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 Martin’s and Mr. Lecky’s are most valuable, but they preconceive

a considerable amount of knowledge on the part of the

student . Books such as Rosser’s ‘ 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 ‘ Day’s

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 Master’s 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.
Norie’s 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 .

Mercator’s

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

Star’s

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

Moon’s

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 un’s L wer

iL mb .
pp The Sun’s 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 n’s Lower

iL mb .

xv i

A BBREVIA TIONS

oo M n’s 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 N’O 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 E’CIMAL

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 2‘i

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

i‘lf .

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 . C1u‘I 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

RAI‘JCR 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

n’s 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, Noric’s Log. Horary An gle T able correspond s to Inman ’s Log. Haversine Table and to Raper’s 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 Noric’s 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 Norie’s Tab es, c n a n ng a c

e e Ha ers ne Tab e, has been s sued .

2

A PCITED H‘Z 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 lOVV‘l 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 RITHME’I‘IC

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

ARI’I'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 .

8°
'

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'I‘HMS

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 Mariner’s 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 ariner’s 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 ship’s 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 Kelvin’s 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 Kelvin’s 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 Tafi’rail 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 ship’s 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 Ship’s 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

0°

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 Ship’s 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 .

Don’t 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 Napier’s

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

1°

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 Ship’s Head North, and under it write the Correct Magnetic Bearing ; the difi erence

is the Deviation with the Ship’s 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

5°

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 Ship’s 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

fl
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 Napier’s Diag ra

A Napier’s 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 Ship’s Head in any position.

The principle of Napier’s 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