above important theorems.
These limitations, the reader will bear in mind, are intended to
apply only to that particular quantity of elementary matter which
is assumed to be necessary and sufficient for the scale of attain-
ment contemplated in the present volume.
In the Table of Positions many points of information of con-
sequence to seamen are expressed by means of a new system of
Symbols. In these days little apology is required for introducing
a scheme which a few years ago would have been deemed a rash
innovation. But a growing tendency to the use of symbols mani-
fests itself on all sides. Efforts have been made to represent, as
far as possible, all matters of instruction under a form addressed
to the eye ;* and symbols effect this object in an eminent degree,
for their distinct and cons])icuous forms, contrasting with the mono-
tonous aspect of alphabetic writing, arrest and fix the attention,
while their extreme conciseness admits the insertion of matters to
which, for want of room, no allusion could otherwise be made.
The employment of symbols, therefore, on a more extensive
scale than we have yet been used to, and that at no distant period,
may be considered inevitable ; and the present system, which bos
occupied my attention for several years, is proposed as so far
deserving consideration that it is constructed with rigid adherence
to principles.f The number of sigus which I have ventured to
â€¢ The Physical Atlas Is an example.
f The necessity for a uniformity in hydrographic sym'iols has already shewn its.Mf.
Symbols similar in character denote, on the French charts, rocks 'ibuce the water, <>ui1
oil the Kussian charts rocks below the water.
ATJVF.RTISl'.MKNT TO TlIK THIRD EDITION, \\
iiifrodiipe is small, since, in matters waiting tlie sanction of experi-
(Mice. it is better to move too slow than too fast.
The introduction of symbols has necessarily modified the original
design of the work, as described in the preface, and has justified allu-
sion to many matters which otherwise would not have found a place
The chief labour of this eaition tas, indeea, ofthe two former)
has been tlie Table of Positions, which, in consequence of the nu-
nierous references made to my labours in this country and abroad,
I was desirou3 to extend. The list now contains 8,800 places; and
as the degree of accuracy is indicated wherever I have found the
means of forming a judgment, and as many physical details are
supplied, â€” such as the dimensions of islands, heights, and the depths
of shoals, â€” the table may be considered as representing the state
of maritime geography at this day. The number of voyages,
charts, and surveys, which it has been necessary to consult, â€” the
labour of digesting and comparing the mass of materials collected,
and the introduction, by a new method, of numerous details im-
portant to navigation, â€” will, it is hoped, excuse tlie long delay in
the appearance of this edition, and account for the work having
remained out of print for nearly three years.
In conclusion, I gladly express my obligation to the draftsmen
and other gentlemen of the Hydrographic Office, whose patience
during many years I have sorely taxed in the inspection and re-