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P RE FA CE
THE RE are many books on Navigation available for the use of the student, and among them some are exceedingly good. Why, then , add yet another volume to a mass of literature already sufiiciently, and more than sufficiently, large ? Well, it seemed to me that for many reasons another work designed on somewhat novel principles might be useful. Most writers have treated the subject from the point of View of addressing themselves either to the highly educated or to the totally uneducated, and there is I think, room for a treatise designed to meet the requirements of those Who lie between the two extremes men who, while ignorant of mathematics and astronomy, possess intelligence and a certain amount of rudimentary kn owl e dge .
Navigation is in many respects a peculiar subj ect. All the problems being based upon the higher mathematics and astronomy, the solutions of them can be calculated and formulated only by men thoroughly conversant with those sciences ; but Navigation has to be put in practice by men who are not, and cannot be expected to be possessed of much knowledge of those matters. More over, mariners have to work their problems in a hurry, and frequently under adverse circumstances. To sit in a
P RE FA CE
THERE are many books on Navigation available for the use of the student, and among them some are exceedingly good. Why, then, add yet another volume to a mass of literature already sufficiently, and more than sufficiently, large ? Well, it seemed to me that for many reasons another work designed on somewhat novel principles might be useful. Most writers have treated the subj ect from the point of view of addressing themselves either to the highly educated or to the totally uneducated, and there is, I think, room for a treatise designed to meet the requirements of those who lie between the two extremes men who, while ignorant of mathematics and astronomy, possess intelligence and a certain amount of rudimentary knowledge.
Navigation is in many respects a peculiar subject. All the problems being based upon the hi gher mathemati cs and astronomy, the solutions of them can be calculated and formulated only by men thoroughly conversant with those sciences ; but Navigation has to be put in practice by men who are not, and cannot be expected to be possessed of much knowledge of those matters. More over, mariners have to Work their problems in a hurry, and frequently under adverse circumstances. To sit in a
vi
PREFA C E
comfortable chair in a warm and cosy room, and leisurely
work out abstract calculations from imaginary observa
tions, i s quite a different thing from taking real observations on a wet, slippery, and tumbling deck, and working them
in a dimly- lit cabin full of confusion and noise, and with little time to spare for the operation.
T herefore, for the i conven ence ' of the practical m an ,
it is necessary that the scientific man should reduce the
formulas to the simplest possible dimensions. With
those formulas the practical man can find his way about
all right if he learns and remembers them, and how to
work them ; but, as it i s very difficult t o remember a
lot of formulas learnt by heart, it is highly desirable
that the practical man should have some idea of what he
is doing and why he does it.
Few things appear to be more difficult than for one
well up on any subj ect of a scientific character to impart
his knowledge to another who is scientifically ignorant. A
thorough past-master may succeed in explaining matters popularly in language which can be understood by the
many but the expositions of writers on highly technical
subj ects—
whether
connected
with
Science,
Art ,
Philosophy,
or anything else— are frequently rendered so obscure, by
the lavish employment of highly technical language, as to be unintelligible except to the educated few.
All
the Epitomes
N i
- or e s ,
Inm an s ,
Raper s,
and
many other books give explanations of the various pro
blems in Navigation somewhat too minute and too diffuse,
I
venture
to
think,
to
be
attractive
to
the
ordinary
reader ,
with the result that the formulas are generally learnt
by heart. A man must be gifted with a gigantic memory
if he can remember how to work everything from
Logarithms to Lunars. Moreover, in most works the
definitions, though of course absolutely scientific and
correct, are so scientific and so correct as to be somewhat
unintelligible to the unscientific person, whose ideas on geometry are very hazy. Books such as Captain Martins and Mr. Leckys are most valuable, but they preconceive
a considerable amount of knowledge on the part of the
student . Books such as Rossers Self- Instructor are
equally valuable in their way, but they seem to have been
written on the supposition that everything must be learnt
by heart and nothing understood by brain. So it occurred
to me that an attempt to give— conversationally— as if Pupil and T eacher were talkingfl suffi cient expl anation
of navigational problems to throw some light upon the
meaning of the formulas used, and some additional in
formation for the benefit of those desirous of obtaining it,
might be useful ;
and ,
having myself
started
to
study
Navigation somewhat ignorant of the sciences upon which
it is founded, I determined to try and impart to others in a
similar plight what knowledge I have gathered to gether. My definitions and explanations may be sometimes
scientifically inaccurat e. Let that pass . My purpose
is gained if they convey an accurate idea.
That portion of the work which treats of the Days
ork -V
,
the
Sailings,
and
so
on,
contains
a
very
short
treatise on plane right- angled triangles, by the solution of which all such problems are worked . The student need not
read it if he d oes not want to, and if it bothers him
he had much better not do so. The method of working
every problem is given, and for all practical purposes
it. i s sufficient if he learns and remembers that. The
learning is really easy enough ; it is the remembering
v iii
PRE FA CE
that i s diffi cult.
But ,
if
the
i m agi n ar y
person I am
en
deav ouring to instruct will read the chap ter on Plan e
Trigonometry, I think it will help him greatly in
learning how to work the problem ; or if he learns the
working of the problem first, and then wants to know the reason why, a perusal of it may giv e him sufficient insight to enable him easily to remember how every
problem is to be solved. If my reader wishes to obtain an Extra Masters certificate of competency he must
learn enough of Plane Trigonometry t o enable him t o
construct plane triangles and solve th em, for that will be
required
of
him .
Of cour se if he is well up in T ri gon ometry,
or has time to master that angul ar science, so much the
better but if such is not the case, I think he will find in
the foll owing pages all the information necessary for hi s
p urp o s e .
In the same way Nautical Astronomy is preceded by
a sketch of the movements of the heavenly bodies and ,
contains a short ch apter on Spherical Trigonom etry it i s
not the least necessary for the student to read it but if he
does so before or after tackling the various problems, it will , I think, help him to understand their n ature and the methods by whi ch they are solved. Be it remembered
that even a v erv little an d very hazy knowledge of thi s
kin d i s sufficient to ensur e that you do not forget how a problem i s to be work ed. Moreover, shoul d a blue
ticket be the obj ect of ambition, the aspir ant to such
honour s will have to solve some spherical triangles, and
to draw the fi gures appropriate to some of the problems.
In this instance also it is better that the subject should
be thoroughly studied and understood ; but if the pro spective Extra Master h as not t he time nor inclin ation
to do so, I think that the little I say will answer all the requirements of the case.
Most problems can be solved in various ways. I have given the formula which is, in my opinion, the simplest but I claim no infallibility for my opinion.
Nories Tables are used throughout, except in some portions of the Double Altitude and Lunar problems, because I happened to be taught with those Tables, and have always used them ; every reference to a Table therefore refers to Norie, but as many men prefer Inman or Raper a comparative statement will be foun d on page xxiii, giving the equivalent in Inman and Raper to every Table in Norie.
I have treated what may be called the mechanical part of the business— for instance, the use of the lead an d the log— very shortly. Such matters can be learnt only by practice, an d if information is required concerning them, are they not fully and clearly explained in the Epitomes and in manuals and books of instruction innumerable
I have not touched upon the Rule of the Road at sea, though it is scarcely necessary to mention that it is of the first importance that a seaman shoul d be intimately acquainted with it. Such kn owledge comes only from habit and experience. I would only say that before going up for examination , a candidate shoul d be thoroughly drilled on this subj ect by a competent instructor. A man whose knowledge and judgment may be perfectly reliable at sea, may be much puzzled when he finds himself seated opposite an examiner playing about with small toy ships on a table. Captain Blackburne has published a little book on the subject, which will be found of great service to the student or candidate.
PREFA CE
I have endeavoured to take the simpler problems first,
and lead graduall y up to the more difficul t ones ; but this
is not easy of accomplishment, as the problems overlap
each other so frequently. And I have treated of the whole subj ect, from a Mate s to an Extra Master s work, whi ch has n ot, I think , been attempted in any single work.
I have also tried to explain , as far as p ossible, how
every portion of a problem is worked as the case crop s up
in the problem ; for nothing is more bothersome than having
to constantly turn back and refer to some previous explana
tion.
The explanation of every diagram is, wherever possible, placed on the same page with the diagram or on the oppo
site page, for I have found it very troublesome to have to
tur n
over pages
to fin d
what
angle
an d so-
-so,
or
line
this
or that is an d I opine that others also must have foun d
it equally troublesome . T his method of treating the sub
j ect involves muc h repetition, b ut repetition is n ot vi cious on the contrary, when something h as to be remembered, it is good, and I have taken some pains not to avoid
rep etition .
I do not flatter myself that the difficul ty of self
i nstruction is entirely got over in thi s work, but I hope it may go some way towards at tain in g that desir able en d.
As far as practical work at sea is concerned, very little, i f any, supplementary instruction woul d be necessary in
order to enable anyone to find his way about but for the
B oard of T rade Examination the personal instruction of a
good master is certainly desir able, for in most cases the problems, as given in the examination , are far more puzzling than as they present themselves at sea. F or one
thing, at sea you know whereabouts you are, and any
PREFA CE
xi
large mistake manifests itself in the working of a problem
but in the examination room no such check upon inaccuracy
exists.
As an amateur I have written mainly for amateurs ;
but if this book proves of any assistance to those whose
business is upon the sea, I shall indeed be pleased.
For convenience sake the book is divided into two
volumes, a big volume being cumbrous to handle. T he first volume contains Logarithms, the Sailings, a Day s Work, the Use of the Compass, some chart work, and the simpler nautical astronomical problems . The
second volume treats of other nautical astronomical pro
bl ems ,
and magnetism ;
it
gives
further information
on
the subj ect of charts, an d shows how the working
formulas are deduced ; and i t con tains numerous exerci ses,
together with the data from the Nautical Almanac of
1898 necessary to work them.
HINTS TO CANDID A TES
No particular and regul ar sequence i s, I believe, foll owed in the examination papers in the order in which
problems are given ; but I fancy they generally come in something like the following somewhat appallin g pro
cession :
M M For
a tes a nd
a s ters
1 .
Mul t i p li c a ti on
by
comm on Logs .
“7 Divi sion by common Lo gs.
Day s 3 .
Y k
Vor .
4 .
L a t it ude
by Meridian
Al titude
of the
Sun.
5 .
Parallel
S ail i n g .
6 .
Mercators
Sailin g.
1 . Time of High Water.
8 .
A mplit u d e .
9 .
Time Azimuth .
10 .
Longitude
by
S un
Chronometer
an d Al titude
Azi m uth .
11 .
Time
of
Stars
Meri di an
passage.
12 .
To find
names
of
S tars
from
Nautical
Alman a c
within a given distance of the Meridian at a certain time,
and also the distance they pass North or South of the
Z enith .
13 .
C o mput e
the
Obs.
M er .
Al t .
of
a
Star
for
a
gi v en
pl a c e .
14 .
Latitude by Meridian Al titude
of a Star.
15 .
Star Time Azimuth.
16 .
Latitude by Reduction
to
the Meri dian .
17 .
S umner .
HINTS TO CA NDIDATES
18 .
Latitude by Pole
Star.
19 .
Latitude
by
Moons
Meri di an
Altitude .
20 .
Correction
for
soundin g s .
M fi For E x tra. a s ter s Cer ti ca te
21 .
Lon gitude
and
Error
of
Chronometer
by
Lun ar
Observati on .
22 .
Latitude
by
D o u bl e
Altitude .
23 .
Position of
Ship
by Double Chronometer
Problem .
24 .
Great
Cir cle Problem .
25 .
Error of Chronometer
by Altitude
of
Sun or that
of any other heavenly body.
26 .
Solution of a right-angled plane trian gle.
27 .
Solution of an oblique-angled plane triangle.
28 .
Solution of a right-angled spherical triangle .
The manner in which problems are presented is con
stantly varied ; different expressi ons an d different words
are employed to denote the same facts. Y ou may be told
that the Sun is bearing North, or that the observer is
South
of
the
Sun ,
or
that
the
Sun
is
South
of
the
Z enith.
You may be given the date in Astronomical Time, or in
Civil Time, in Apparent Time or in Mean Time at Ship
or at Greenwich. Y ou may be given the absolute date
such and such a time, Mean or Apparent at Ship , or you
may be told that a Chronometer showed so many hours,
minutes, seconds, which Chronometer had been foun d to
be so much fast or slow on Apparent Time at Ship at
some earlier period, since when the Ship had run so many miles on such and such a course, and you would have to
find the Ship date by allowing for the Difference of Longi
tude due to the run. In fact, the Examiners ring the
changes as much as possible, and very properly so, for it
is but right that candidates should not only work the
problems but also show an intelligent knowledge of what
they are doing. Nevertheless, these changes are apt to
HINTS TO CA NDID A TES
be puzzling. T hey would not puzzle anyone in actual
practice at sea ; but the nervous conditi on of most men
i s apt to fall below the norm al, and the brain to become
unn aturally c onfused when they are shut up i n an ex
amination room for long hours, and so much depends
upon their efforts. T herefore, read the statement of each
problem very carefully, and if you notice anything um
usual, anything you do not quite understand in the word
in g, j ust think it ov er quietly until y ou quite understan d
what you have got to do ; tran slate it, as it were, in your
head in to the language y ou have been accustomed to.
Don t hur ry over your work. Remember that it takes a
long time t o discover an error in a problem returned, an d
that
havin g
found
it ,
you
may have to
do
most
of
the
work ov er again .
ABB REVIATIO NS
p oi t o p The
n s of the c m ass
o p m the c
as s card .
i i t nd ca ed t i by he r initial l tte ers. Vida
Par
in
Alt .
Colat i o D ff. L ng.
Mer. D ifi . Lat.
V OL . I.
tit La ud e. o pl t C m emen of the
t it La ude or Go. tit La ud e. o it L ng ude. i t D s ance. iD fference. i ti D fference of La t ud e. iD fference of Lon
gitud e .
p t D e ar ure.
i i M M er d ian or
er
dional .
i io l i Mer d na D ffer
tit ence of La ude.
i t i o R gh Ascens n.
li tio D ec na n.
i i t m m Se
D -
a e er.
Horizontal l Para
M T S .
.
.
i Sid .
T
me
Sid . Time
Z i en th ol P e. it i t Zen h D s ance. ol i t P ar D s ance. Al t i t ud e.
ll Para ax in Alt
t ud e.
The Sun.
o The S uns L wer
iL mb .
pp The Suns U er
iL mb. i t i o R gh Ascens n of
M th e ean S un.
ltit True A ude. pp t A aren Altitude. v ltit Obser ed A ude
pp t i A aren T me at
Ship. i ip Mean T me at Sh . Equation of Time
pp t A aren Time at
i Greenw ch . i Mean T me at
i Greenw ch . i l i S derea T me (the
t i same h ng as
Na uo
ti l l ) m ca A anac . i l i S derea T me of
v tio Obser a n t i m (sa e h ng as
RA . of
i i ) Mer d an .
ol l P ar Ang e.
o l H ur Ang e.
o o M T he
n .
oo M ns Lower
iL mb .
xv i
A BBREVIA TIONS
oo M ns Upper
t l t A S ar or P a ne .
Far Limb.
Y
ear
Limb .
o o Sun and M n s
t o M S ar and
o n s
i far
L
mb .
t M S ar and
oo n s
nea r
Limb .
lP us.
ference .
is to or to .
il l y aux iar ang e.
SY MBOLS
l q E ua . x Multiplicat ion.
iv io D
is
n .
w D if
o so is . x An unkn wn qua ntitv . 0 An unknown
V O ABBRE IATI NS OF TRI-GO NO METRICAL RATIO S
S in
iS ne ; 0 0 5
S ec
t Secan ; Co sec
o c sine ; Tan
t Tangen ; Cot
t Cc - a ngent ;
t V Cosecan ;
ers
Versine ; Ha v
v Ha ersine
C ONTENT S
T HT Z FI R S T V' O L U M E
PREFA C E
D D HINTS T O CAN I ATES
V LIST OF ABBRE IATIO NS US ED
N ComBARAT IV E S T AT EMENT OF
O RIE,
TABLES
IN MAN,
AND
PAGES
RAPER
i xxii —xxv
PART I
CHAPTE R I
ARITHMETIC
U PRO PORTION, O R R LE or TH REE
F DD D ECIMAL
RACTIONS ; A
IT I ON,
S
U
BTRAC
TIO
N ,
V CATION, AND D I ISIO N
D RE U CTION OF D EOIMALS
U M LTIPLI
1 4—1 8
CHAPT ER I I
LOGARITHMS
H M C ARACTERISTIC O R IND Ex ;
ANTIS SA
U LOGS . OF NAT RAL NU MBERS A U NATU R L N MBERS or LOGS .
MULTI PLICATION AND D IVISION BY Loo s .
CONTENTS OF
U F U LOGS . or N MBERS CONS IST I NG o r MO RE THAN
OR
U FIG RES
U D m LOGS . or N MBERS COMPO S E or INTEGERS AND B Ec ALs,
Y OR or D ECD IALS ONL
H MU T E IR
L TI PLICATION AND
D IVISION
PROPO RTIONAL LO GS.
H T EORY or Loo s .
PA G E*
25 —29
CHAPT ER III
INSTRUMENTS USED IN CHART AND COMPASS
WORK
TRE HARTNER S COMP AS S
Azmum MIRROR
D D TEE LEA AND LEA LIN E
m TEE Loc s p AND LOG LINE ;
HA RPO O N AN D
LOG
PARALLEL BULERS ,
U TEE PELO R S
D WI DEBS ,
PRO T RACTO R S
STATIO N POIN TER
TAFFRAIL
CHAPTER IV
THE PRACTICAL USE OF THE COMPASS
VARIATION AND D EvI ATI ON
U U To FIND CO MPAS S CO RSES FRO M T RUE CoU Rs ES AND TR E
U CO URSES FROM C OMP AS S CO RSE S
U T o T R N POIN TS INT O D EGREES , M D OVABLE COMPAS S CAR .
&c . ,
A ND
I'I CL' VERS A
To AS CERTAI N TEE D EvIATIO N
NAPIER S D IAGRAM
Emm zs
53- 56
CHAPTER V
THE SAILINGS
TEE TRAvERSE TABLE S AND HO W T Q U SE THE )!
T AB LE
X XV .
AND
HOW TO
U SE
IT
xx
CO NTENTS OF
CHAPT ER VIII
CHARTS
EAC H;
H ow T o FIN D T HE S HI P S PO SIT ION ON THE C HA RT
D H D T o KIN THE S I P S PLA C E BY BEARI NGS O F THE LAN
U T O ALLow ' FOR -A C RRE NT
D D S UMMA RY OF POIN T S T O BE C ONS I ERE
I N C HAR TI NG
PROBL EMS
Y M THEO R OF
C '
ERC AT O R S
H ART
M U D To C ONS TR U C T A
E
R
C
A
T
O
R
S
CHA RT
Fo R A
GIvEN LATI T
E
2 1 6—22 1
DEFINITIONS
TRI GO NO AIET RICAL
3226 —22 7
CHAP TE R N .
INSTRUMENTS USED IN NAUTICAL ASTRONOMY
HORIZ ON GLAS S
H ow T O D ETERMINE
H Y T E O R OP SEETAN I
D IN EX
E RRO R
ARTIFIC IA L H O RIE O N
245
THE FIRST VOLUME
CHAPTER X I
MOVEMENTS OF THE HEAVENLY BODIES
Y T H E T E RMS E AS T ERL A ND V RO T AT IO N A ND RE OLU T IO N OF T H E E ART H
IN C LINAT IO N O F E ART H S AX IS AND IT S E FFE C T S
D D LAT ITU E , LO NGIT U E , RIGHT AS C E NS ION , D EC LINAT IO N
APPARENT TI ME
D SI EREAL A ND S OLAR T IME
T HE PLANET s
THE MO ON
Y D U GREENWIC H T I ME ALWA 'S T o BE U SE WIT H THE NA T IC AL
A M '
'
L
A
N
A
C
E NGLIS H S GLO B E S T AR-FIN D E R
257— 2 60
CHAPTER X II
LATITUDE B'Y -MERIDIAN ALTITUDE OF . THE SUN
THE SIMPLE PROBLEM WIT HOU T ANY C ORREC T I O N S
C O RREC TIO N FOR D IP
S EMI- D IAMET ER
RE FRAC T ION
PA RALLAx
CHANGE OF D ECLI NAT I ON S INC E GREE NWIC H
NOON '
F D Y L IN ING T HE
A _ 'I-IT U D E ,_ A PPL ING ALL
D D D AS R EQUIRE BY BOAR OF TRA E
T HE
C O RREC T IO NS
V D U D PRINCIPLE INV OL E IN D ETERMINAT IO N OF LATIT E
2 73—2 74
C HAPTER XIII LONGITUDE BY SUN AND CHRONOMETER
M A-PPA RE NT AND
EAN TIME
D M U D LONGITU E
EAS RE B Y ARC O N EQU AT O R O R ANG LE AT
PO LE
D NAMING LO NGIT U E
C ORREC TIO N OF ELEME NT S
286— 287 2 8 8 — 2 89
xx ii
CONTENTS OF THE FIRST V OLUME
U FO RM LAS
US E
O F TABLES
XXXI .
A ND
XX X- II .
W U EAS T AND
ES T H O R AN GLE S
CORRE CTION OF CHRO NOME T ER
U D W F M LL S TAT EMENT OF
ET HO OF
O RK ING
PROBLEM AND
E XAMPLES
U D H O R AN GLE S BEYON
LIMI T S
O F TAB LES
XXXI .
AN D
XXXII .
J OHN SO N S TAB LE S
F D H EXAMPLE OF IN IN G S I P'S PO S IT IO N
M D U D PRINCIPLE OP
ET H O O F R A T IN G LO NGIT E
TEE C O NVE RS E PRO BLEM : T O FIND T HE
U METE R FRO M A KNOWN LO NGIT D E
H ERRO R OF C RO NO
322 —324
CHAPTER X IV
OBSERVATIONS USED FOR MAKING COMPASS
CORRECTION
D U ALTIT U E AZ IM' TH OF THE S UN H TIM E AZ IMU T OF THE S U N BY TABLES U TI ME AZ IM T HS OF A S T AR O R PLANE T U FO RM LA PO E TIM E A Z IMU TII
3 3 7—341
CHAPTER XV
REDUCTION TO THE MERIDIAN
D U Y D M F ETHO AND
O RM LAS T o BE EMP LO E
U F M RIGO RO S
O RMULA FO B Ex - E RI D LAN S
D M D U LA TITU E BY
E RI IAN AL TIT D E OF A S TAR
US MM ARY OF T HE O RD LNA RY PRO BLEMS AND
NEC ES S ARY
352 4354
COMPARAT IV E STATEMENT OF NORIE , INMAN AND RAP ER TABLE S ,
oN m a ED . 1696
INMAN ED. '92 I RAPER ED. '9 8
j Sub ec t
l Tab e
Pag e
1" 1“ 29
o v t i to iT m To c n er Arc n
e
oo vtot i i to I To c n er T me n Are
Fac t r
c orrect V ar. in 1" O
R. A.
I.
II .
III .
III.
t tio of D 2°
m D and
D ec .,
a nd
E.
T .
I Aug en a n
S.
.
d tio fi of t 2i
Re uc
D n of
B . P. for gure
Ear h
v l oi 1- 16
T ra erse Ta b e for P nts and Quarter
oi t P n s
v l d 18-107 idio l t 108- 1 14
tio of o t 1 15
Tra erse Ta b e for eg rees
Mer
na Pa r s
00 1 rec n
Mean Ref. for Bar me er
IV .
V.
V V I.
I
I
I .
IX .
X.
K LA
X II.
116 1 16
116 1 16 117
1 18—121
1 18 121 ,
119 1 20 ,
XIII. XIV .
1
2
2 ,
123
124
a nd T lIérmonIet er
f tio Mean Re rac n
of Ho i o Dip Sea
rzn
ll Para ax in Alt.
of o o Dip S h re HOI Iz n
tio of to Correc n
Ob s. Al t. 6)
fi nd T r. Alt.
f o to (9 Dec . r m 1888 1890
o t oo To c rrec 6) D ec. at N n at Sea.
o t iod p To c rrec 6) Dec. for changes in er
s
qof ftoio of i 8 E
ua
m ur y ears
n T e 18
8- 1 895
of i i i d t l Mean P ac es
p l Pr nc a F xe
S a rs
X
IV
.
XV .
125 126
o tio of t to Mean R. A.
C rrec n
Ob s. Alt. S ar
find T r
o tio of XVI. fi di i of Hi t XVI
.
126 1 26
I Alt. C rrec
For n
W n
Mer. Pas sag e .
ng T me
gh a er
V of ol X II.
o tio of d d d f o l KV
II .
127 127
m T o fi nd the Lat . by Alt .
Pe
C rr ec n
Lat . e uce r
T ab e
V X
II .
V { { fz of lt of — III f 128 130 .
COT-e iou Sa
App. A .'s
S un and
d o i to i XIX.
1 31
For re uci ng L ng. n T me and the
vre erse
D of Ho i o Diff t XX. H i d t i XXI. D d i i XXII.
131
I ista nc e
S ea
r z n for
eren
e gh s
1 32-1 213 For re ucing (9 D ec . to a ny Greenw c h
a te
1 34- 1 35 For re uc ng G R. A . to any Greenw ch
D t to v iX
III .
1 36
o otf oi t (XIV .
i t t XXV .
t l i (X
V
I .
V o ti X II.
1 38-1 51 1 5 2-265 266- 274 2 15-280
V fi dii t to L
III .
i i (XIX .
ll XXX .
of ili l X
X
X
.
l ZKK I.
I p XX
II .
o i D t to (XIII.
2 81 -284 285- 29 1 29 2- 308 29 3- 309
310- 318
31 9 320 ,
S a e
.
Log . S ine , Ta ns, and S ec s
e ery
Quar er P n
L gs.
Numb ers
Log . S nes, Tangen s, S eca n s , Ge Na ura S nes
(A ) and (B ) for c rrec ng Lo ng. and
n ng A z.
(C) Az mu h for Lots.
68°
Log . R s ng
I Para a x in Alt .
orrec t ion
A ux
a ry
A ng
e
A .
) m Log . HoraI y Ang e (In an, Log. Hav
(Ra er, Log. Sin Sq .)
L
gs . for I ed ucing
G) D ec.,
R A .
.,
are.
any Greenw ch a e
6 3—6 5
17 18
35
395 -422
64
706—723
32 44
1 1 9-304 66 67 68
ASS- 5 1 7 “
,
,
65
726-821 725
39
676 - 684
69
82 8-89 2
xx iv
COMPA RATIV E STA TEME NT OF
l Tab e
NORIE ED . 1 896
j Sub ect
34 11- 3 62
op tio l Pr or na Lo gs. o t D t T o c rrec Ap p. Lunar
is . fo r Pa r .
a nd
Re
f .
V i t l ers nes ( Na ura )
d tio Re tar a n
l tio Acce era n
o it i L ga r hm c D ifi erence
o of o v d C rr.
Lo g.
D iff .
w h en
G)
is
b ser e
.
of t Co rr.
Lo g. D ifi .
wh en
a
s ar
i .
oh
t d Ampli u es
Time Amplitnd s
of p i ip p i To fi nd
A .
T .
di Meri an
r nc
l ti l o jet P i TO fi nd when ce es a
la st a rs as s ng
m b
c
'
15
on
r
e
V ti l er ca .
of o j t To find A lt.
c ela t ial b ec “ hen o n
i V ti l Pr me er ca
o fi d o L gs. for
n ing c rr
p
R .
A .
and
D ec .
D t for a ny Greenwi c h a e
ll l t Para
ax
"
I II
Alt .
for P a ne
s
t l t To correc
Lo o
Difi .
when
a
P ane
is
o d b se rv e t l T o correc
A anx il ia n a ng e
“ hen a
l t o d P a ne 15 bse rve
d D w To re
uc e
E .
T .
a n) Gree nn-Ic li
p t q tio of Lo gs. for co m u ing the E ua
ate
n
q l lt E ua
A
s .
o ti fo m For c nver ng reig n eas ures
fi d t o o d M For n ing exac
T G .
.
.c
rres p
n
D t ing to T rue Luna r
H D To re duce
P
.
.
t m Gree
'
nc
h
Da
e
is .
a nd
S .
o . t
a ny
q t i l i fi of Corr. of
E ua o r a Par . to ' gure
t Ea r h
i p l o H o H l d Pr nci a P rt s,
a rb urs
ead a n s ,
n p i m W &c .
T i e of 11 .
. at
P .
a nd C
a
dS
r ng
INMAN ED . 92
l Ta b e
Pa ge
4 6 7—486
RAP EB ED . 9 8
l Ta b e
Pa ge i
— 4 6 2 -4 6 5
462 16 5
73
59 ,
59A
26
90 0- 90 8
69 66
61—, 6696
7 3
2 7 27A
6 64 665 ,
0675- 6711
666-6 70
a
900 90 "
10
540—1 634
l T ab e
I NMA N
ED .
1 89 2
B A PER ED . '98
l T ab e
oi t of p P n s
m Co
as s. & c .
t d d t Leng h o f a egree o f Lo ng . in
ilf .
La
s.
o v i D p t t For c n ert ng
e
ar
ure i n o
Di if .
Lo ng .
o i i to D p t For c
nv ert
ng
D ifi .
Lo ng
n
i tit d Fur
co rrect
ng
Mid .
La
ue
e a r ure
of j t D is ta nce
Ob ec s see n a t sea
o itio i P s
n
by
N
o
Bea r
ng s
a nd
Di .
t a nc e
r un
o o SI whols a nd A b b rm ia t ions use d in N Int ica l As t r n my
tio idi Re duc M n ( 0 t he er a n
of d Le ngt h
a eg ree in La t . a nd Lo ng .
l T rue Bea rinzf o f Po e t ta r
-4 7
I”
688- 69 1
ALo g . Ha v en -ines
tio of d Red uc n
La t it u e
D t of Red nf: tio u ( I .7
on a cco un
.
Ref.
of a ili d C orrect ion
ux a ry a ng le A fo r Ba r. a n
m Ther .
m i — m orra po nd ing T ne r o
et e r Se nes
m A n n -O no ic a l l« a t . l
m A i I I e T int io ne in A l ira l t . C1uI rt s
D t l 5 e1 . is a nce Ta b es
635 —6 43
l Tab e
NORIE INMA N A ND RA PER TA BLES
,
,
XXV
RAIJCR ED . ”499
j t Sub ec
v p l l S h eri ca Tra erse Tab e
of f t t di di t di t No.
ee sub e n ng l at f eren
s ances
i do i fo d o o t i d lP aces at wh ch i l I Time S gna s t p E ac s
l p l m cks or s s ay be un and c a
t q itl m Se imens rual Ine ua y
o t i of d pp ll A m r xi a e R se and Fa
Ti es
o d i d i v i tio l L gs. for re uc ng a y ar a
ns
b a ne
fi di q tio of d Diff For n ng the E ua n
Seco n
erences
o o di H of oo p C rres n ng
P
.
and
S -D
.
.
Mn
o ti of t C rrec on for red ncmg Tr. Al t . to A pp. A lt.
Su nor S ar
oi ttio od i o di to of of oio t of ti p M m m C rrec n for re uc ng Tr. Alt.
App. Alt.
A z u h and c rres n ng change
Alt .
in
1
n nu e
o t oo o d o of Di t p m For c
M u ing the
ns S ec n C rr.
s ance
o i to o Di t p pl Err r of Sh s ace due
1 err r in Lunar s anc e
me
oid p l l S her a Tab es
o o ti o of o t p L gs. for c m u ng C rr.
Lil -t . b v acc un
NOTES ON THE TABLES '
It will be seen that Norie, Inman, and Raper all contain the Tables essential to the work of the Navigator.
But some Tables are more convenient than others. For
example, Norics Log. Horary An gle T able correspond s to Inman s Log. Haversine Table and to Rapers Log .
Sine Square Table, but the two latter are more convenient than the Log. Horary Angle of Norie for two reason s . The first i s that while Inman and Raper each give a. complete Table, Norie, for some reason known best to
himself, limits the Horary Angle to 8 hours, an d con
sequently it
might
v ery
well
happen
that
a
bright
st ra ,.
such as V ega, or Capella, might be rendered useless for
finding Time if the observer was ignorant of other
methods than Norics for calculating the Hour Angle. An d the second reason is that Norie, unfortunately, does not give the arc corresponding to Time in his Horary
T able. The three T ables, though bearing different names , deal with the same thing, for the Log. Horary Angle is
really 3. Log. Haversine ; and Log. Haversine of any
angle is the Log. Sine Square of half the same angle.
i ov itt i p ov itio m 1 S nce the ab e was wr en a. new and
uch m r ed ed
n of
l o t i i o pl t v i l i m Nories Tab es, c n a n ng a c
e e Ha ers ne Tab e, has been s sued .
2
A PCITED HZ TIQ
1 8000 .
It w ould b e mere waste of time
make two sums of it, thus
1 231
5 678 901 1 121 3
i7i as
as
o ne
sum ,
thus
2ud .
Under
ord inarv circu mstanc e s
of mul tipli ca tion
by L ogs . one woul d put t he numbers, or angles, or time on
the l eft, t he L ogs . equal to them on the ri ght, and t he num b er, angle, or time equal t o the resultant Log. t o the
i h g t“
r
.
of
it,
t hu s
1 2-3 Log . = 2 -0899 05
456 Log.
-6 5 896 5
But t he exi gencies of space, and general convenience frequent ly rend er i t. necess arv to put the answer als o on t he left and the above sum would be written thus :
1 23 LQg . 456 Log.
Preport ion
'
or
Rul e
of
Th ree
A s t im e an d arc are menti one d in t he l lOVVl D f
examp le s, it is w ell t o s t at e t hat t irne is c oun t e d in h our s
m h an d d re mi u s i ( m n ut es , d C e con s
.
s ),
a rc in
g e
es ,
n te ,
s econds
There are c ixty seconds of t ime or of arc
ARITHMETIC
3
in a minute, sixty minutes of time in an hour, sixty minutes of are in a degree.
A simple proportion takes the following form As 2 is
to
4
so
is
3
to
6 ,
or
s ub s t i t u ti n g
the
abbreviations,
as
2
4: 3
6 .
All simple proportions consist of four parts or terms .
In
this
case
these
terms
are
2 ,
4 ,
3
and
6 .
Of these 2 and
6 are called the extremes, and 4 and 3 are called the
means.
The fact upon which the solution of problems in pro
portion rests is, that the product of the means is equal to the product of the extremes .
For instance, in the above proportion, 4 and 3 are the
means, 2 and 6 the extremes. And 4 multiplied by 3
equals
2
m ul ti p li e d
by
6 .
4x
and 2 x
This
form of simple proportion you will not have much
occasion to use ; but you will have to use simple pro
portion to find an unknown fourth term from three
known terms. If you have any three terms of a prop or
tion you can find the fourth term by the following rules (1 ) If two means and one extrem ' e are known, the
product of the mean s divided by the known extreme , gives the other extreme.
(2) If two extremes an d one mean are known, the product of the two extremes divided by the known
m mean gives the other mean. This is easy enough. You must re ember, however, that the first and second terms in a proportion must be of
the same n ature, that is, they must be multiples of the same quantity or measure, and that the fourth term will be of the same nature as the third . Thus , suppose you were given the following proportion, :1: representing the extreme you want to find
m As
1
h .
10
. : 1 8 111 1 : 1 2 °
x .
4
A R ITHMETIC
B efore multiplying the two means together you must
make the first and second terms of the same nature, that
i s, mul tiples of the same quantity, which in this case can
be
easily
done
by
t ur ni n g
1
h .
10
111 .
into
mi nut e s
of
time.
Ai m ,
to
avoid
the
trouble
of
c o mp o u n d
multiplication ,
it is best to reduce 1 2 48 into minutes of arc.
to work out the problem
As l h l om zl s 2 2 1 2 3 43 2 9:
As 70m
70 70
(1 9 7 2 9 or 3 : 1 7
Here we mul tiply the t wo means together, an d the
product i s 1 3824 thi s we divide by the kn own extreme,
m 1 0
.
which gives us as the other extreme 1 9 7' an d i i
over.
Turn
the
34
into
s econds ,
and
divide
by
70 ,
and
we
ha v e
1 9 7 29 " divide the 1 9 7' bv 60 to tur n them in t o d egrees,
and we g et 3° 7 29 Remember always that what you
get in the fourth term is of the same natur e as the “third
et rm ,
whether
it
be
degree s
mil es, feet , tons, or anyt hing
You will fin d lat er on the utilitv of thi s rule in determining, among other t hing s , the amount a h eavenl y bo dy will ri se or fall in a cert ain time if vou know how
A RITHMETIC
5
much it has risen or fallen in a given time. For example,
suppose at 9 h .
s 1 8 m 28
.
.
the Altitude of
some
heavenly
bod y was 32 °
an d
that
at
9
h.
m 35 .
30 s .
the
Altitude of the same body was 35° 1 4' 1 8 and that you
wanted to
know what
its Altitude
was
at
9h .
m 2 2 .
s 1 4 .
How would you proceed In the following way.
First find out ho w much the body rose in the first
i n terval .
At 9“ At 9 35
o Theref re in 0 1 7
28” 30
2
it or se
Next you must find how much it would rise in second interval. What i s the second interval ?
9 h 2 2 m l 4a 9 18 28
o i t v l 0 3 46 is the sec nd n er a .
Now
you
h av e
three
known
terms,
17m .
2
s.
(the
first
i n t e rv a l),
s 3 m 46
.
.
(t h e
second
i n t erv a l ),
and
2° 55' 58"
(the increase of Alti tude in the first interval), and require to find the fourth unknown term.
W As
2 I]
5
3m 468
2° 55' 58" x 60
60
1 0558 secs . o f arc. 226
63348 21 1 16 21 1 16
2044
secs. of arc. or
3 42 1 3066
3 550 3066
4848 4088
l (near y)
6
A RITHMETIC
Therefore 38 55” i s the amount the body will rise in
m 3 .
6 4
s.,
and
this
amount
added
to
32°
1 8
the
known Altitude
at
h m 9 1 8 .
2 8 .
s.,
gives
the Altitude
at
the time
required,
namely
at
9
h.
22
m .
14
s.
T ime
A l t it ud e
ltit o y At 9h 1 8m 2 89 the A ude of the b d was 32 ° 1 8 2 0
In
it o 3 46
i se
38 55
o ltit Theref re at 9 2 2 1 4 the A ud e was
32 57 1 5
This
is
a
long
sum ,
but by using proportional Logs ,
as will hereafter be explained, the work i s very much
shortened.
Decimal Fractions
A vulgar fraction consi sts of two parts, the numerator and the denominator the numerator is above the line and
the denominator below it. The denominator expresses
the value of each equal part into which any unit is divided, and the numerator expresses the number of such parts .
Thus % is a vulgar fraction ; the numerator is 3 and the
denominator
4 .
The denominator shows that each part
is one-fourth of the whole, and the numerator shows that
there are three such part s . T ake another fraction, g for
example. Here the unit i s divided into 5 equal parts
the denominator shows this ; and there are 3 of these
parts, as indicated by the numerator ; the value of the
fraction is therefore three-fifths .
The denominator of a vulgar fraction may be any
number you like the denominator of a decimal fraction
must be ten or some multiple of ten, and therein lies the difference between a vulgar and a decimal fraction. In
decimal fractions the denominator is expressed by a dot,
thus : 1 i s one-tenth. The figures after the dot are
called decimal places. The number of decimal places
A RITHMETIC
7
shows the value of the denominator ; thus 1 i s $3 , '01 is
1 0 0,
i s w l ru :
12
is
12
T3 3 ,
123 is
and so on.
Y ou can always, of course, express a decimal fraction
as a vulgar fraction ex actly, but you cannot always express a vulgar fraction as a decimal fraction exactly.
T he decimal equivalent of a vulgar fraction is often self
evident thus 4 is evidently the same thing as and 1 11 i s written decimall'y as 5' ; and even in those cases in
which the conversion is not self-evident, the process of turning vulgar fractions into decimals is very simple. All
you have to do is to divide the numerator by the denom
inator— this will give you the decimal exactly if the vulgar
fraction can be turned exactly into a decimal fraction,
and if it cannot the process will give you the decimal very
n early .
Thus
3
13
is
a
vulgar fraction,
and can be expressed
exactly as a decimal fraction thus 10 3-0 3
Some vulgar fractions , as for instance 4, cannot be expressed exactly as a decimal fraction .
3
fi m ' 333 etc . a d tn nttu .
9
10 9
10
Such a decimal fraction i s called a recurring decimal ,
and
is
written
thus,
3 ,
with
a
dot
over the
3 .
In turning vulgar fractions into decimals, you may
arrive at a decimal containing three or more, and perhaps
a lot more figures . Console yourself by the reflection that .
for navigational purposes, one decimal place, or at any
rate two decimal places, are good enough. Thus 1 234
would be called ' 12
or
probably
1' .
If the figure to the
right of the second or
of
the first
decimal
place
is
5 ,
or
bigger than
5 ,
increase the
second
or
first
figure
by
one ,
8
A RITH BLETIO
thus
1
26
should
be
called
-1 3 ,
and
' 36
should
be
call ed
4 , because in the first case ' 1 3 i s n earer to the truth than
“ 1 2 an d in the secon d place 4 i s nearer to the truth
than 3
The immen se advantage of the decimal system is, that compoun d addition , subtraction, mul tiplication , and divi sion are don e away with. Its weakn ess is, that some
fractions cann ot be expressed absolutely by its means,
but they can be expressed quite nearly enough for all
n avigational work . Decimals are wt mderfully useful in
n avigation, as you will appreciate ful ly later on in fact, problems coul d not be worked without them .
Ad dition of Decimals
The quantiti es to be added t ogether must be written
down so that the decimal points are all in the same
perpendicul ar lin e, un der one another. T hen proceed to add as in ordinary ari thm etic, and place the decimal
point in the sum in a line with and un der the decimal
points in the quantities added.
For
example, add
together
1 7 89 ,
78 0 1 ,
0 26
1 0000 ,
11
002 ,
and
1 00 01 .
There you are.
The 1 01 90 ,
being
a whole
num ber,
is
to the left of the decimal point , and the fr action 837 is
to the right of i t.
lo
ARIT HMETIC
Here are a few examples, to which I would ask your
closest attention
(1 )
Mul tiplv
1 8 -5
by
1 9 -2 .
There is one decimal place in each of the two factors,
185
and
19
2 ,
that
i s,
two
decimal
places
in
all,
so
that
you p oint 0 11 two decimal places from the right of the
p ro du c t ,
and
the
dot
comes
between
the
2
and
the
5 .
Of
course, zeros on the right of a decimal without any di git s t o the ri ght of them are of no value, but they must n ever be st ruck off a product till the decimal point has
been placed.
(2)
Mul tiply
1
042
by
198 .
Here
there
are three
d e cima l
plac es
in
1 0 42 ,
and
none
in
1 93 .
T herefore we po int off three decimal p la ces
from the ri ght ih the product.
(3)
Mult iplv
79 -89
bv
0042 .
Here t here are two decima l plac es 111 1 9 -8 9 . and four
in
0 042 ,
therefore six
d e c i ma l
p la ce s
are
point e d
off
in
t he pro duc t .
A RITHMETIC
11
(4)
Multiply
0 045
by
10 .
0045 10
0 156
Here we have altogether four decimal places to point
off in
the
product
450 ,
and
so
a
zero must
be
placed
to
the left of 450 to make up the number. Z eros required
to make up the number of decimal places in a product
must be placed to the lef t of the left-hand digit .
Although the last zero is valueless , it must be counted
when pointing off the product .
(5)
Mul t i p l y
0 001
by
0 002 .
'000 1 ' 0002
0 0000002
This is rather an extreme case. We have eight decimal places in the factors, and therefore we must add seven zeros to the left of the product 2 before we can place the decimal pomt.
(6) Multiply 79 8 9 by 1 2 1 2 .
79 i ! i 1 2 1 -2
1 5 978 798 9 1 5 978 79 8 9
9682 668
Three decimal places in the factors, therefore three in
the prod uct .
In such a case as this, after the decimal place has been
put in according to the rule, you can check the result by
taking two simple n umbers nearly equal to those in the
question, an d multiplying them in your head. Thus in this
case, instead of
79 8 9
take
80 ,
and instead of
121 2
take
1
20 .
12
A RITHMEIIC
The product
of 80 and
120
is
9 600 .
Thi s is suffi ciently
near to 9682 668 to show that the decimal point has been
put in correctly. If you had made a mistake and put
down
9 68 -2668 ,
you
woul d
have
found
it
out.
So much for multiplication of decimals. Y ou will
have to do plenty of it in the course of your navigational
studies, so I will pass on to
Division of Decimals
D ivision of decimal fractions is managed exactly in
the same way as division in ordinary arithmetic. As in
mul tiplication, the only di fficulty consi sts in plac ing the
decimal point correctly in the quotient. Y ou must place
in the quotient that num ber of decimal places which,
added
to
the
n um b e r
of
decimal
places
in
the
d i i z' s o r ,
equals the number of decimal plac es in the d ir id end . It
is really the same rul e as in multiplication, because the
product of the divisor and quotient is, of course, the
divi dend. Here are a few examples
(1 )
Divide
461 43
16
by
31 2
2 .
312 2 31 22
1 4923 1 2488
2435 1 2 1 854
1 478
Here we have three decimal places in the divi dend, and
onl y one in the divisor. It i s to nec es s aiw add two
decim al places t o those in the divi sor to m ake them equal
to t he number of decimal places in the dividend ; you
con s equen t ly
have
t wo
in t he
quotient ,
an d
here it
is ,
1 4 78 .
A RITI—IMETIC
13
As a check on the result, notice that, roughly speaking,
you
are
di vi d i ng
4600
by
300 ,
so that
14
or
15
is
evidently
pretty near the answer.
(2)
Divide
702
by
0 09 .
-009 -702 78
Now you have three decimal places in the dividend,
and three in the divisor, therefore you want none in the
quotient,
and
the
an swer
is
78 .
(3) Divide 63675 by 849
84-9 6 3675 (75
5 943
42 45 4245
Here are five decimal places in the dividend, and only
one in the divisor, therefore there must be four in the
quotient . But we have only two figures, and to make up
the four necessary places two zero s must be put to the
left of them, and then the decimal point. So the answer
is
0 075 .
As in the product of a multiplication sum, so
in the quotient of a division m su , zeros to make up the
number of decimal places required must be placed to the
left of the left- hand digit.
Check.— If in doubt about the position of the decimal
point,
multiply
84
9
by
0 075 ,
and
the
result
6
3675
shows
the decimal point is correctly placed.
(4)
Divide
5
by
2 50 .
250 5-00 2
In
this
case, in order
to
make
five
divisible
by 250 ,
you must add two zeros after the decimal point which ,
makes no difference to the value of the dividend. Then
you have two decimal places in the dividend, and none in
the divisor
you
must
therefore
have
two
in
the
quotient ,
and
here
you
are,
02 .
14
ARITHMETIC
In all cases where the divisor will not go into the
divi den d, add zeros to the dividend, placing them to the
right of the decimal point if there is no fraction, or to the
right of the fraction if there is one. These zeros make
no difference to the value of the divi dend, but they count as decimal places when placing the decimal point in the
quo ti en t .
(5) Divide
17
by
50000 .
5 000 0
1 -70000 34 1 50000
200000
2 00000
Here there are five decimal places in the top line of
the dividend, and we borrowed another in the third line, makin g six 111 all . But there are none in the divisor, so
we must have six decimal places in the quotient, and four
ze ro s
must
be
placed
to
the
left
of
the
34 ,
and
the
answer
is
0 00034 .
(6) Div ide
1
by
0 00001 .
No decimal point in the dividend and six in the
divisor. Add 6 zeros to the right of the 1 in the dividend ,
an d divide out .
0 00001 1 0 00000 1 000000
Red uction of Decimals
You must understan d the reduction of decimal frac
tions The subj ect naturally divides itself into tw o .
branches, the one dealing with reducing ordinary quantities into decimals, and the other with reducing decimals into
ordinary quantities. Let us first deal with turning ordinary quantities into decimals.
S upp o s e
you
were
asked to turn
1 0l .
1 2s .
6d .
into
pounds and decimals of a pound. The first step would
be to find what decimal of a shilling Sixpence is, and the
A RITHMETIO
15
second to find what decimal of a pound the shillings and
decimal of a shilling i s. In expressing a penny as the
decimal of a shilling, consider the penny as a vulgar
fraction of a shilling
one
penny
is
1 T?
of
a
shilling
then
turn the vulgar fraction into a decimal by dividing the
numerator by the denominator as has been already ex
pl ain e d .
First then turn the 6 pence into decimals of a shilling
by
di v i d i n g
them
by
12 ,
thus
:
5
Sixp ence is 5' of a shilling, and we now
pounds and 1 2 5 shillings. Next turn the
into
decimals
of
a
p oun d
by
dividing
by
20 .
20 1 2-50 -625 1 20
50 40
1 00 1 00
Here we have three decimal places in the dividend, having borrowed a zero in addition to the two decimal
places in the first line and, as there are no decimal places
in
the
divisor,
we must
have
three in
the
quo ti en t ,
which
is
therefore
6
25 .
1 2 5 of a shilling is therefore 6 25 of a
pound, and tacking this on to the pounds, we find that
1 02 1 2s
.
.
Now suppose you want to reverse the process and ,
turni ng decimals into ordinary quantities, require to find
the
value
of
1 0'625l .
You must first turn the decimals of
a
p oun d
into
shillings
by
mul tip l yin g
by
20 ,
thus
0 25 20
1 2 500
Therefore 6 25 of a pound x
shillings .
Then
16
ARII'H METIC
turn the decimals of a shilling into pence by multiplying
by
12 ,
thus
T herefore, 5' of a shilling x
p ence. And you fin d
that
1 2s.
6d .
It i s not improbable that you will spend more time at
sea in dealing with are and time than with money, unless you happen to hit upon a treasure island, so I append a few examples here.
(1 ) Turn 37° 48' 00” into degrees and decimals of a
degr ee.
In one degree there are
Therefore, divide 48 by
60 to brin g it into decimals of a degree. 48 -1
of a
degree. The answer, t herefore, is 378 ° T o reverse the above an d express 378 ° in degrees and
minutes . To turn 8' of a degree into minutes you must
mul tiply it
by
60 .
'
therefore, 37 8°
3 7°
(2) Fin d what decimal fraction of a day 1 4 hour s
1 8 minutes i s .
T here are 60 minutes in an hour, therefore, to
t ur n
18
min ut e s
in t o
d ec im al s
of
an
h our ,
divi d e
by
60 .
18
Therefore 1 8 minute s = 3' of an hour.
\' ow t o fin d what decimal fraction of a day 1 4 3 hours
i s . There are 24 hours in a day, therefore divide 1 4 3 by
24 1 4-3 5 9 58 1 20
18
ARITHMETIC
Thus, suppose you want to know what decimal fraction of an hour ten minutes i s. You proceed thus
6 1 0000
-1 666 & c. & c.
i s the correct an swer. X—Vell, 1' 7 i s near enough for you .
Remember always to add 1 to the last digit if the next
one is
5 or more
than
5 .
Thus
1 66
must
be
called
'1 7 ,
b e c aus e
'1 7
is
nearer
the
truth
than
1
6 .
It i s generally easy to place the decimal point, even in division , by using a little common sense. If the number
to the left of the decimal point in the divisor is less than
the number to the left of the decimal point in the dividend , there must be at least one whole num ber in the quotient .
If ,
on
the
c ont rary ,
the whole num ber
in
the dividend
is
less than that in the divi sor, the decimal point must c ome
fir st in the quotient.
I V hen the decimal place has been put in according to
the rule, look at the result and see that it i s roughl y ab out the ri ght amount .
CHAPT ER II
LOGARITHMS
L O GARIT HMS
are
the
invention
of
a
most
talented
man ,
John Napi er, of Merchistoun. L ogarithms, or, as they
are called for convenience sake, Logs , enable us to sub stitute addition for multipli cation , and subtraction for division — an immense boon to the mariner. If the
wretched sailor had to multiply and divide the long rows
of figures and the numerous angles which abound in great
profusion in hi s calculations, he would not be done work ing one set of sights before it was time to begin working
another set, and every sea- going ship would have to be fitted with a private lunatic asylum. But with the help
of Logs , Navigation becomes easy, for addition and sub traction are simple operations, which do not consume much time, or cause any great amount of chafe of the brain.
Every natural number, that is to say every number
in the natural ordinary sense of the word, has a Log. 5 and
p er contra every Log. has a natural number. If you have to multiply two numbers or two dozen numbers together, or i f you have to divide two numbers or two dozen numbers, all you have to do is to find the appropriate Logs , and add or subtract them the result will be the Log. of a natural
number, which i s the result of the multiplication or divi sion of the numbers . W hat you have got to learn
therefore
is :
1 st ,
how
to
find
the Log.
of
any natural
9
-l
20
LOGA RITHMS
number ;
2 md ,
how to
fin d the natur al
number of any
L og.
3rd ,
how
to
add
Logs. together ;
4th ,
how
to
subtract Logs. from each other.
A Log. generally consists of two parts, a whole num
ber containing one or more digits— this is called the Cha
racteristic or In dex — and a number of digits separated
from the characteristic by a decimal point thi s decimal part of the Log. is called the Mantissa. Though
Characteri stic is the proper term to employ, Index is more generally used, and for the future I shall speak of
the
n d e I
x
.
For in stance, take any Log , say 2 9 44483
2 is the In dex , an d 944483 i s the Mantissa.
Natural numbers and L ogs. are tabulated in Table
X X IV .
headed
Logarithms
of
Numbers.
In the left
hand column, headed
you will find natural numbers
from
1 00
on
page
1 37 ,
to
999
on
page
1 51 .
Zeros in
n atural numbers make no difference to the Mantissa of a
Log. For instance, the Mantissa or decimal part of the
Log. of
1 ,
of
10 ,
of
1 00 ,
of
1 000 ,
and so on,
is
the
same ;
the Log.
of
15 ,
of
1 50 ,
of
1 500 ,
&c. is
the
same ;
the Log.
o f 1 72 ,
of
1 720 ,
of
1 7200 ,
&c.
is
the
same.
T herefore
you need take no notice of that portion of Logarithms
of
Numbers
from
1
to
1 00
contained
on
page
1 36 .
It is
useless and confusing, so leave it alone. To fi nd f the L og . o a na tura l n mnbmz— Remember
that the T able gives you the Ma ntissa only, and that having first got that you must afterwards find the Index.
S up p o s e
you
require
the
L og.
of
a
single
n umb er ,
say
of
2 .
Look for 200 in the left- hand column headed
and
to the right of it, in column headed
you will find
301 030
that
is
the
Mantissa
of
2 .
Suppose you require
the
Log.
of
a
number consisting
of
two
fi gur e s ,
say
23 .
Look
for
2 30
in
the
No . column ,
and
in
column
0
you
will fin d 361 728
that
is
the Mantissa of
23 .
S up p os e
LOGA RITHMS
21
you want the Log. of a number containing three figures,
s ay 2 34.
Look
for 2 34 in
the
N
o.
column,
and
in
the
0
column you
will
find
36921 6
that is the Mantissa
of
234 .
Suppose you want the Log. of a number contain
ing
four
figures,
say
2 341 .
Look
for
2 34
in
the
N
o.
column, and in a line with it, in the column headed
you will fin d 369401 ;
that
is
the Mantissa
of
2341 .
If
you wanted the Log. of 2342 you would find the Manti ssa
in
the
2
c o l umn ,
by following
along
from
234
in
the
N
o.
column .
If
you
wanted
the
Log. of
2 343 ,
the
Mantissa will be in the 3 column. If you wanted the
L og.
of
2344 ,
the
Mantissa
will
be
in
the
4
column, and
so
on
to
2349 .
Now to find the Index.
The Index is always one less than the number of
figures in the natural number. If the natural number
consists of one figure the Index will be zero (0) if the
number has two figures the Index will be 1 if the num
ber has
three
figures
the
In d ex
will
be
2 ,
and
so
on.
Consequently, in the case of the natural number 2 which
I have used above, as 2 consi sts of one figure the Index
is
0 .
The
Mantissa
of
2
is
301 030 ,
therefore
the
L og.
of
2
is
0 301 030 .
It i s useless expressing the zero, an d you
would
wri t e
the
L og .
of
2
as
3 01 030 .
23
contains
two
fi gu re s ,
the
In dex
is
therefore
1 .
The
Man ti s s a
of
23
is
361 72 8 ,
therefore
the Log.
of
23 is
1 361 728 .
The
Mantissa
of 2 34 i s
36921 6 ,
and the
L og.
of
234
is
2
36921 6 ,
because
234
c ont ain s
thr e e
figures,
and
the
Index
consequently
is
2 .
The Mantissa of 2341 is
369 401 ,
and the
Log .
of
2 341
is
3 369 401 ,
because
2341
~ contains four figures. To fi nd f the na tura l number s o L ogs . -L ook out the
Mantissa of the Log. in the table in the columns
&c. &c., and, wherever i t may be, you will find
its
natural
number in the
same line with it
in
the
N
o.
LOGA RITIl MS
column . The value of the Ind er will show you how many figures there are in the natural number. You
know that the Index of a Log. i s always one less than the natural number of the L g o ., an d p er contra. the natural number must always be one more than the Index
of its Log.
C o n s e qu e n t l y,
if
the Index
is
0 ,
the
natural
number will consist of one figure .
If
the
Index
is
1 ,
the
natural number will contain two figure s. If the Index i s
2 ,
the
n atur al
number will contain
three figures,
and
so
on and so on. If the natural number belonging to the
Mantissa of a L og. does not contain one figure more than
the Index of the Log. you must add zeros till it does. If
the Mantissa of a Log. gives you more figures in the
natural number than there ought to be according to the
Index of the L og , then the n atural number contains a
decimal fraction, and you must put a dot after the proper number of figures as determined by the Index. T ake any
Log , say 6 9 89 70 ; yo u want to know its n atural number.
Look for 698970 in the T able in one of the c olumns
headed from 0 to
Y ou wi ll fin d 6989 70 in column
O
on
p.
1 43 ,
and
alongside
to
the
l eft
in
the
No
c o l um n
you
will
see
5 00 .
Your Log.
was
6 9 89 70 .
It had zero
in the In dex, therefore its natural number must consi st
of
one
fi gur e
:
therefore
the
natural
number
is
50 0 ,
or
5 .
Suppose
the
Log.
to have
been
1 6 9 89 70 .
1 in the
Index shows there must be two figures in the natur al
n um b e r ,
therefore
the
natur al
number is
500 ,
or
50 .
If
the L og. had been 2 6 9 89 70 the natural num ber woul d be
500 .
If
the
L og .
had
been
36 9 89 70 ,
3
in
the
In d e x
requi res four figures in t he num ber, but there are only
thr ee
in
500 .
Y ou
must
therefore
add
e r o Z “
3.
,
and make
it
5000 ,
and
that
is
the natural
number of
36 9 89 70 .
An d
so on.
Take
another
Log ,
say
2 6 62663 .
Look for the
LOGA RITHMS
23
Mantissa 662663 in the Table— you will find it in column
p 9
,
.
1 42
—and alongside to
the left, in the
ol mn N o. c
u
,
you will
see
459 .
The Mantissa being in the 9 column
of
course
9
must
be
added to
the
numb er
in
the
N
o.
column, so 4599 i s the natural number. The Index of
the
L og.
is
2 ,
and
there
must
be
three figures in
the
natural number ; therefore cut off three figures by a
decimal
point,
and
you
have
the
natural
n um b e r
459 9 .
If
the
Index
had
been
3 ,
the
number
w o ul d
have been
4599 .
If
the Index
had
been
1 ,
the
n atura l
number would
have
been
459 9 .
If
the
Index
had
been
0 ,
the
natural
u number wo
ld have been
4 599 .
Now ,
having
seen
how
to find
the
Log. of
a number
and the number of a Log , let us consider multiplication
and divi sion . Multip lica tion a nd D ivision by L ogs — T o multiply
two
n
um
b
e
r
s ,
find
the
Log.
of
each
num b er ,
add them
together and find the natural number of the resul tant
Log. To divide one number by another. T ake the Log.
of the Divisor from the Log. of the Dividend, and find
the natural numb er of the resulting Log. For instance,
4x
by ordinary multiplication 4
by ordinary
division ; now Work the same sum by L ogs . The L og.
of
4
is
6 02060 .
The
Log.
of 2
is
3 01 030 .
Add them
together.
The
n a t ur a l
numb e r
of
90309 0
is
8 00 .
Zero in
Index gives one figure in the number, therefore
numb e r
is
80 0 ,
or
8 .
Subtract
6 01 030
from
6 02060 .
' 602 0 60
301 030
301 030
24
LOGA RITHMS
The natural number
of
301 030
is
200 ,
and
the In dex
being
z er o ,
it
is
20 0 ,
or
2 .
Suppose you wish to multiply
8197 by
5 32 9 ,
and
also
to
divide
8197 by
5 329 .
The
Mantissa
of
8197is
91 3655 ,
and
the Index
is
3 ,
because
there
are
four figures
in
the
number,
therefore
the
Log.
is
3
91 3655 .
The Mantissa of
5329
is
72 6 646 .
The
L og.
will
be
3
72 6 6 46 ,
because
there
are four figures in the number.
3 91 3655 3 726646
76 40301
You will not find the exact number 640301 in the Tables ,
but you will find something near enough to it, namely,
640283 in the
8
column
on
p.
1 42 ,
and
that
will
give
you
436 in the
N o.
c o lumn
the n atural number, therefore, is
4368 .
7 in the Index requires eight figures in the natural
number, but you have only four, and you must therefore
add four zeros
and the natural number
is
436 8 0000 ,
Therefore 81 9 7 x 5329 43680000 nearly. Now for the
division .
39 1 3655 3 726646
0 1 87009
Y ou will not
find
1 87009
in
the
Tables,
but
you
will .
find
something
near
enough,
namely,
1 869 56
in
the
8
c olumn
on
page
1 37 ,
with
the number
1 53
in
the
N
o.
c o lumn .
The
natural
number, therefore,
is
1 538 .
Z ero
in the Index gives one figure in the number, therefore the
n atur a l
n u mb e r
is
15
38 .
Therefore 8197
5329
1 5 38 ,
or 14 very nearly.
Whenever you can check the answers easily as far as
number of figures or position of the decimal point goes, do
so . For example, as in the last case you were multiplying
8000 by 5000 roughly speaking, the answer would be
40 000000 .
Thi s agrees with 43680000 sufficiently to
26
LOGA RITHMS
to the Log. of the first four numbers already found, and
the result i s the Log. required . Remember tha t a zero
fi counts a s a
g ur e.
F or instance, suppose you want the
Log. of
1 23456 .
Tick
off
the
first
four
figures
thus ,
and find the Log , or, to be accurate, the Mantissa
of the Log. of
1 2 34 .
It
is
09 1 31 5 .
In the same line in
the
D i
ff
.
column
you
will
find
352 .
Multiply 352 by
56 (the remaining figures in your number).
35 2 56
2112 1 760
1 9 71 2
From the product 1 971 2 cut off from the right as many
fi gur e s
as
the
mul tip lier
contained,
n am e l y
two .
That
leaves 1 9 7 to be added to the Log. of the first four figures.
09 1 3 1 5 1 97
0915 1 2
091 51 2 is the Log. required. The reason for this process is very simple. The
numbers in the column Diff. are the differences between the Logs. of two con secutive numbers . The di fference
between the two numbers is 1 00 the difference between
the number whose Log. you have taken out and the
number whose Log.
you require is
56 .
The differe nc e in
the
D iif .
column
between the Mantissa
you
have taken
out
and
the
n ex t
larger
is
352 .
It i s a simple sum in
proportion, as 1 00
56
352
m .
Now for the Index. You must count all the figures
i n your number. T here are six figures, therefore the
In d ex
is
5 .
Therefore
the
Log.
of
1 2 3456
is
50 91512 .
In all questions of this kind it is advisable after
the answer has been obtained to check it by seeing that
the Log. found lies between the Log. of the right two numbers . The Log. of 1 23456 should lie between Log.
LOGA RITHMS
27
1 234 and Log. 1235 and since 091 51 2 is between 091 31 5 and 091 667 it is evident that no mistake has been
made. fi m T o nd the na tur a l nu ber cor resp on d ing to a Log . to
f fi more tha n our gures — Now supp o se you are o ccupi ed
in the reverse process, and having the Log. 50 9 1 51 2 you
want to fin d its natural number. . Look for the Log. in
the T ables. Y ou won t find 091 51 2 anywhere. In such
a c ase you must take out the natur al number to four
figures, for the nearest less L og , and write it down . Then find the difference between this nearest less L og.
and your Log. divide thi s difference by the figure in the column, adding as many zeros to the difference as
may be necessary“, and add the quotient to the first four fi gures of the natural number already taken out and
written down.
You want
the
n atural
numb er
of
50
91 51 2 .
The
nearest
less
Mantissa in
the
Table
is
0 9 1 31 5 ,
of
which the natural number is 1234 ; write that down .
Next find the difference between 09 1 31 5 (the nearest Log .)
an d 09 1 51 2 (your L og ).
09 1 5 12
091 315
1 97
T he
difference
is
197 .
In
a
line
with
091 31 5 ,
and
in
the
Di f
f
.
c o lum n ,
you
Will
find
352 .
Y ou have got to
divide
197
by
352 ,
adding
zeros
to
197 .
l 352 1 9 70 y 56 near
1 760
2 100 21 12
56
is
to
be
tacked
0 11
to
the
four
figures
already taken
o ut ,
namely
1 234 ,
and
the
natural
number
r e quire d
is
therefore
1 23456 .
Y ou will note that the division of 1 97 by 352
did not
come out
exactly,
but
the pro duct,
56 ,
was
much
more nearly correct than 55 ; and as you knew by the
Index that you only wanted two more additional figures ,
28
LOGA RI'IHMS
it was useless proceeding further. Had you proceeded further, the sum would have worke d out thus
352 1 970 (559
1 760
2 100
1 760
3400 3 1 68
2 32
This would have given you 559 t o tack on to 1 234
already found, an d your natural number would be
1 234559 .
B ut
as
the
In d ex
of
the
L g o .
was
5 ,
there could
only be six whole figures in the natural number, which
would
therefore
be
1 2345 5 -9 .
All you wanted was a
number consi sting of six figures, and 1 23456 i s nearer than 1 23455 with a useless { 17.
Here are some examples
Find
the
L og.
of
79841 2 .
ti Man ssa. of 7984 t Par s for 1 2
8972 9 7 Difi . 5 5
7
12
Log. of 79841 2 5 89 7304
660
Find
the
L og.
of
548208 .
ti Man ssa of 5482 t Par s for 08
i 738939
D ff .
79
6
08
ly 7 near .
Log. of 548208 5 -738945
Find
the
L g o .
of
400006 .
ti Man s s a. of 4000 t Par s for 06
602 060 6
Log. of 400006
56 02066
D ifi .
632
1 08 06
648
Find
the
number
whose
Log.
is
4
9 02030 .
43902 030
t Nat. No. 7980 902003 Neares Log.
D ifi 54 2 70 5 . 2 70
The
number
is
79 805 .
Find
the
number
whose
L og.
is
6
01 2839 .
6 01 2839
t Nat. No. 1 030 01 2 837 Neares Log. v D ifi . 420 2 000 005 ery nearly
2 1 00
The num ber i s 1 030005 very nearly.
LOGA RITHMS
29
Find
the
number
whose
L og.
is
56
39 48 6 .
Log. 4360
5 6 39486 6 39 48 6
T he
number
is
436000 .
Hitherto we have considered and used numbers com
posed entirely of integers or whole numbers, but you may
require the Log. of a number consisting partly of integers
and partly
of
decimals,
such
as
23 ,
or
composed
entirely
of decimals,
such
as
23 .
f f L ogs . o numbers comp osed o intege-rs a nd d ecima ls .
Use the whole of the number, d ecima ls an d all, to find
the Mantissa of the Log. T hus to fin d the Log. of 1 2
Look out the Mantissa
of
12 ,
which,
as
you
know,
is
the
same
as that
of
1 20 ;
it
is
079 1 81 .
Now for the
Index. You have only one integer, and therefore the
Index is zero and the Log. of 1 2
is
0 79 1 81 .
In the case
of numbers composed of integers and decimals, the Index i s always either 0 or a p ositive or p lus quantity.
In the c ase of numbers consisting entirely of decimals ,
the Index will be a nega tive or minus quantity. As one
integer gives zero in the Index, it is obvious that
no integer will give an Index one less than zero or ,
minus
1 .
The Index of a decimal, say 2 or ' 23 or 234
and
so on,
is
—1 ,
and
the
Index of
02 ,
or
0 23 ,
or
0 234 ,
and so on, is
2 ,
and
the
Index of
0 02 ,
or 0 02 3 ,
or
i s 0 0234
3 & & & — ,
c.
c.
c .
B ut ,
as
in
adding
and
sub
tracting, it would be awfully confusing to mix up minus
and plus quantities, the arithmetical complement
of
the minus Indices i s always used. 1 0 — 1 9 1 0
10— 3
7 ,
and
so
on
therefore 9 is the arithmetical
complement (ar. co ). of 1 8 i s the ar. cc . of 2 7 i s the
ar. co.
of
3 ,
and so on ;
and
9 ,
8 ,
7 ,
& & c .
c.
in
the
Index
are always used instead of
—1 ,
—2 ,
—3 ,
& & c .
c.
f L og. o a f m d eci a l ra ction.— Suppose you want the
30
LOGARITHMS
Log. of a decimal fraction. V ery well. Look for the
figures in the decimal fraction in the Table in the same
way as i f they were integers, and take out the Mantissa .
Remember that zeros have no value in finding the Man
tissa, unless they occur between digits. The Mantissa of
2 ,
or
20 ,
or
2 00 i s
the
s am e ,
namely
301 030 .
The
Mantissa
of
23 ,
of
0 23 ,
or
0 023 &c . is
the
same,
namely
361 728 .
But introduce a zero or zeros among the digits
and the Mantissas are by no means the same ; the Man
tissa of
2 03 i s
not
361 728
but
30 749 6 ,
and the Mantissa
of
2
003
is
301 681 .
New for the ' Index. If the decimal p oint is followed
by
a
digit,
the Index will
be
minus
1 ,
which you
will
call
9 .
If the decimal point is followed by one zero, the
Index
will
be
minus
2 ,
which
you
will
call
8 .
If the
decimal point is followed by two zeros, the Index will be
minus
3 ,
which
you will
call
7 ,
and
so
on .
Thus the Log.
of
2
3
is
9
361 728
;
the
Log.
of
0 23 is
83
61 728 ,
and
so
on .
What you do is, in fact, to borrow 1 0 for the use of the
Index when it i s minus, an d c all the balance p lus. T hi s
is the reason why, when you come later on to deal with
cosines and such things, you will have to drop tens in the
Index. You will be giving back tens, which you have borrowed in order to turn minu s Indices into plus Indices
for the sake of convenience but you need not bother your
head about this now.
Now suppose you want to reverse the operation, and
find
the
n atural
number
of
a
Log
,
say 9
361 728 .
361 728
gi v e s
you
23 ,
the
Index i s
9 .
Therefore if the 9 is really
a
plus
9 ,
the natural
number
must
have
ten
figur es,
and
woul d be 2300000000 but if the Index 9 represent s minus
1 ,
the
n atural
number
must
be
a
decimal,
23 .
If your Log.
is
83
61 728 ,
the
natural
number
is
either
2 30000000
or
0 23 ,
an d
so
on
and
so
on .
Well, you may say,
how
LOGA RITHMS
31
am I to know which i t is The nature of your work will
tell you. The difi erence between 2 3 (twenty-three hun
dredths) and
(two thousand three hundred
millions) i s so great that you cannot very well make a
mi st ak e . Here is how the Logs . of a natural number decreasing
in value from four integers or whole numbers to decimals
would look carried right through the scale.
T ake
any
number,
say
3456 .
The Mantissa or decimal
part of the Log. will of course always remain the same
the Index only will change.
3456 345 -6
Log.
345 6
3 456
3 456
0 3456
0 03456
3 5 38574 2 5385 74 1 5 38574 0 5 38574 1 or 9 5 38574 2 or 8 5385 74 3 or 75 38 574
and so on and so on.
Take any Log. and reverse the process. Take
Mantissa
60659 6 .
i v 36 0659 6 g es nat. number 4042
2 6 065 96
4042
1 0 06596
0 6 06596
40 42 40 42
1 or 9 606596
404 2
2 or 8 6 06596
0 4042
3 or 70 0659 6
0 04042
the
and so on and so on.
To multip ly a nd d ivid e mix ed numbers — To multiply
and divi de mixed numbers— that is, numbers consisting of
integers and decimals— ad d and subtract the Logs as has .
been
explained before ;
the operation i s quite
simple ,
and
the only possible difficulty you can experience i s in respect
of the Indic es.
In addition of the Logs , as the Indices are either zero or plus quantities, the Index of the sum is either zero or p lus . But in subtraction the result may be a minus
quantity. Therefore in subtraction of the Logs , if the
32
LOGA RITHMS
L g o . of the divi sor exceeds the Log. of the dividend, you
will have to borrow 10 for the use of the Index of the Log.
of the dividend. If you paid the ten back, the Index
would be minus, but you keep the ten in order to make
the In dex plus, as already explained.
Here are some examples
I .
Multiply
68 2
by
17 8 ,
by
Logs.
6 82 Log . 08 33784
1 78
1 2 50420
2 0 842 04
The
nearest
less
Mantissa
to
0
84204
is
0 83861 ,
which
gives
the
natural
number
1 21 3 ,
and
the
d iffer en c e
between
them
is
343 .
T hi s 343 divided by the D ifi . in the Tables,
35 7 ,
gives
9
to
be
tacked
on
to
1 21 3 ,
and
the
Index
being
2 ,
there
must
be
three
integers
in
the
answer.
2 0 842 04
t Nat. No. 1 2 1 3 0 838 61 Neares Log.
i to t to D
ff .
357
3430
9
be acked on
1213
32 1 3
and 121 3 9 is the answer.
Check — D ecimal point i s right , because 6 x 1 7
1 02 ,
and
is
pretty
near
121 .
It is unnecessary to go through all the steps in every
example for the future, as you must h ave got it well into your head how to add and subtract Logs. If not, turn
back an d study ' that questi on a little more. In the next
example, therefore, I merely give the figures.
II .
Multiply
1 82
7
by
6
49 5 .
1 82 7 Log. 6 495
2 2 61 738
30 7431 7
Nat. No. 1 1 8 6
t 0 74085 Neares Log.
i to t t o D ff 366 2 320 6 .
be acked on
1 1 86
2 196
The ans wer i s 1 1 86 6 .
Check —
An s w e r
should
be
somewhere
near
200
x
6 ,
or 1200 thus decimal is evidently in the right place.
LOGA RITHMS
borrowed and not returned. So much for quantities composed of integers and decimals.
f To multip ly a nd d ivid e numbers consis ting entir ely o decimals — Un der these circumstances the Indices are
always minus. You have, therefore, to borrow ten for each Log. Pay back both the tens if you can, in which case the In dex of the result i s a plus quantity. But if
you can only pay back one ten, the Index, though really a minus quantity, is converted into a plus quantity by retaining the ten.
I .
Multiply
2 34 by
0 234 .
Log. of 2 34 i s 9 36921 6 (the Index i s really
1 ,
because there i s no integer in the number). The Log. of
0 234 is 83 6821 6 (the In dex i s really
2 ,
because,
if
such
an expression is permissible, there is one less than no
integer in the number).
9 3 692 1 6 83 692 1 6
1 7 738432
You have borrowed twenty, namely ten on each Log.
retain ten to preserve a plus Index, and pay back ten, and
you
get
the
L o g.
77 38432 .
738432 gives you the natural
number 5476 nearly and near enough, which with 7 or
3 ,
in the Index, gives you “ 005476 as the product of 2 34 x
0 234 .
Check — 2 x 0 2
0 04 .
II .
Multiply 7
by
0 25 .
-7 Log. 9 845098
8 25
9 9 16454
Nat. No. °5 775 97 61 5 52
T he
answer
is
5
775 .
Here also you have borrowed two tens and only
returned one, therefore the Index of the Log. of the product represents a minus quantity.
Check — 7 8 x
“ 56 .
LO GA RITI—IMS
35
111 .
Multiply
0 49
by
0 063 .
0 49 Log. 80 9 01 9 6
0 063
Nat. No. 0 003087 6 4895 37
T he
answer
is
0 003087 .
For the same reason as in the two preceding ex
amples the Index of the Log. of the product represents a minus quantity.
Check — 0 5 x 0 06 0 0030 .
IV .
S up p o se
you want
to
divide 0 234
by
3 45 .
The
Log.
of
0 234
is
8
369 21 6 ,
and
the Log. of
' 345
is
9
53781 9 .
83 692 1 6 9 5 3781 9
8 831 397
831 39 7
gives
the
natural
number
6 78 3 .
8 in the Index
makes
the
number
0 6783 ,
which is
the quotient
of
0 234
“ 345 .
In this case you have borrowed ten for each L ogu
they neutralise each other ; and you have borrowed an
additional ten in order to be able to subtract, and you
retain this ten to provide a pl us In dex.
But if you do not require to borrow ten to preserve a
plus Index , it will be a positive one. T hus
v .
Divide
2 24 by
0 35 .
2 2 4 Log. 9 3502 48
0 35
85 44068
Nat. No. 6 4 0 8061 80
Ten has not been borrowed, and the Index zero, as
above.
The
answer
is
6
4 .
VI .
Divide
1'
by
0
001 .
1
Log. 9 0 00000
0 001
60 00000
Nat. No. 1 000 30 00000
and 1 000 is the answer.
To
sum
p u .
In division by Logs , (1 ) when the Index
of the Log. of the dividend is greater than the Index of
D2
36
LOGA RITH MS
the Log. of the divisor, the Index of the Log. of the quotient is a p lus quantity. (2) When the In dex of the Log. of the dividend is less than the Index of the Log. of the divisor, the Index of the quotient i s a minus quantity, and has to be turned into a plus quantity by borrowing a ten .
Proportional Log s. and how to Use them
T able
X XX IV .
gives
Pr op orti on a l
Logs.
for Time or
A
rc
from
h m 0 0
.
.
or 0° 0 to 3 h .
or
The hours
and minutes, or degrees and minutes, are at the top, and the seconds are given at the sides. L ook out the
time or arc, and write down the appropriate Log.
Find the arithmetical complement of the Log. of the
first term.
The
arithmetical
c o mp l em en t ,
or
ar.
co .,
is
found
by
taking
the
L og .
from
1 00 000 .
Then add
together the ar. co. Log. of the first term and the Logs.
of the second and third terms the result, rej ecting ten s
in the Index , is the L og. of the answer at.
Fer
exa mp l e ,
take
the
sum
we
have worked
on p.
5 ,
namely
As 1 7m 24 3m 46.
2 ° 55 ' 58
1 7m
(23328 1
2 Log. 1 '0240
L g = ar. co .
o.
8 -9 760
op 46“ Pr
L .
og .
op 2 ° 55 58 ” Pr . Log.
38' 55
009 8
op Pr
L g .
o.
00 651
T hi s, you will admit, is a simple an d expeditious working a sum in proportion.
That is all there is to be said about Logarithms, and quite enough too. I could never see the obj ect of requiring such an intimate knowledge of Logs. in all their twists and turns and subtleties on the part of candidates for a
LOGARIT HMS
37
certificate of competency, seeing that all the problems
given for a master can be solved if you know how to find the Log. of a natural number of four integers, and to take out the natural number of four integers of the nearest Log. But so it i s ; the knowledge i s required, and must be acquired. It i s a puzzling subj ect, and the student should work a lot of exercises in it. For this reason any amount of exercises are given in the second volume.
f I n ca s e you
s hould
like
to
know
no
w ,
or
at
s o me
utur e
f f time,
wha t
Loga rithms
r ea lly
a
r
e ,
here
ollows a v ery br ie
d escrip tion ; bu t d on t bo ther to rea d it unless you ha ve a
mind to .
The Logarithm of a number is the power to which the
base must be raised to produce that number. Any number
may be the base, but in all Nautical Tables 1 0 is the base.
With
the base
10 ,
s up p o s e
the Log.
of
1 00 i s
wanted.
10 x 10
1 00 ;
10
x
1 0 is ten squared,
or
1 02 ,
that is 1 0
raised to power 2
therefore
2
is
the
Log.
of
1 00 .
Suppose you
w an t
the
Log. of
1 000 .
10 x 10 x 10
1 000 ;
10
x
10
x
10
is
1 03 ,
10
raised to
power
3;
therefore
3
is
the
Log.
of
1 000 .
Now you will see why a d d ition of their Logs . i s the s ame as multip lica tion o f m nu bers .
10
x
10
x
10
x
10
x
10 is
1 05 .
(1 0 x 1 0) x (1 0 x 1 0
x
1 0)
is
1 05 .
10 x 10 is
10 x 10 x 10
is
1 03 .
2
is
the
Log.
of
1 02 ,
and
3
is
the
L og.
of
1 03 .
2
3
5
therefore the addition of the Logs. of 10 x 1 0 and of
1 0 x 1 0 x 10 produces the same result as the multiplica
tion of the
numbers
10
x
10
x
10
x
10
x
1
0 ,
namely
1 05 .
Also you will see why subtra ction of their Logs .
produces the same result as division of numbers .
Suppose you want
to
divide
1 000
by
1 00 .
The Log.
38
LOGARITHMS
of
1 000
is
3 ,
and
the
Log.
of
1 00
is
2 .
1 000
1 00
3
2
=
1 ,
which
is
the
Log .
of
10 .
The
Log.
of
1
is
0 .
1 00
18
2 .
2 —2 = O .
1 00
1 00
1 .
The Log. of
Suppose you want to raise a number to any given
power. All you have to do is to multiply the Log. of the
number by the given power. F or instance, suppose you
wi sh to rai se 1 02 to its fifth power, that i s to say to
The
Log
of
1 02
is
2 ,
an d
5
is
the
power
to
which
1 02
is
to
be raised.
2
x5=
10 .
1 02
x
1 02
x
1 02
x
1 02
x
1 02
So
you
see
that
1 02
multiplied
together
five
times
is
1 010 ,
and
that
2 ,
the
Log. of
1 02 ,
multiplied
by
5 is
the
L og.
of
10m .
The Logs. of all numbers which are not tens or
multiples of tens are obviously fractional. From what
has been said it is also obvious that Logs. of numbers
between one and ten must lie between zero and one, and
that the Logs. of numbers between ten and one hundred
must
be
more
than
one
and
less
than
two,
and
so
0 11 .
Hence it i s that the Index of a Log. is one less than the
number of digits in its natural number. The Logs. of
fractions must always be of a minus description. If you
divide the less by the greater, the result must be less than
unity. Ten divided by one hundred expressed in Logs. i s
one minus two. 1 2
1 .
Hence the minus Indices
already spoken about, which are for convenience sake
expressed as plus Indices by using their arithmetical .
complements.
CHAPTER III
"INSTRUMENTS USED IN CHART
COMPASS WORK
instruments which are nec essary for th e purpose of navigating a ship by D ead Reckoning are the follow
1 .
M
a
rin
e
r
s
C ompass .
2“ .
Instrument for
taking Bearings
in
connect ion with
fl the Mariners Compass.
3 .
Lead.
;
s
Log.
r
e
Parallel Rulers .
w
o
D ividers .
P rot rac t or s .
The .
following. i nstruments ,
though
necessary, are extremely useful, namely
a bsolutely
lor 8 .
Pe
us .
9 .
Sta tion Pointer.
The M ariners Compa ss con si st s of a Compass Card under whi ch are sec ured one or more magnets lying exact ly
m paralle l with a. line ]o1n1ng the North and South points on
the Co pass Card and with their Positive or Red Pole s -
towards the North p oint . Thi s Card is fitted under i ts cent re with a cap of agate or some similar hard stone
40
INSTRUMENTS USED IN
which rests upon the hard and sharp point of an upright metal rod, firmly fixed to the bottom of the Compass Bowl. By this means the card is accurately and delicately balanced upon its centre. The Compass Bowl i s made of copper, because that metal does not affect the Needle. The bowl is hung on gimbals, so arranged that it always remains horizontal, no matter at what angle the binnacle to which the gimbals are fastened may be canted. The binnacle is generally a hollow wooden column, fitted with slides inside for the compensating magnets, and having some arrangement on either side at the same height as the Compass Needles for supporting the soft iron correctors ; it should also have perpendicular slots on both its forward and after sides, in the fore and aft line for placing a Flinders Bar, should it be required.
The essentials of a good Compass are, that its Magnets should be extremely powerful, and as light as possible. The cap in the Compass Card should be perfectly smooth, not rough or cracked, and the pivot on which it is balanced should also be quite smooth and free from rust. The Card, if deflected mechanically, should return to exactly the point from which it was twi sted. It must be divided into points, half points, and quarter points and degrees with the greatest accuracy. The point of the pivot should be in the same plane as the gimbals of the bowl when the ship is upright. In the case of a Standard Compass, a clear view of the Horizon all round should if possible be obtainable, so that the bearing of any obj ect can be taken with the ships head in any position. The vertical line, called the Lubber Line, marked on the Compass Bowl, must be exactly in the fore and aft line of the ship .
In choosing a Compass go to a good maker, and pay a good price for a good article.
42
INSTRUMENTS USED IN
which the reflection of an object can be projected on to
the rim of the Compass Card . To take a bearing with the Azimuth Mirror, turn the
instrument round until the object is roughly in a line with
your eye and the centre of the Compass Card. Then, looking at the rim of the Compass Card through the lens, revolve the prism till the image of the object falls on the
rim of the Compass Card ; read off the degree on which
the image appears, and you have the bearing of the object. Some little difficulty may at first be experienced in using
the instrument ; in this case, as in so many others, practice makes perfect, and after a few trials and the
exercise of a little patience you will find that you can get
the bearings
of
obj ec t s
on
shore,
of
s hi p s ,
and
of
the
s un ,
moon, and stars with very great accuracy and case. It is
not advisable to take the Azimuth of a star whose Altitude
exceeds
As the prism inverts the obj ect observed, ships, objects
on shore, or a coast-line appear upside down, but you will
soon become accustomed to that.
See that the Compass is level by putting pennies, or sovereigns if yo u have them, on the glass till the air
bubble is as near the centre as possible.
A shadow pin— a pin placed perpen dicularly over the
pivot of the Compass— affords an easy way of getting
Azimuths
of
the
S un .
Take the bearing of the shadow
of the pin and reverse it, and you have the bearing
of the Sun .
The Lead and Lead Line
There are two descriptions of ordinary Leads, namely,
Hand Leads and Deep Sea Leads. Their names indicate
the difference between them. Hand Leads are of different
weights,
but
they
rarely
exceed
9 lb .
Deep Sea Leads
CHA RT A ND COMPA SS WORK
43
often weigh 30 lb. and even more. Hand Leads are hove by one man, and are no use except in shallow water. When a ship is going 9 knots it takes a good leadsman to ge t bottom in 9 fathoms .
Deep Sea Leads are for getting soundings in deep water, 1 00 fathoms an d more sometimes. It is necessary when using an ordinary Deep Sea Lead to heave the ship to. The line i s reeled off until there is a s ufficient amount of loose line to reach the bottom. The Lead, which has an aper ture in the lo wer end of it, in which grease is put (this is called the arming), is taken on to the lee cathead or fore tack bumpkin ; the end of the lead line is passed forward from the lee quarter, where the reel is, outside everything and secured to the lead. A line of men is formed along the bulwarks, each of whom has a coil of
lead line in his hand. When all is ready the man at the
cathead heaves the Lead from him as far to leeward as he is able, calling out Watch there, watch. E ach man as his coil runs out repeats this to the next man astern until the bottom is reached, or until all the lin e is run out if the Lead has not reached the bottom.
T his clumsy operation is nowadays almost completely superseded by Lord Kelvins Patent Sounding Machine. It depends for accuracy upon the increase of pressure in the sea as the depth increases , which the instrument records thus
A glass tube descends with the lead. It is her metically closed at the upper end and open at the lower its interior surface is coated with a chemical pre paration, which becomes discoloured when salt water touches it. As the depth of water increases the pressure becomes greater, and the air in the glass tube is com pressed as the salt water is forced into it the discolora tion of the chemical coating shows exactly how high the
44
INSTRUMENTS USED IN
water rose in the tube, and by means of a scale applied to the side of the tube the d epth of water which causes
that pressure is read off. To accelerate the descent of the Lead, piano wire is
used for the lead line ; the wire is wound upon a drum
fixed to one of the ship s quarters, which enables a few
men to b an] in the Lead after a cast, instead of, as under
the
old
system,
very
often
requiring
the
whole
s
h
i
p
s
company. With Lord Kelvins machine bottom can be
reached at 1 00 fathoms, with the ship going, it i s said, as
much as 1 5 or 1 6 knots.
The old- fashioned lead line is marked as under
t o p i l t it At 2 fa h ms a ece of ea her w h two end s
3
three
5
white lca ico
7
ti red bun ng
10
l t ea her with a hole in it
13
lb ue serge
15
white lca ico
17
t i red bun ng
20
t it ot a s rand w h two kn s
25
one
30
th ree
35
one
40
of ur
45
o ne
50
fi ve
o ne
six
one
v se en
o ne
eight
o ne
in ne
1 00
one
p i t i a ece of bun ng
and then the marking is repeated for the secon d hundred. The difierence of materi al i s to enable the leadsman
at night to identify the sounding without reference to the
colour.
CHA RT A ND COMPA SS WORK
45
In heaving the Hand Lead, the leadsman must use
his own judgment as to the depths obtained by reference
to the position of the marks . He reports the sounding
by the follo wing cries
and so on .
o di S un ng t o 5 fa h ms
6
61
iCr es
By the m ark fi ve
p By the d ee six l And a h a f six t l i t q A uar er ess e gh q t And a uar er ten
The Log ship and Log Line
The old-fashioned Logship is generally a piece of wood
in the form of the segment of a circle . It has lead run into
its circular p art, so that when in the water it will float up right with the rim down . A hole i s bored in each corner,
and it is fastened to the L og line with three cords, in such
a fashion that its plane is perpendicular to the pull of the line. ~One of these cords i s so fastened to the Logship,
that when a heavy strain
is
put upon
it,
it
comes
loose ,
which allows the Logship to lie flat in the water when it
i s being hauled on board after use .
Sometimes a conical canvas bag is used for a Log
ship, arranged so that it presents its mouth to the direction
of the pull of
the Log
line while the Log
is
being
hove ,
and its point when it is being hauled in.
The idea in each of these cases is to make the Log
ship as nearly stationary as possible while the line is run
ning out, and to offer the least possible resistance when
it is being hauled on board.
The principle involved in the Log line is a simple
proportion. The ordinary length of the Log glass is
28 seconds. o As 2 8 sec nd s
o 1 h ur
l li the ength o f ne
run out in 2 8“
l t li t th e eng h of ne ha t o l w u d run out in l n
46
INSTRUMENTS USED IN
Now, supposing the ship to be travelling one mile in
one hour, we have the following proportion
o 2 8 sec nds
o 1 h ur
60
a:
t l l 1 nau ica mi e
2 040
i t 60 m nu es 60
2040 yard s
3
o o 2 8 sec nds 3600 sec nd s
t 61 20 fee
61 20
28
48960 1 2240
ft . in.
v l 3600 ) 1 71 360 (47 7 y y er near 1 4400
2 7360 2 52 00
2 1 60 12
25920
2 5200
That is to say, if a ship is travelling at the rate of 1 knot per hour, she will run out 47 ft. 7 in . of line, very nearly, in 28 seconds . It i s, therefore, quite clear that if she is sailing at the rate of 2 knots per hour, she will run out in 28 seconds twice 47 ft. 7 in. if she is going 3 knots
,
three times 47 ft. 7 in. Again, if a 1 4-second glass is used, she will clearly
only run out half the line she would, had a 28-second glass been used, and therefore if she ran out 47 ft. 7 in. in 1 4 seconds, she would run out twice 47 ft. 7 in. in 28 seconds. In other words, she woul d be goin g two
knot s. The Log line is marked thus About 1 0 fathoms of stray line are allowed between
the Logship and the first mark on the line, which consists
of a piece of white bunting or rag. At the di stance of 47 ft . 7 in . from this mark a piece of twin e with 1 knot
is placed ; at a further di stance of 47 ft. 7 in . a piece of
twine w' ith 2 knots i s placed ; at a further di stance of
CHA RT AND COMPA SS WORK
47
47 ft. 7 in. a piece with 3 knots is placed, and so on
generally up to about 7 knots. Halfway between these
knots a single knot is placed. So we have the following
marks, at a distance of the half of 47 ft. 7 in. apart : A
white piece of rag, 1 knot, 1 knot, 1 ' knot, 2 knots, 1 knot ,
3 knots, 1
knot,
4
k
n
o
t
s ,
1
knot, 5 knots, 1 knot, 6 knots,
1 knot, 7 knots, 1 knot. If the 28-second glass is used,
the knots run out indicate the speed of the ship ; but if
the 1 4- second glass is used, the number of knots run out must be doubled to give you her speed.
In practice the Log is hove thus
A man stands with the reel, on which the Log line is
held above his head, so that it can run clear of everything. Another man holds the Log glass, seein g that the upp er bulb is clear of sand. The man heaving the Log sees
that the Logship is properly fastened, and asks if the
Log glass is clear. He then throws the Log as far to
leeward as he can, and lets the Logship run the line off the reel, till the white mark passes through his hands,
when he says Turn to the man holding the Log glass,
who instantly reverses it. When the sand has run out, the
man
holding
the
glass
calls
t p S
o
,
and
the
L og
line
is
seized and prevented from running out any more. The
numb er of knots r un out gi ves the speed of the vessel, as
explained already.
Patent Logs, which indicate the number of miles the
sh1p has gone through the water, possess so great an ad
vantage over the ordinary Log, which only tells you the rate of the ship at the moment of heaving the Log, that
the latter has become quite out of date, patent Logs being
now invariably used at sea.
The patent Logs most commonly used are two in
number. One is called the Harpoon Log, and the other
the Tafirail Log.
INSTRUMENTS USED IN
The Harpoon Log is shaped like a torpedo, and has
at
one
end
a
metal
loop
to
which
the
Lo g
line
is
fastened ,
and at the other, fans which cause the machine to spin
round as it is drawn through the water. The spinning of the instrument sets a clockwork machinery in motion
,
which records the speed of the vessel up on dials, the rotation of the instrument being, of course, dependent
upon t- he rate at which it is dragged through the water .
When you want to know the distance your ship has
run , you must haul in the Log and read it off on the
dial.
The Taffrail Log is called so because the dial which contains the recording machinery is secured to the taffrail.
It is connected by a long line with a fan towing astern, which revolves when dragged through the water, and
makes the line spin round. This causes the machinery in
the dial to indicate on the face of the dial the distance
travelled . The advantage of using the T affrail Log is
that it can be consulted at any time without having to
haul the line in and, as it is usually fitted with a small
gong which strikes as every one-eighth of a mile is run
out, it is a simple matter to find out the speed of the ship at any moment by noting the time elapsing between
two successive strokes of the gong.
Paral lel Rulers
For chart work parallel rulers are indispensable. They are simply rulers so arranged that you can move them over. a chart and their edges will always remain p arallel to any line from which they may have started. Of course there i s some danger, if the distance to be moved is considerable, of the ruler slipping, particularly when a ship is knocking about . And I strongly m reco
50
INSTRUMENTS USED IN
Pr ot r a ct o r s
The most useful form of protractor for chart work is made of horn or celluloid . It is very convenient to have a thread or piece of silk attached to the centre as the measurement of angles is greatly facilitated thereby. The ordinary protractor is divided into degrees radiating from the centre. It i s usually a semicircle, the horizontal line passing through the centre being marked 90° at each end, and the vertical line 0°
T o measure a Course ruled on the chart, place the centre of the protractor on the point where the Course cuts any Meridian, and see that the zero on the vertical line of the protractor is also on the same Meridian. You can now rea d off the angle of the Course where it passes under the semicircular edge of the protractor.
The above instruments are essential, and the following will be found very useful
The Pelorus
A Pelorus is a dumb Compass Card— that is, a card without a needle— fitted with sight vanes for taking bear ings. It i s usually placed on a stand, and so mounted that the Card can be turned round to any desired position, and there fixed by means of a screw. The sight vanes can also be turned round and fixed to the Card at any required b e arin g .
It is a handy instrument for determining Compass Error, and also for placing the Ship s Head in any positi on that may be wished. Its use will be more fully explained later on in the chapter on Magnetism and Compass Cor rection.
CHART AND CO MPAS S WORK
5I .
A Station Poin t-s r is an instrument with three legs
by whi ch, when used in conjunction with a chart the
angular distance between three obj ects on shore is known
either by measurement or by their bearings. With the
three legs measuring the an gular distanc e between the
three obj ects and clamped, plac e the instrument 0 11 the
Chart in such a position that the legs are exactly even
with
the
three
obj ect s
on the
C ha rt
;
the
s
h
i
p
s
position
is
indic ated by the Centre of
CHAP TER IV
THE PRACTICAL USE OF THE COMPASS
A COMPAS S Card is , like 5 11 other circles, divi ded into 360 degrees . E ach degree consists of 60 minutes and
each minute contain s 60 seconds
It has four Cardin al
P oin ts, Nort h, S outh, East, an d IVest ; four Q uadr antal
Points,
NE ,
SE ,
S IV ,
and
NI V ;
an d
twenty - four inter
mediate Points, as shown in the figure, thus making
thirty-two Point s in all . As there are 360° in any circle,
eac h Point c ontain s 1 1 ° 1 5' that i s, 360° divide d by
Each Point is subdivided into half and quarter Points .
As the C omp ass Card moves freely on its pivot. the
North Point of the Card is caused by the C ompass Needle
to point towards the North Pole of the earth;
In speaking of the direction of any object from the
ship, or of the direction in which a ship i s proceedin g, it i s equally accurate to use Points, half Points, and quarter Points, or Degrees, Minutes, and Seconds but as in many cases their use simpli fies calcul ation very much,
it is advisable for the student to use D egrees and parts
of D egr ees . The B earing by Compass of any obj ect is the angle, at
the c entre of the Compass Card, between the North and S outh lin e on the Card and an imaginary straight line
drawn from the centre of the Compass Card to the obj ect.
A Bearing is measured along the circumference of the
Compass Card, so many D egrees and parts of a D egr ee
THE PRA CTICAL U S E OF THE COMPA SS
53
from the North or South Points on the Compass Card to where the imaginary line cuts the circumference of the C ard .
On the inside of each Compass B owl a vertical line i s marked, indicating the line of the keel of the vessel. This is called the lubber line . \V hatever D egree, Point, half
D F 1 IG.
.
-COMPAS S CAR
Point, or quarter Point is opposite the lubber line, is the Compass Course you are steering.
Va ria tion — The Compass Needle is supposed to point North and South with unswerving fi delity true as the Needle to the Pole is the idea. But unfortunately the idea is inaccurate, for the Needle very rarely points to the North and South Poles of the earth if it did, the mariner
54
THE PRA CTICA L U S E OF THE COMPA S S
would be relieved of much anxi ety and bother. It points towards what are called the North and South Magnetic
Poles of the earth, situat ed in about Latitude 70° N and
L on gi tu de
9 7° IV ,
and in Latitude
74°
S
and
L on gi tu d e
1 47° E I t .
it points in that direction goodn ess onl y
knows
but it
does—
that
is
to
say ,
it
does when no
di s
turb ing
causes
affect
it .
Wh en the Needl e does not point T rue North and
S outh it makes a certain an gle wi t h the Meridian or True
North and South line.
Thi s angle
is
called
the I i
'
ar a
tion of the Compass. V ariation varies in different parts
of the globe, and is also constantly changing, but as the
fl change is slow and the V ariation l S given on all charts,
you can alway s find what it is bv looking at your chart, unl ess you are using an antedil uvian one. The C ompass Needl e affected b y V ari ation and by nothing else is said t o p oint Cor rect Ma g netic.
D evia tion — But another an d very inc onveni ent in u
ence comes into operation in most ships, and in all vessels buil t of ir on or steel. The ship itself i s a Magnet, and its M agn etism affects the Comp ass N eedl e, causing it to diverge from the Correct Magnetic Meri dian. The angle which it makes with the Correct Magnetic Meridian is
called the D evia tion of the C omp as s.
Thus it wi ll be seen that any obj ect may have three
different bearings from a ship— namely, fir st, a True B earin g. T his i s the an gle formed by an imaginary line dr awn from the obj ect to the Compass , and the T rue Meri di an whi ch p asses through the Compass . Second, a Correct Ma g netic B earing, whi ch i s the angle forme d by an imaginary line drawn from the obj ect to the Compass, and the Magnetic Meridian which passes through the Comp ass . T hird, a Comp a ss Bearing, whi ch is the angle
formed by an imaginary line dr awn from the obj e ct to the
THE PRA CTICA L USE OF THE COMPA SS
55
Compass and the North and South line of the Compass
Card.
If you want to know how an obj ect bears for a- ny
charting work, you must first take the Bearing by Compass , and then correct the Compass B earing for the Deviation
d ue to the position of the ship s head ; thi s correction will give you the Correct Magnetic B earing. This i s suffi cient if you are using a Magnetic chart, that is to say, a chart the Compasses drawn on which show the Magnetic
Points. But if you are using a chart showing only the
True Points, or if for any other reason you want the T rue B earing, you must correct the Correct Magnetic bearing
for V ariation. This will give you the True B earing of the
obj ect, whatever it may be. The way of making these
corrections will be explained later on.
Now, as to Courses, the same facts and considerations
apply.
The True Course of a ship is the angle between her
track through the water and the Meridian— that is to say, the True North and South line. To find it from a Com
p ass Course, three allowanc es— namely, for Leeway, D e
viation, and V ariation— must be m ade . To a C orrect Magnetic Course, V ariation only must be applied.
The Correct Magnetic Course of a ship i s the angle between the ship s track and the Magnetic Meridian, that is, the line j oining the North an d South Magnetic Pole of
the earth . To fin d it from a Compass Course, Leeway
and Deviation must be applied.
The Compass Course of a ship i s the angle between
the line of the ships keel and the line of the North and
South Points on the Compass Card .
If
you
know
the
T rue
Course
between
two
p
l
a
c
e
s ,
and
want the Correct Magnetic Course, you must apply the
V ariation to the True Course, and there you are. Then
56
THE PRA CTICA L USE OF THE COMPA S S
if you want the Compass Course the Devi ation, if any, appli ed to the Correct Magnetic Cour se, will give it t o you ; and if your ship makes no Leeway, and there are no currents, y ou will get to your destination if you steer your C omp as s Course thus found . But if y ou are makin g Leeway , or if ti des or currents are setting y ou across your C ourse, allowance must be made for them.
It i s in making these corrections and allowances that the whole system of steering by Compass an d using the Chart consist s.
The converse, of cour se, holds true. If you know your Compass Cour se between two places, and w ant the Correct Magneti c Cour se, you must correct the former for D evi ation ; and if y ou require the T rue Cour se you must correct the Correct Magn eti c Cour se for V ariation .
m Correct ion of Co pass Cour ses
As in workin g all problems in the various sailings T rue
Cour ses must be used, it i s very necessary to un derstand h ow to turn a Comp as s Cour s e into a T rue Cour se.
f To fin d a T r ue Cou rse rom a Comp a ss Course — In the first place bear in min d always that as the rim of the
Compass Card represent s the Horizon, you must always imagine your self to be looking from the centre of the
Card out towards the rim in the direction of the Course
to be corrected. The first thing to do i s to correct y our Compass
C our se for Leeway if the ship has made any . Leeway i s
the angle between the line of the keel and the track of
the
s hip
through
the
t e wa r ,
and
is
caused
by the wind
forcing the vessel sideways as well as forward. The
amoun t of Leeway can only be j udged by experience.
The correction for Leeway must always be made in the
58
TIIE PRA CTIC A L USE OF THE COMPA S S
Magnetic into a T rue C ourse— that i s to say, if the
V ariation
is
W l t es
er
'z ,
apply it
to
the
right ; i f it i s
E a s ter ly,
apply
it
to
the
f le
t .
The next operation is to allow for Deviation, and here
comes the difficulty. You d o kn ow the Deviations on
every position of the Ship s Head by Comp a ss , but you do
not know the Deviation for the Ship s Head on any given
Cor-res t Ma gne tic Cours e, and you have to find it out. The simplest plan is to find it by inspection— by drawing a small p ortion of a Napier s curve, as explained later on, and measuring off the Deviation from it ; but you must
also know how to calculate the Deviation, and the best
way of doing so is as follows. Judge, by reference
to your Devi ation Card, whether the Deviation appli cable to the Correct Magneti c C our se which you wish
to convert into a Compass Cour se will be to the right or
to the left ; then write down three Comp a ss Cour ses, within the limits of which the Compass Cour se to be
derived from the Correct Magnetic Course you are dealing
with is pretty certain to be included . To these Compass
C ourses apply their respec tive D eviation s, which, of course, you know. You have now thr ee Correct Magnetic Courses. If the Correct Magnetic Course you are correcting i s the
same as one of these three Correct Magnetic Courses,
then the Deviation which you used to find that Correct
Magneti c Course is the D eviation to be applied to the
C orrect M agnetic Cour se you wi sh to convert into a
C omp as s C our se. D on t forget that in tur ning your three Compass Courses into C orrect Magneti c Courses,
you apply the Deviation directly, that is, East to the
right, West to the left ; and that in converting the Correct Magnetic into a Compass Course, you apply
the Deviation indirectly, that is, East to the left, West to
the ri ght .
THE PRA CTICA L USE OF THE COMPA S S
59
But it may, and probably will , happen that not one of
the three Compass Courses you have turned into Correct Magnetic Courses coincides exactly with the Correct Magnetic Course you have to turn into a Compass Course.
In such an event you must do a little sum in simple pro
porti on . Y ou have got three Correct Magnetic Courses, on
which you know the Deviation. Y ou find that the Correct Magnetic Course you have to convert to a
Compass Course lies between two of them. Take the difference between these two Correct Magnetic Courses,
and
call
it
A .
T ake the difference between one of them
and the Correct Magnetic Course you are dealing with,
and
call
it
B .
T ake the difference between the Deviations
on the two Correct Magnetic Courses used, and call it
C .
Then as A is to B so is C to the answer.
Multiply
B
by
C
and
divi de
the
result
by
A .
The result gives you
the portion of Deviation to be added to or subtracted from
the Deviation belonging to that Correct Magnetic Course
from which B was measured whether it is to be added or
subtracted, will be apparent on the face of the case.
It may also h appen that, having turned your three Compass Courses into Correct Magnetic, you will find that the Correct Magnetic you desire to turn into Com pass does no t lie within their limits, but i s less than the least of them, or greater than the greatest of them ; in which case you must select one or two more Compass
Courses to convert until you have two Correct Magnetic
C ourses, one greater and the other less than the Correct Magnetic Course you are dealing with, or, if you are
lucky, one of which coincides exactly with it. This is a long explanation, and sounds complicated, but it really is
simple, and its simplicity will best be shown by one or two examples, worked with the following Deviation Card.
60
THE PRACTICA L U SE OF THE COMPA SS
It will be seen that the Deviation i s given for the Ship s Head on every Point by Compass.
s u r rox CARD
H d l p Shi s ea by t d d o p S an ar C m ass
Devia tion
D iev ation
Now suppose you want to s ail from any one place t o
another,
let
us
call
it
from
A
to
B .
You lay the edge of
your parallel
ruler
on
A
and
B ,
and working
them
to
the
nearest Compass on the Magnetic Chart you find that the
Correct Magn etic Course to steer is, let us say, S b W }1 W.
On looking at the Deviation Card you see that the
Deviation with the Ships Head on S b W 5 W is chang
ing very rapidl y. On a S b W C ompas s C o urse it i s 6° 1 8 '
W,
and
on
a
SSW
C omp a s s
Course
it
is
56' E .
It i s
probable that by applying the Deviations to these two C ompass Cour ses youwill get the two Correct Magneti c
Courses between which the Course you wish to steer lies.
Proceed thus. Turn the Compass Course into degrees
and parts of a degree, and apply the Deviation.
o p o W C m ass C urse S b D ev i at i on
o p o W W S 1 1° 1 5
C m ass C urse S S
W 6° 1 8'
D ev i at i on
W S 22° 30
0° 56' E
o t ti o o t ti o W M W S C rrec
agne c C urse S 4° 57'
M C rrec
agne c C urse 2 3° 26 '
Now the Correct Magnetic Course we want to steer i s
S b W 5 W, which i s S 1 6° 52' 30" W, and thi s lies between
THE PRA C" ICA L USE OF THE COMPA S S
61
the above Correct Magneti c Cours es , namely, S 4° 57 W and S 2 3° 2 6 W.
To proceed.
i i F nd the d fference be t t ween th e two Correc t i o Magne c C urs es
W S 4° 57 W S 23° 2 6'
w 1 8° 29 '
i i F nd the d fference be
t t ween the neares Cor
t ti o M rec
agne c C
l W y name
S 4° 57
,
W S 1 6° 5 2 30"
urs e ,
a nd
W S 4° 5 7
W S 1 6° 52 30”
11 ° 55 30
i i F nd the d fference be
t vi tio ween the D e a ns
t o o d ue
the C mpas s
oC urses you ha ve con
v ter ed
W 6 ° 1 8
0° 56 E
7° 1 4 '
Then, as 1 8° 29 1 1 ° 55 30
7° 1 4'
a
.
To Simplify the sum, use the nearest decimals of a
degree, and say : as
cc .
Multiply the second and third term, an d divi de by the
first term.
1 8 -5
4-6
Therefore,
or 4° 36' is the c orrection to be applied
to the Deviation on the nearest Course, which is S b W,
or S 1 1 ° 1 5 W, and it must be subtracted, because the
Deviation Westerly is decreasing.
vi tio o p o W ( ) D e a n on S 1 1 ° 1 5
C m a ss C urse
(Correct ion)
W 6 ° 1 8
4° 3 6 '
W
1° 42 ' i s therefore the Deviation to be applied to the
Correct Magnetic C ourse S b W 5 W.
S b Wé W
Deviation
W S IB° 52 BO
1 ° 42
W
SbW
S w 1 8 ° 34' 30'f
W o p o to t is the C m ass C urse
s eer.
T ake another case. Suppose you find from the chart that the Correct Magnetic Course to the place to which
you
want
to
go
is
N
40°
E ,
and
you want
to
find
out
what
Compass Course to steer.
62
THE PR ACTIC A L U SE OF THE COMPA SS
Take two Compass Courses from the Deviation Card,
and correct them for Deviation.
o p o m C
ass C urses
D eviation
NE = N 45 ° E
o t c rrec ed for 2 ° 35 E
o t NE b N = N 33° 45 E c rrec ed for 6 ° 42 ' E
o o M C rr.
g a . C urse
N 47° 35 E
N 40° 2 7 E
Here you have hit so nearly upon the Correct
Magneti c Course that no sum in proporti on is necessary, and in steering NE b N by Compass, you will be within 1 ° of the Correct Magnetic Course you require, and
goodness knows that is near enough.
Again, suppose you want to find the Compass Course to steer in order to sail S 42° E Correct Magnetic.
o p o C m ass C urses
Deviation
S SE SbE
o t W S 2 2° 30' E c rrec ed for 2 6° 1 6 o t W S 1 1 ° 1 5 E c rrec ed for 20° 5 8 '
o o M C rr.
g a . C urse
S 48° 46 E
S 31 ° 43 E
T he Cor rect Magnetic Course you require to convert
into a Compass Course lies between these two, and a sum in proportion must be done.
48 ° 46 3 1 ° 43'
42° 0 ' 3 1 ° 43
1 70
3
1 0° 1 7'
o Th eref re, 1 7° 3 '
1 0° 1 7
5 ° 48
Or put for convenience sake decimally,
17
1 0-3
5 -8
on
Multiply the secon d and third terms and divide by the
first
17)
51
87
(3 5
o ti o i ( q ) the c rrec n re u red
vi tio o ( M ) D e a
n on S 3 1 ° 43 E C rr.
g a .
(C orre ct i on)
v i t i o o t ti ) ( D e a n on S 42° E C rrec Magne c
W 20° 2 8
3° 30
W 23° 5 8 '
o t ti voi tio W M C rrec
agne c C urse S 42 ° 0 E
De a n 23° 58
o p o to t e C m a ss C urs
s eer
S 18° 2' E
THE PRA CTICA L USE OF THE COMPA SS
63
Now for B earings. To turn a True Bearing into a
Compass B earing, first convert True into Correct Mag
netic, by applying the V ariation, and then apply the
Deviation d ue to
the p osition
of
the
S
hip
s
H d ea .
Remember that the D eviation due to the B ea ring has
nothing whatever to d o with it. In all these cases
you will find it conveni ent to work with Degrees and parts of a Degree, therefore accustom yourself to turn Points and parts of a Point into Degrees and parts of a
Degree.
Y Q A TAB LE or THE ANGLES WHIC H EVER POINT AND
UARTE R POINT
M D or T HE COMPAS S MAKE S WIT H T HE
ERI IAN
o t N r h
Points
Poi tn s
o t S u h
NbE NNE
Nbw
0
0_
1
1 — 31 1 —s Ls
2
2 —i
Nw b N
2— i
g1
I
NE b E Nw b w
h
u
k
b
— H
N
kl b
w l
Fa
FI
!
0
SE b S Sw b s SE b E Swb w
ENE
EbN
West
Eb S
64
THE PRA CTICA L U SE OF THE COMPA SS
The scale upon the preceding page shows you the
number of degrees due to any Point, half Point, or quarter Point, an d vice ve-rsa . At s ea you have alway s a Comp ass
with you, wi th degrees in dic ated on the Card ; all the
Epitomes contain T ables giving degrees for points and
points for degrees, and the Board of Trade Examin ers
will provide you with a compass card containing a T able
of Angles similar to the one overleaf, so calculation is
really unnecessary ; but at the same time there is no
harm in knowing how to calculate for your self the
number of D egrees cont ained in any Course given in
Points and parts of Points, and the Points and parts of
Points equivalent to any number of D egrees.
To turn P oints into D egrees, eta — If y ou w ant t o ex
press Points in Degrees as every Point contains 1 1 °
all you have to do i s to multiply the Compass Course by
11°
For example, if
the Course
is
E
3
4
N— that
is,
74 Points from North, or in decimals — this multi
plied by 1 1 °
or in decimals
will pro duc e the
number of D egrees in E g N . Thus
EN
1 1° 15'
7% Poi tn s from North
4 1 1 D egrees
72 5 Points D egrees
362 5 1 450 725 72 5
8 1 5 625 Degrees
5625
But the 5 625 must be turned into minutes
60
i t 33 7500 m nu es.
75
There remains '75 to be turn ed into secon ds 60
45 -00
an d 81 ° 33 40 i s the are required. Therefore, E N is
equal
to
N
81 °
33' 45"
E .
To turn D egrees etc. into P oints .—' Now for the revers e
of this problem— namely, to express Degrees in Points.
66
THE PRACTICA L USE OF THE COMPA SS
Error is caused by V ariation or by Deviation, or by both combined. \Ve will consider the effect of Error from
whatever cause it arises.
Consi der yourself to be in the mi ddle of the Compass, looking towards its circumference. Suppose the North
seeking end of the Needle to be from some cause or other
drawn to the right. The Error will be Easterly. Y ou
can see this for yourself. Set the movable Compass Card
pointing true North suppose the Needle to be deflected
two Points to the right, the Error will be t wo points to
the
rig
h
t ,
and
the
E rro r
is
in
scientific
works
call ed p lus
;
but I presum e, because the Error i s towards the East when
you are looking North, it i s commonly called E a s terly
Error. It is called Easterly Error when either end
of the Needle is dr awn towards your right, even i f i t
is drawn towards the West for instance, leave the
Compass Card in the same position, and look toward the South . The South- seeking end of the Needle has been
drawn towards your right hand, and the Deviation i s Easterly, though the South—seeking Pole of the Needle i s
deflected towards the “ 7est . Hence the rule always to be
ob served i s, that when the Needle is drawn to the right, D eviation i s E a sterly when the Needle i s drawn to the
f le t ,
it
is
l Ves terly.
Another rule never to be forgotten is, that when the Tr ue B ea r ing i s to th e right of th e m C np a ss B ea-r ing ,
the Error i s E asterly. \Vhen it i s to the left it i s
\Vesterly. T hi s sounds odd in connection with the fore
going rule, but a glance at the Compass Card will Show
it is true. Make the North Point of the movable card to coincide
with the North Point of the fixed card ; now shift the
movable card round two Points to the right : the Needle
i s now pointing to NNE (True), NNE i s to the right of
THE PRACTICA L USE OF THE COMPA SS
67
North, therefore the Error i s two Points easterly. Shift
the card in any way you like, say till the North-seeking
end
of
the
Needle
points
to
WNW .
WNW is six
Points to the left of North, therefore the Error i s Six Point s Westerly. Now if you look the other way towards
the South, the South-seeking end of the Needle will point to E SE (T rue). E SE i s six Points to the left of South , and the Error i s of cour se six Points Westerly, although
the South-seeking end is actually drawn to the East)
The only thing to be absolutely remembered is, that looking from the centre of the Compass towards any part
of the circumference, if T r ue Bearing is to the right
of the Compass Bearing, D eviation i s Easterly ; if it ' is to the left i t i s Westerly. And if the Needle i s
drawn to the right of True it gives Easterly Deviation
if it i s drawn to the left of True it gives Westerly
Deviation.
Supposing you know that with the Ship s Head in a
certain direction there is such and such an Error, and you want to find out what Course to steer in order to counter
act that error and make the required True Course. Let
us imagine you want to steer NE (True), and you know that with the Ships Head NE you have 1 4 Points Westerly
Error. Fix the movable card pointing North an d South (True), then the Compass NE will of course be pointing NE (True). But the Needle is deflected to the left, because the Error is Westerly 1 4Points. Revolve the Card till the
NE Point points to NE b N AN if, therefore, you steer NE by your C ompass, you would be steering NE b N gN
(True), which w ould
not do
at
all .
You would have to
steer
NE
b
g E E ,
or
11 ,
Points to
the
right of
NE
by
your Compass. Therefore, it is plain that to allow for an Error, if the Error is Westerly, you must steer the amount of Error to the right of the Course wanted, as shown on
68
THE PRA CTICA L USE OF -THE COMPA SS
your Compass . If the Error i s E asterly, steer the amount
of Error to the left of your Comp ass . Here comes another
golden rule in finding what Course to steer. Knowing
the True Course and the Error of your Compass Easterly
Error must be allowed for to the Left, Westerly to the
Ri ght .
Dont forget these three important facts.
i f l s t,
T rue
is to the right, Error is Easterly ; and if True is to the left,
Error is Westerly.
2nd ,
if
the needle
is
deflected
to the
right of True the Error is Easterly, and if to the left of
True the Error is Westerly . 3rd, knowing the Error, steer the amount of it to the left if the Error is Easterly, and
to the right if it i s Westerly, in order to counteract the
Erro r . (In the ordinary Masters Examination it i s required
that the candidate should be able to ascertain the Correct
Magnetic Bearing by taking the Compass Bearings of a distant obj ect with the S hip s Head in the Cardinal and
Quadrantal Points, and to draw and un derstand a Napiers
Diagram.)
To Ascertain the Deviat ion
In order to ascertain the Devi ation of your Compass, it is necessary to know how to find the Correct Magnetic
Bearing of a distant object at sea, so as to compare it
with its hearing by Comp as s . The following method i s
usually adopted.
Take the Compass Bearings of an object not less than
5 or 6 miles distant, with the Ship s Head on the four
Cardinal and on the four Quadrantal Points by Compass
by swinging the ship.
If the Bearings are all the same ,
the Compass has no D eviation . But if they differ, write
them down and turn them into degrees . If they are all in
the same Q uadrant, their sum divide d by 8 will give the
THE PRA CTICA L USE OF THE COMPA SS
69
Correct Magnetic Bearing of the distant object. For
example :
H d p Shi 8 ea by t d d o p S an ar C m ass
of D t t Bearing
is an
853; z Ob]6
s andai d
1 as
H d p Shi 8 t d d o S an ar
ea by
p C m ass
t t Bearing of Dis an
gpyifg Obj ec
nd ard
ru s
Here we have 8 Compass Bearings, all in one Quadrant and their sum divided by 8 will give us
,
the Correct Magnetic Bearing of the distant obj ect.
Thus
8 )285
ov 35° a nd 5 °
er
50
60
Therefore
the
C o rr e ct
Magnetic
B e ari n g
is
N
35°
38
E .
If the Bearings are not in the same Quadra nt , but are
all Easterly or all Westerly, while some are North and
some are South, see which of the Bearings are the more numerous, those from North or those from South change
the names of the less numerous Bearings by subtracting
each from 1 80° so as to make all the B earings of the same
name — that i s, all from North or from South towards
East, or towards West, as the case may be. Add them
together, and divi de by 8 and the result i s the Correct Magnetic Bearing of the di stant obj ect . If it
is 90° the Correct Magnetic Bearing will be due East
or West. If it is more than 90° take it from 1 80° and
change m its ' na e from North to S outh, or i v ce versa .
THE PRA CTICA L USE OF THE COMPA SS
Thus N 90° E will of course be East, and N 1 00° E
will
be
S
80°
E .
Here is an example
i H d p Sh
s
e a by
t d d S an ar Compm
oN rt h
i of p Cam as Bear ng Di t j t s -a nt Ob ec
S 84° W
i H d p Sh
s
t d d S an a r
p ea by
Com ass
o i of p C m ass Bear ng Di t t j t s an Ob ec
Here we have
some of
the Bearings in
the
N\V ,
and
some in the S\V Quadrants ; of course \Vest i s N' 90° “ T
or
S
90°
7
,
whichever
you
li ke .
There are more Bearings
in the N\V Quadr ant than in the W S ' Quadrant, and
therefore we will change the
Bearings into N\V
S 84° W
t Wes
N 96° W
N 90° N 8 1 ° IV N 76° \V
N 79° W
S 83° W
S 79 ° \V
o t ti i M C rrec
agne c Bear ng
N 97° W
N 101° W
8 708
N 88—37° W
T ake another combinati on . Suppose all the Bearings
are in the Northern or all in the Southern half of the
Compass, but some of them are East and some
In
such a case add the Easterly ones together, and add the ones together ; then take the difference of the
sums,
and
divide it
by
8 ,
and name
the product
E ast
or “ Test according to whether the sum of the Easterly
or “ v esterly B earings i s the greater. The resul t i s the Correct Magnetic Bearing.
Shoul d any of the C ompass B earings be due North or
due South, they are to be reckoned as zero in the additions,
but the difference is still to be divided by eight. Here i s
an example
TIIE PRACTICA L USE OF THE COMPASS
71
i H d p Sh s ea by t d d o p S an ar C m ass
o p C m ass B Di t t s an Obj ec
i H d o i of p Sh 's ea by
t d d o Di t t j p m S an ar C
a ss
p C m ass Bear ng s an Ob ec t
t SE Q uadran
s S° E S 4° E S l o° E 8 9° E
S 28° E
t W Q S
ua dra n
s w 2 ° s no w s 6°W W S l 9°
S28° E
w s1 9°
8 )S 9° E
s 1 ° 7 E
T hus
the
Correct
Magnetic
Bearing
is
S
7
E .
To find
the
D
evia
tion
_
.
—Having
t h us
found
the
Correct
Magnetic Bearing of the di stant obj ect, the next proceeding
is to find the D eviation of your Compass on the eight equi
distant positions oi the Ship s Head from the observations
on which you have derived your Correct Magnetic Bearing.
You can begin where you like. It does not matter.
Suppose we begin on North. Write down the Bearing of
the obj ect by Compass with the Ships Head North, and under it write the Correct Magnetic Bearing ; the difi erence
is the Deviation with the Ships Head North by Compass.
fl If the Compass Bearing and the Correct Magnetic
Bearing are both in the same Quadr ant, you have only to
subtract the less from the greater. Thus, suppose the
di st an t
obj ect
bore
by
Compass N
75 °
E ,
and
the
Correct
Magneti c
Bearing
was
N
80°
E ,
the
differe n c e ,
namely
i s the Deviation . But the B earings may be in di erent
Quadrants.
Suppose
the
obj ect
bore
by
Compass
N
75°
E ,
and
the
Correct Magnetic Bearing was
S 80° E .
Well, from
N 75° E to East i s
and from East to S 80° E is
in this case y ou must Obviously add them together,
and the Deviation is
Or, if you like, take one Bearing
from 1 80° so as to make them both of the same name, and
72
THE PRA CTICA L USE OF THE COMPA S S
then take the less from the greater. Thus S 80° E taken
from
1 80°
is
N
1 00° E .
N 1 00° E
N 75° E i s
which is the Deviation.
B ut ,
a gai n ,
the
Bearings may lie on
opposite
sides
of the North or South Points. Suppose the Compass
B e arin g
of
the
distant
obj ect
to
be
N
E ,
and its
Correct
M a gn eti c
B e ari n g
N
1 3° “ 7 ,
obviously
you
must
add
them together. Y ou have 5° on one side of North, and
1 3° on the other side, therefore they are 1 8° apart, and
the Deviation is
To na me the D evia tion — Fancy yoursel f situated in the
mi ddle of the C ompass Card and looking out to the rim
and towards the Bearings then if the Correct Magnetic
is to the right of the Compass, the Deviation is Easterly i f Correct Magnetic i s to the left it is \Vesterly.
Having thus foun d the Deviation and n amed it
correctly for the Ships Head North by Compass, proceed
to find the D eviations, and name them with the Ship s
Head
NE ,
East,
SE ,
South,
SW ,
\V est,
and
W N .
If
you c an do one you can do all. It only requires a little
care in naming them correctly. Here are the examples
given above completed. No. 1 is
i H d p Sh s ea by
t d d p m S an ar Co
a ss
De ation
With the Ship s Head North by Compass, the Compass
bearing
of
the
distant
obj ect
was N
40°
E ,
an d
the
Correct
Magn etic Bearin g N 35° 38 E ; the differen ce, 4°
is
the D evi ation, and it i s \Vesterly, because the Correct
74
THE PRA CTICA L USE OF THE COMPA SS
m Napiers Diag ra
A Napiers Cur ve is a most ingenious and useful
invention, for which the author deserves the thanks of all
those who go down to the sea in ships, and especially of those who go up for examination . It offers the simplest
of all methods of turning Compass Courses into Correct Magnetic C ourses, or Correct Magnetic C ourses into Compass Courses, and of ascertaining the Deviation of
the Compass with the Ships Head in any position.
The principle of Napiers D iagram is very diifi cult to
explain, and I give up the attempt. Y ou have got to
imagine as best you can the circular rim of the Compass
Card represented as straight.
The diagram consists of a straight line marked North
at the top and bottom, and South in the middle, and divided into the thirty-two Points of the Compass. The
degrees are given from zero at the top to 90° at East,
from 90° at East to zero at South, from zero at South to
9 0°
at
YV est ,
and
from
90° at “ T est
to zero at
North
at
the bottom. Lines are drawn forming an angle of 60°
with the medial line of the Diagram, and intersecting each
other at every Compass Point. The right - hand side of
the medial line is East, the left-hand side is West . The lines drawn from right to left downwards are plain, those from left to right are dotted. A glance at the accompany
ing diagram (fi g. 2) will show thi s at once.
To d ra w a f curve o D evia tion — Tu practic e you woul d
of course have first to find the D eviation on the four
Cardinal and the four Quadrantal Points but these will
be given you at the Board of Trade Examination.
With a pair of dividers measure anywhere on the
medial line the D eviation with the Ship s Head North ;
then, if the D eviation is Easterly, measure it on the d otted