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By Lt. Comdr. A. E. Redifer, U.S.M.S. 

Lifeboat Training Officer, Avalon, Calif., Training Station 

*Indispensable to ordinary seamen preparing for the Lifeboat 

*A complete guide for anyone who may have to use a lifeboat. 
Construction, equipment, operation of lifeboats and other buoyant 
apparatus are here, along with the latest Government regulations 
which you need to know. 

163 Pages More than 150 illustrations Indexed $2.00 


By Peter E. Kraght 
Senior Meteorologist, American Airlines, Inc. 

*Embodies author's experience in teaching hundreds of aircraft 

flight crews. 

*For self-examination and for classroom use. 
*For student navigator or meteorologist or weather hobbyist. 
*A good companion book to Mr. Kraght's Meteorology for Ship 

and Aircraft Operation, 0r any other standard reference work. 

141 Illustrations Cloud form photographs Weather maps 

148 Pages W$T x tl" Format $2.25 

By E. A. Turpin and W. A. MacEwen, Master Mariners 

"A valuable guide for both new candidates for . . . officer's 
license and the older, more experienced officers." Marine Engi- 
neering and Shipping Re$etu. 

An unequalled guide to ship's officers' duties, navigation, 
meteorology, shiphandling and cargo, ship construction and sta- 
bility, rules of the road, first aid and ship sanitation, and enlist- 
ments in the Naval Reserve all in one volume. 

03 086 

3rd Edition 

"This book will be a great help to not 
only beginners but to practical navigating 
officers." Harold Kildall, Instructor in 
Charge, Navigation School of the Washing- 
ton Technical Institute, 

"Lays bare the mysteries of navigation 
with a firm and sure touch . . . should make 
a secure place for itself." A. F. Loomis in 

"Covers the field thoroughly." Tbe 

Thousands of students of sea and air 
navigation have agreed with this testimony 
to the Primer. 

They have found that the Primer takes 
into account all the stumbling blocks, and 
moves progressively from the simple funda- 
mentals to the complex problems, covering 
each step clearly. 

Astronomy, Time, the Astronomical Tri- 
angle, Trigonometry and reliable procedures 
for Position Finding are explained under- 

An added feature of this Third Edition 
is the author's own method for determining 
more exact time of transit of any naviga- 
tional body reducing Dutton's method from 
sixteen lines to ten. "You have made quite 
a contribution," Herbert L. Stone, editor of 
Yachting, wrote the author regarding this 
new method. It is published here for the 
first time* 

267 Pages 21 Tables Fully Indexed 

Kansas city |||| public library 

Books will be issued only 

on presentation of library card. 
Please report lost cards and 

change of residence promptly. 
Card holders are responsible for 

all books, records, films, pictures 

ecked out on their cards. 

Primer of Celestial Navigation 

Primer of 
Celestial Navigation 

By John Favill, M.D. 

Associate Member, U, S. Naval Institute 

Navigator, U. S. Power Squadrons 
Colonel, Inactive Reserve, U. S. Army 

Third Edition 
Revised and Enlarged 

New York- 1944 

Cornell Maritime Press 

Copyright, 194O, 1943, 1944, by 
Cornell IVTa/rltime !Press 

IDesigpnted t>y Reinnold Frederic Genner 
Coixxposed., printed and bound In the XJ. S. 


For answering bothersome questions or for helpful sug- 
gestions, the author's sincere thanks go to P, V. H. Weems, 
Lt. Comdr., U. S. Navy (Ret.), Professor Harlan T. Stetson 
of the Massachusetts Institute of Technology, Mr. Alfred F. 
Loomis, Secretary of "Yachting," and Selwyn A. Anderson, 
Master Mariner. 


Preface to Third Edition xiii 

Introduction xv 

Part I: Fundamentals 

1. Astronomical 3 

2. Time 24 

3. The Nautical Almanac 51 

4. Altitudes 55 

5. The Sextant 61 

6. The Compass 69 

7. The Astronomical Triangle 79 

8. Trigonometry 82 

9. Logarithms 92 

Part II: Procedures 

10. Introduction to Position Finding 99 

11. Latitude 103 

12. Longitude and Chronometer Error 1 14 

1 3 . Azimuth and Compass Error 117 

14. Sumner Lines of Position 123 

15. The Saint-Hilaire Method 128 

16. Short-Cut Systems 134 

17. Special Fixes 141 

18. Polar Position Finding 148 

19. Identification 155 

20. Tabular Summary 157 

Part HI: Supplementary 

21. The Sailings, Dead Reckoning, and Current 161 

22. The Day's Work 180 

23. Essential Equipment 182 

24. Practical Points 183 

25. Navigator's Stars and Planets 196 

26. Reference Rules for Book Problems 205 

27. Finding G. C. T. and Date 207 

28. Abbreviations 209 

29. Forms 212 

30. Problems 224 
Selected Bibliography 259 
Index 261 

List of Illustrations 

1. Declination , 8 

2. Excess of One Rotation 13 

3. Diagram for General Orientation facing page 1 3 

4. Phases of the Moon 18 

5. Precession of the Equinoxes 20 

6. The Earth's Orbit 23 

7. The Solar Year 27 

8. Time Diagram 33 

9. Time Diagram 33 

10. Time Diagram 33 

11. Time Diagram 34 

12. Time Diagram 34 

13. Time Diagram 35 

14. Time Diagram 35 

15. Time Diagram 36 

16. Time Diagram 36 

17. Time Diagram 37 

18. Relations Between Zone Time and Local Civil 

Time 40 

19. Zone Time of Mean Sun Noons in Zone 40 

20. Time Frame 44 

21. The "Opposite-the-Sun" Meridian 45 

22. Time Frame and O. S. M. 45 

23. Change of Date 46 

24. Refraction 57 

25. Parallax 58 

26. Sextant 61 


27. Sextant Angles 62 

28. Arc and Vernier Scales 64 

29. Reading the Sextant 66 

30. Compass Card 71 

3 1 . The Astronomical Triangle 79 

32. For Napier's Rules 87 

33. Angles in the Different Quadrants 89 

34. The Four Cases of Latitude from Meridian Alti- 

tude Observation 104 

35. Latitude by Phi Prime, Phi Second 1 1 1 

36. Longitude by Time Sight 1 14 

37. Azimuth ^ ng 

38. Zenith Distance and the Radius of the Circle of 

Equal Altitude 125 

39. The Saint-Hilaire Method 131 

40. Rules for H. O. 211 136 

41. Plane Sailing 163 

42. Middle Latitude Sailing 166 

43. First Current Problem 176 

44. Second Current Problem 178 

45. FixbyH.O.214.D.R. Position 242 

46. Fix by H. O. 2 1 4. Assumed Position 244 

List of Tables 

1. The Solar System 4 

2. The Brightest Heavenly Bodies 7 

3. Declinations of Navigational Stars 17 

4. Time 30 

5. Calculations of t 36 

6. Zone Time 41 

7. Compass Errors 74 

8. Finding Deviation 75 

9. For Magnetic Steer Compass 76 

10. Compass Points and Quarter Points 78 

1 1 . Finding Parts o the Astronomical Triangle 80 

12. Definition of the Trigonometric Functions of 

Plane Right Triangles 83 

13. Equivalents of the Trigonometric Functions of 

One Acute Angle of a Plane Right Tritegl' 

when Hypotenuse = 1 &P 

14. Trigonometric Functions o Any Angle with the 

Sign for Each Quadrant 86 

15. Equivalent Trigonometric Functions of Angles 

in the Different Quadrants 88 

16. Examples of Logarithms 96 

17. For Correcting Polar Position Lines 152 

18. Summary of Methods facing page 157 

19. Rules for Time and Angles 206 

20. For Finding G. C. T. and Date 208 

21. Working Form for Azimuth by H.O. 214 233 

Preface to Third Edition 

PEARL HARBOR and all its consequences have come 
since this Primer's first birthday. Thousands o officer 
candidates have taken up with grim determination the 
study which to me for years has been a delightful hobby. 

In preparing a new edition, the chief problem was 
whether to cut to the bone and leave only the bare essen- 
tials for practical purposes or to retain what seemed, and 
still seems to me, essential for a secure understanding of the 

Pruned down to lines of position only and plotting on 
charts or plotting sheets, it is possible to get along without 
the following: 

Meridian altitudes, reductions to the meridian and ex- 
meridian sights for latitude, interval to noon, time of 
transit, time of sun on the prime vertical, time-sights for 
longitude, sights for determination of chronometer error, 
trigonometry, logarithms, much of the sailings, basic 
astronomy, and sidereal time. (I realize one need not even 
know the meaning of declination in order to use the almanac 
andH. O. 2 14 effectively.) 

The original purpose of the Primer however was to help 
clarify obscurities. It seems to me this can best be done by 
a build-up from simple to complex, and by giving some 
procedures which, though no longer needed, have been 
strong links in the chain leading to present practice. So the 
main framework of this book remains unchanged. 


xiv Preface to Third Edition 

Some omissions and many more additions will, I hope 
increase usefulness. While written primarily for surface 
work, the principles here given are, of course, basic foi 
aerial navigation. No attempt is made to supply detail foi 
the latter, which is so adequately covered in Dutton. 

I am still convinced that years of custom have made it 
more natural and easy to look, in imagination, at this world 
and the universe from above the north rather than from 
below the south pole. Hence I have retained the style of 
diagram in general use until the years 1936-39, when the 
leading texts shifted their viewpoint to face Antarctica. 
Probably the important thing is to get the habit of using 
some time diagram not a particular type. 

In this third edition, about twenty-five minor correc- 
tions, changes or insertions have been made without 
change of page number. An important addition on Si- 
dereal Time has been added on page 249 and the author's 
new "Uniform Method for More Exact Time of Local 
Transit of Any Body' 1 will be found on pages 254-258. 

Having been found physically ineligible for active duty, 
I launch this new Primer with a bold hope that somehow 
its influence, however small, will count in the score of ulti- 
mate Victory. 

J. F. 

February, 1944 
Chicago, Illinois 


K1SCUE AT SEA of human beings from a ship in 
distress is usually the result of a radio message. One 
or more vessels respond by hastening to the locality of the 
trouble. Few of us on shore, reading the press accounts 
of such a rescue, fail to feel thrilled that the genius of 
Marconi has again added to the large total of lives saved in 
this way. But I wonder how often it is realized that many 
centuries of development of the art and science of naviga- 
tion led up to the ability of the master to state the position 
of his ship on this globe within a radius of about one mile. 
Without this ability, there would be little help in wireless. 
The term Navigation covers a number of items. It is 
broadly divided into (1) Geo-navigation and (2) Celo-navi- 
gation. Geo-navigation includes the methods of locating 
the ship's position by earth landmarks or characteristics. 
It is subdivided into (a) Piloting, which has to do with 
bearings, buoys, lighthouses, soundings, radio beams and 
chart study and (b) Dead Reckoning which deals with 
methods of estimating the distance covered and point 
reached in a given interval by means of compass observa- 
tions, log readings, record of engine revolutions and a few 
calculations. Celo-navigation or Celestial Navigation is 
also called Nautical Astronomy. It is the subject concerned 
with position finding away from all landmarks when a 
ship is at sea. This is done by sextant observations of the 
sun or moon or certain planets or stars, with notation of 


xvi Introduction 

exact time of each observation and with the aid of the 
Nautical Almanac and certain tables. 

The professional seaman learns these things as part of 
his job and probably gets very tired of their routine prac- 
tice. The occasional cruising yachtsman or even he who 
seldom leaves land can find the study of navigation a very 
fascinating one and may even let it become an absorbing 
hobby. Such at any rate has been my happy experience. 

Some years ago on Cape Cod a friend who had a sextant 
helped me take an observation of the sun and worked out 
the longitude via Bowditch. I tried to understand the 
various steps but could not. Buying a Bowditch I began 
to study these matters from the ground up. Many more 
books were soon accumulated. A sextant was purchased 
and sights were taken over Lake Michigan and worked 
out, with gradually increasing accuracy. 

Some of the by-products of this activity have been: an 
appreciation of the ingenious powers of spherical trigo- 
nometry; an interest in astronomy with realization for the 
first time that there actually is some reason, for one who 
is not an astronomer, to know certain of the stars; some 
understanding of the various kinds of time-keeping; a 
growing taste for sea stories and sea-lore in general; enjoy- 
ment of that fine magazine, U. S. Naval Institute Pro- 
ceedings, and the popular yachting monthlies; the fun 
of having a paper on a small overlooked point accepted 
and published; correspondence with several men in dif- 
ferent continents as a result of this; the diversion of work- 
ing out text-book problems; glimpses of the history of 
navigation; and, probably most important of all, the 

Introduction xvii 

mental refreshment of studying and doing something 
utterly different from one's professional work. 

The following compilation is limited to off-shore posi- 
tion finding and so omits all consideration of the subject 
of Piloting. It is intended in no way to supplant for the 
beginner such splendid texts as Bowditch or Button but 
rather to smooth the road to those books, to which fre- 
quent reference will be found. I have used new ways in 
presenting some of the old subjects. But my main purpose 
is to prevent certain confusions which come to the ama- 
teur when first attempting this study. The best-known 
books are written by men of much learning who often 
seem to have forgotten certain stumbling blocks which 
they themselves passed long ago. I have done some of my 
stumbling quite recently and am now trying to show how 
such can be prevented. The student who wishes to learn 
a method as quickly as possible and who does not have 
internal distress at following rules blindly and knowing 
nothing of their origins will have no need of this manual. 
But those who enjoy understanding why things are done 
and what they mean, and who like to have all important 
steps included will, I hope, find in these pages a certain 

I : Fuinclamentals 

1. Astronomical 

tain of the so-called heavenly bodies to locate his posi- 
tion. These are the sun, the moon, four planets (Venus, 
Mars, Jupiter, Saturn), and fifty-five stars. 

It is vital for a clear understanding of the uses of these 
bodies to consider their positions in space and to orient 
ourselves. A long story in the history of science lies behind 
our present knowledge. 

The earth was known to be a sphere in the fourth cen- 
tury B. c. 

Eudoxus of Cnidos (about 360 B. c.) taught that the sun, 
moon and planets all moved around the earth, which was 

Aristarchus of Samos (310-230 B. c.) considered that the 
sun and stars were stationary and that the earth revolved 
around the sun. 

Hipparchus (130 B. c.) preferred and developed the geo- 
centric system of Eudoxus. However, he invented plane 
and spherical trigonometry and observed the phenomenon 
known as precession, to be described later. He was the 
first of the Greeks" to divide the circle into 360 degrees 
and attempted to determine the positions of places on the 
earth by measuring their latitude and longitude. 

Ptolemy of Alexandria (127-151 A. D.) expounded the 

4- ^ 

ro W a 

+ 1 + ^ 

rt c* 

. 2 B 3 g 


CL 5? 

Ol vH ^t <O O CS O 



111 1 -f 1 

+ + 




*"H T-H >-H 

* - 


* G 

. s s s 

O oo ijQ *o t-- O\ cs O 


J2 2 

T* >* 

00 vO 







Distance from 
Sun in Millions 
of Miles 

O ^O *- CO i CN CO VO 

co VO O\ "^ co OO OO 
^ CO -^ 00 


OO O\ 





cu ?? 

O VO *O OQ Ir^ O OO O 





xfT rT *>^ Jt*--" ^ vo" o-T 




VD OO *^ 


CO co 


O X> o e "b ct A 




' W) 

& - s c 

3 co _e <u g 
g g t | a3-tj| 

3iSj^,5*S" a rt 
coS>wS<i >c/5 

Uranus .... 
Neptune. . . 



Astronomical 5 

geocentric system so well that it acquired his name and 
was held by scholars for fourteen centuries. 

Copernicus (1473-1543) wrote a great book developing 
the ideas of Aristarchus, then eighteen centuries old, that 
the sun was the center of our system, and stationary, with 
the earth and planets revolving around it. This book was 
published only in time for Copernicus to see a copy on 
his deathbed. 

Tycho Brahe (1546-1601) observed the planetary mo- 
tions with improved instruments and recorded a great 
number over many years. 

Johann Kepler (1571-1630)vafter years of calculations, 
using much of Tycho's data, found the three laws of 
planetary motion that bear his name. 

Galileo Galilei (1564-1642) was the pioneer of modern 
physics. He made the first telescope, discovered the moons 
of Jupiter, and proved the conclusions of Copernicus by 
actual observations. 

Isaac Newton (1642-1727) worked out the laws of motion 
and gravity. His Principia published in 1687 "marks 
perhaps the greatest event in the history of science" (Dam- 
pier- Whetham) . 

An ellipse is a closed curve such that the sum of the 
two distances from any point on 'its circumference to two 
points within, called the foci, is always constant and equal 
to the major axis of the ellipse. 

The solar system consists of a group of bodies all revolv- 
ing around the sun. They go counterclockwise if seen 
from above the north pole, in ellipses not circles with 
the sun at one of the foci of each ellipse. (See Table 1.) 

6 Astronomical 

Perihelion is the point on a planet's orbit nearest the 
sun. It is reached about January 3 for the earth (91,500,000 
miles). The earth moves faster along its orbit when nearer 
the sun. 

Aphelion is the point on a planet's orbit farthest from 
the sun. It is reached about July 3 for the earth (94,500,000 

The so-called "fixed stars" lie far away from our solar 
system, so far in fact that only very slight shifts can be 
found in the positions of the nearer ones when we observe 
them from opposite ends of the earth's orbit. Their own 
actual motions appear negligible. The nearest fixed star is 
Proxima Centauri, 4i/ light years away or 26,000,000,- 
000,000 miles. Deneb, a bright star often used in navigation, 
lies 650 light years away or 3,900,000,000,000,000 miles! 

Brightness of bodies is recorded by numbering the 
apparent magnitude. From minus quantities through zero 
into plus quantities represents diminishing brightness. 
Plus sixth magnitude is just visible to the naked eye. Each 
magnitude is 2i/ times brighter than the next fainter one. 
The brightest heavenly bodies are shown in Table 2. 

The Nautical Almanac gives the necessary data for the 
exact location of 55 of the brightest stars and less full data 
for an additional list of 110 other stars for occasional use. 

The earth rotates on its axis daily as well as revolving 
in its orbit around the sun in a year. The earth's axis of 
rotation is inclined to the plane of the earth's orbit at 
about 66 1/ degrees and the north pole points approxi- 
mately toward the north star or Polaris. The plane of the 

Astronomical 7 

earth's equator is therefore inclined to the plane of the 
orbit by about 23i/ degrees. 






Moon (full) 


j3 Centauri 



- 4 to -3 




3 to +2 




- 2 












to +1 















Rigel Kentaurus 




7 Crucis 


A meridian is an imaginary line on one half of the earth, 
formed by the intersection with the earth's surface of a 
plane which passes through both poles and is therefore per- 
pendicular to the equator. 

The longitude of a place on the earth is the arc of the 
equator intercepted between the place's meridian and the 
Greenwich, England, meridian, being reckoned east or 
west from Greenwich to 180. 

The latitude of a place on the earth is the arc of the 
place's meridian intercepted between the equator and the 
place, being reckoned north or south from the equator 
to 90. 


2T * 

* H 

-J i= 

S I 

'* s =, I 

O C3 

i I 

*-J Cj 

^ 51 o= 
55 r~ a. 

Astronomical 9 

A parallel of latitude is an imaginary circle formed by 
the intersection with the earth's surface of a plane passed 
parallel to the plane of the equator. 

Declination is one of the most important things to un- 
derstand in navigation. It is the angle that a line from 
the center of the earth to a given heavenly body makes 
with the plane of the earth's equator. Remember it has 
nothing to do with where you are on the earth. It is de- 
scribed as north or south in reference to the plane of the 
equator. The declinations of the stars change very little in 
the course of a year. There are great changes however in 
the declinations of the sun, moon, and planets. 

Equinoxes. As the earth travels around the sun from 
winter to summer, the (extended) plane of the equator 
approaches nearer and nearer the sun finally cutting its 
center at an instant known as the Spring or Vernal equi- 
nox on March 21. The plane passes through the sun and 
beyond till the earth turns back in its swing around the 
sun when the process is repeated through the Fall or 
Autumnal equinox on September 23 and so on to win- 
ter. The days and nights are equal at the equinoxes except 
at the poles. The sun's declination is then zero and it 
"rises" in exact east and "sets" in exact west. (See FIG. 1.) 

Solstices. There are two instants in the earth's journey 
around its orbit when the sun attains its highest declina- 
tion north or south. They are designated Summer solstice 
and Winter solstice according to the season in the north- 
ern hemisphere, and occur June 21 and December 22, 

Celestial Sphere. Declination alone is not enough to 
locate a body in the sky. We require some means of regis- 

10 Astronomical 

tering its east or west position as well as its north or south. 
For this purpose we imagine a great hollow "celestial" 
sphere to lie outside our universe with the earth at its 
center. The heavenly bodies can be projected onto the 
inner surface of this sphere as can also the plane of the 
earth's equator, or any meridian or point on the earth as 
though observed from the earth's center. Likewise the posi- 
tion of the sun's center at the time of the Vernal equinox 
on March 21 as seen from the earth can be projected. This 
point is also called the Vernal equinox or ''First Point of 
Aries." Its symbol isT. It is taken as the zero of measure- 
ment around the celestial equator or equinoctial which is 
the projection of the earth's equator. The extension of 
the plane of the earth's orbit to the celestial sphere pro- 
duces a circle on the latter known as the ecliptic. The 
two points where the equinoctial and ecliptic intersect 
mark the two equinoxes. The angle of about 23 1/ of 
these intersections measures the obliquity of the ecliptic. 
Right Ascension is a measure of angular distance around 
the celestial equator, eastward from T . It is expressed in 
hours (and minutes and seconds) up to 24. By giving a 
body's declination and right ascension, we pin it down to 
a definite location on the celestial sphere just as a place on 
earth is fixed by giving its latitude and longitude. The 
declination corresponds to the parallel of latitude and 
the right ascension to the meridian of longitude. We will 
see later that the R. A. of the projected local meridian 
equals the local sidereal time and 24 hours of sidereal 
time measure practically one exact rotation of the earth. 

Astronomical 1 1 

Translating time to arc we have: 

24 h = 360 

1 h = 15 

4m= 1 

1 m = 15' 

4 s = 1' 

1 s = 15" 

Speaking in terms of apparent motion, we could say a 
body's R. A. is the distance (angle or hour) at which it is 
trailing the T in the latter's journey around the equi- 

The importance of correctly orienting ourselves in rela- 
tion to solar system and stars and their real and apparent 
motions and our real motion cannot be overstated. Much 
of the bewilderment of the beginner comes from encoun- 
tering emphasis on apparent motion with inadequate ex- 
planation of the real situation. For instance, "the sun's 
path among the stars" is a most confusing expression. It 
would be plain if we could see stars in the daytime and 
would compare the sun's position at a certain time on two 
successive days. Then we would observe the shift to the 
eastward which is meant. It is just as though we were in a 
train going forward and, looking out a window on the 
left side, observed a tree 200 yards away and a bit of 
woods 2,000 yards away and immediately shut our eyes 
(passage of 24 h) and quickly looked again. The nearby 
tree (sun) then seems to have moved to the left in relation 

12 Astronomical 

to the distant trees (stars). Another expression, the "re- 
volving dome" of the celestial sphere, is also misleading. 
Remember our earth is rotating to the east. The celestial 
sphere therefore only seems to rotate to the west. In north 
latitudes we see the north star as though it were a pin or 
hub on the inside of a sphere which turns around it counter- 
clockwise. Bodies are swinging over it to the left and 
returning under it to the right. 

Another extremely important fact to realize is this: the 
earth because of its progress around its orbit must make 
a little more than one exact rotation between two succes- 
sive noons. From Monday, at the instant the sun is due 
south in the northern hemisphere, till Tuesday, when the 
sun is again due south, there has been something over 
one complete rotation of the earth. It is incorrect to ex- 
plain this by saying the sun has shifted somewhat to the 
east. The truth is that the progress of the earth in its orbit 
at a speed of 30 kilometers per second has altered the 
direction of the earth from the sun. This makes the sun 
seen from the earth appear to have shifted eastward. This 
will explain why a true sun day is a longer bit of time 
than a star day which requires practically only one exact 
rotation. (See FIG. 2.) Owing to the eccentricity of the 
earth's orbit and obliquity of the ecliptic these sun days 
are not exactly equal. 

At this point the student may profit by inspection of 
Figure 3, a diagram designed to make more clear some of 
the matters so far discussed. The following explanation of 
the diagram should be carefully studied. 





FIG. 2. Excess of One Rotation. 

Earth's gain of V* of a rotation when 14 around Orbit. Line on Earth = 
Meridian of some place, say Greenwich, at Noon in each instance. 

Explanation o Diagram 

Outer circumference represents the celestial sphere in the 
plane of its equator seen from north. 

p at top is "First Point of Aries" from which right ascension 
is counted in a circle divided into 24 hours, numbered counter- 

14 Astronomical 

clockwise. These are shown in the 2nd band. Right ascension 
on the celestial sphere is similar to longitude on the earth. 
Decimation on the celestial sphere corresponds to latitude on 
the earth but cannot be shown here in the one plane. 

Fifty-five navigational stars including Polaris whose right 
ascensions and declinations change very little are shown in 
the outer band in approximately correct positions. Those with 
a minus sign added are o southern declination and lie below 
(behind) the diagram while all others are of northern declina- 
tion above (in front of) the diagram. These are the stars for 
which complete data are given in the Nautical Almanac and 
which are most easily used for position finding. 

The 12 "Signs of the Zodiac" are included in the 3rd band 
for popular interest only. They are named from various con- 
stellations of stars which lie in a belt not over 8 above and 
below the plane of the earth's orbit, or the ecliptic on the 
celestial sphere. Because of the slow swing of the north end of 
the earth's axis in a circle around the north pole of'the ecliptic, 
clockwise if seen from above, once in 26,000 years, the equi- 
noxes (points) gradually shift to the westward. Hence the Spring 
or Vernal equinox (T)> which when named about 2,100 yearr, 
ago was actually in the sign of Aries, is now almost through 
Pisces. Note that this point T on the celestial sphere is where 
the sun seems to be when seen from the earth on the 21st ol 
March. At that time (also known as the Spring or Vernal equi 
nox) the sun's declination is changing from south to north 
Dotted lines show this date and also the dates of the Fall equi- 
nox, the Summer solstice and the Winter solstice. 

The 4th band of months is for locating the earth in its orbit. 

The five circles outside the sun as labeled are for the orbits 
of the earth and the four planets used in navigation, with their 
periods of rotation, all counterclockwise. 

The "spokes" are to show two-hourly intervals of R. A. 


T t 


Astronomical 15 

How to Use Diagram 

Make a dot on earth's orbit corresponding to the date. 

Look in N. A. for R. A. of each of the planets for the same 
date and make a dot on each orbital circle at the given R. A. 

Look in N. A. for moon's R. A. for the same date and time 
and make a dot near earth corresponding to this. 

Turn diagram till sun and earth are in a line "across the 
page" with sun on the right. 

Lay a ruler across diagram edge up so edge passes through 
sun and earth. 

All above ruler edge represents stars and planets available 
around evening twilight of day in question at equator. 

Holding diagram stationary, rotate ruler edge counter- 
clockwise with earth as center, through 90. All to left of ruler 
edge represents sky of midnight at equator. 

Similarly rotate ruler edge counterclockwise with earth as 
center through another 90. All below ruler edge represents 
sky of twilight next morning at equator. 

If position of observer were at one of the poles, all stars and 
planets with the corresponding declination would be visible 
during the six months of darkness while those of opposite 
decimation would be invisible. 

As position of observer increases in latitude from equator 
to, say, the north pole, the visible stars of southern declination 
will diminish in number, beginning with the highest declina- 
tions, while the visible stars of northern declination will in- 
crease in number, also beginning with the highest declinations. 

Note: This diagram is of course not to scale and only an approximation. 

Remember the planet dots do not represent exact positions of planets 
in their orbits but rather the planets' positions on the celestial sphere as 
seen from earth and expressed in R. A. The earth's journey around its 
orbit accordingly results in apparent retrograde movement of planets at 
times. (Jupiter, 1937: Jan. 1, R. A. 18* 29**; May 1, R. A. 19^ 56m; Octo- 
ber 1, R. A. 19*> 18m; December 1, R. A. 19* 52m.) 

16 Astronomical 

The orbits are in reality ellipses with the sun at one focus and not circles 
as here given. 

The plane of the earth's orbit, near which the Zodiac signs are 
grouped, and the plane of the earth's equator, along which the R. A. 
hours are measured, actually lie 2$V2 degrees apart but are here supposed 
to be compressed into the plane of the diagram. Likewise the planes of 
the planets* orbits are leveled into the diagram. The months are here 
marked as though of uniform length. Periods of rotation are approximate. 

In order to show how far above and below the previous 
diagram the various stars lie, a list of declinations is pro- 
vided. It shows the order the stars would be met with from 
the north pole to the south pole of the celestial sphere 
(See Table 3.) 

A chart in the back of the Nautical Almanac combines 
data from the preceding diagram and list (R. A. and Dec.) 
in one plane. It is of the Mercator type (which will be 
explained later) and so is somewhat confusing to the 

A celestial globe is a great help in learning star loca- 
tions, constellations, etc. The only difficulty is that it shows 
the celestial sphere from without and one must always 
imagine a given group as seen from the center of the globe 
in order to duplicate the actual group in the sky. 

The moon's motion and "phases" deserve some atten- 
tion here. The moon is about 238,840 miles away. It 
rotates on its axis only once, counterclockwise from above, 
in its trip of revolution around the earth; hence it always 
keeps the same face toward us. It makes a complete revo- 
lution judged by its relation to stars in 27 1/3 days but to 
completely circle the earth (which is traveling in its orbit) 
requires 29 1/ days. It is cold and only shines by light 
from the sun. It revolves around the earth counterclock- 



wise seen from above the north pole, or from west to east. 
As this motion is so much slower than the earth's daily 
rotation, the moon seems to be going each night from east 
to west. Each successive night, however, at a given time, 
its position is about 12i^ farther east. After reaching 































Deneb Kaitos 
























Kaus Australis 




6 Centauri 












Al Suhail 




Al Na'ir 








y Crucis 




a Pavonis 








s Argus 




(3 Crucis 




Rigil Kentaurus 








a Tri. Australis 





"full" it "rises" about 50 minutes later each evening. Pre- 
vious to full it may be seen as early as mid-afternoon and 
subsequent to full it may be seen up to several hours after 
sunrise or even till noon. 

As the plane of the moon's orbit is near the plane of the 
earth's, the moon will appear to observers in the northern 
hemisphere lower in the southern sky in summer and higher 

<t i> 







FIG. 4. Phases of the Moon. 

Diagram is from above North Pole of Earth. Outer row shows how each 
position appears in our North Latitudes. The small projection from the 
Moon is a fixed point and shows that the same face remains toward the Earth. 

Astronomical 19 

in winter. This will be understood if Figure 1 is now re- 
viewed. Figure 4 will explain the moon's phases. 

Miscellaneous Facts 

Precession of the Equinoxes. Each year the equinoxes 
(Spring and Fall) come about 20 minutes sooner. An equi- 
nox can either be thought of as a certain instant in the 
earth's journey around its orbit>when the plane of the 
equator cuts the center of the sun, or as the point on the 
celestial sphere where the sun appears to be at that time, as it 
is changing from S. to N. or from N. to S. declination. 
These points are shifting to the west, or in a direction op- 
posite to the earth's orbital motion. They do so about 50" 
of arc on the celestial sphere per year. 

This all happens because the extended ends of the earth's 
axis are very slowly making circular motions around the 
poles of the ecliptic. (Disregard the ellipses which the axis 
makes in one year because at the distance of the celestial 
sphere these ellipses would be extremely small.) The projec- 
tion of the earth's north pole describes a circle with a radius 
of about 2 3 1/ around the north pole of the ecliptic, coun- 
terclockwise as we look up at it, clockwise as seen from out- 
side celestial sphere looking down, once in 26,000 years. The 
plane of the earth's equator, perpendicular to the axis, must 
likewise shift and, when meeting the sun's center at equi- 
nox, will be intersecting the track of the earth's orbit on 
each side at a point slightly more westward. This may be 
represented (see FIG. 5) by a metal circular ring (plane of 
earth's orbit) with a disc (plane of earth's equator) slightly 

FIG. 5. Precession of the Equinoxes. 

Astronomical 21 

smaller inside the ring and loosely attached at two opposite 
points (equinoxes) and inclined 23i/ to the ring (obliquity 
of ecliptic). A rod projects up from the disc's center (parallel 
to earth's axis). Shifting the attachments clockwise when 
looking down at disc will show rod's tip describing a circle 
clockwise around a point above center of ring (north pole 
of ecliptic). The rod is placed in center instead of at edge 
by earth's orbit because it shows the effect more clearly and 
because the misplacement is negligible when considering 
the extreme distance of the celestial north pole. 

The Vernal equinox (T) called "First Point of Aries" 
was in that constellation when named about 2,100 years 
ago but has now shifted westward almost through the next 
constellation "Pisces." 

As a result of this performance, there is a succession of 
stars called north stars through the centuries. Our present 
one, Polaris, is therefore only playing a temporary role, 
The series is as follows: 

Vega 12,000s. c. 
fi Hercules 7,200 

Thuban 3,000 

Polaris 2, 100 A. D. 

Er Rai 4,200 

Alderamin 7,500 

SCygni 11,500 

Vega 14,000 

Another result of precession is that there is a slow in- 
crease in the right ascensions of all stars. Naturally, if the 

22 Astronomical 

starting point for measurement (T) is moving west and the 
measurements are made to the east, these measurements 
will grow larger. 

The north pole of the ecliptic is at R. A, 18, Dec. 661^ 
N. and the south pole at R. A. 6, Dec. 66i/ S. 

It may be wondered why the hottest part of northern 
summer is not halfway between Spring and Fall equinox 
at June 21 and the coldest part of winter at December 22. 
The explanation is probably that extra time is required to 
warm up the earth in summer and to cool it off as winter 

Earth's perihelion, the position nearest sun, occurring in 
northern winter and southern summer, and aphelion, the 
position farthest from sun, occurring in northern summer- 
and southern winter, might lead one to expect the southern 
hemisphere to show more extremes of climate, hotter in 
summer and colder in winter. However, the eccentricity 
of the orbit is so slight that no great difference is noted. 
The eccentricity of an ellipse is expressed by the following 

Distance from center to one focus 

Distance from center to one end of major axis' 

For the earth's orbit, this is only about y$Q, so the orbit 
is not far from circular. Nevertheless, it is 186 days from 
Spring to Fall equinox, and only 179 days from Fall to 
Spring equinox. The sun therefore is in that focus which is 
nearer to us in December and farther from us in June. 
(See FIG. 6.) 

All the planet's orbits lie within 8 of the ecliptic and the 
four navigational planets lie within 3. 

Prof. Dayton C. Miller reported (Science, June 16, 1933) 

Astronomical 23 

after very exhaustive observations on the speed of light, that 
the entire solar system was moving as a body through space 
at a speed of 208 kilometers per second toward a point in 
R. A. 4 h 56 m , Dec. 70 30' S. This is close to the south pole 
of the ecliptic and about 20 south of the second brightest 
star, Canopus. 


DLC. 2.Z 


FIG. 6. The Earth's Orbit. From Above. 
(True eccentricity is even less than here.) 

Kepler's Laws are as follows: 

1. The orbit of every planet is an ellipse, having the sun 
at one of its foci. 

2. If a line is supposed to be drawn from the sun to any 
planet, this line passes over equal areas in equal times. 

3. The squares of the times of revolution about the sun 
of any two planets are proportional to the cubes of their 
mean distances from the sun. 

2 a Time 

THE SUBJECT OF TIME is one of the tough spots in 
the study o navigation. This is partly because new 
ways of thinking about time are required and partly because 
the explanations are often inadequate. We will try now to 
proceed carefully and put in all the steps so that there may 
be no misunderstanding. 

Apparent Time. We think of time as somehow measured 
by the apparent movement of the sun each 24 hour day 
around the earth, caused of course by the earth's rotation 
with the sun stationary, 

We have seen in Chapter 1 that, while the earth rotates 
on its axis at a uniform speed, it does not travel in its orbit 
around the sun at a uniform speed. Its speed is faster when 
closer to the sun in the elliptical orbit. We also saw that, 
because of progress along its orbit, the earth must make 
somewhat more than a complete rotation on its axis in order 
to bring the sun from a meridian one day back to the same 
meridian the next day. Finally, we noted the obliquity of 
the ecliptic. 

So days measured by sun noons are not uniform because: 

(a) The varying speed of the earth in its orbit varies 
the necessary daily excess over one revolution, and, 

(b) As the plane of the earth's orbit in which the sun 
appears to move, does not coincide with the plane of the 
earth's equator along which movement is measured, equal 
divisions of the former do not make equal divisions of the 
latter when projected onto it. 


Time 25 

No clock can be made to keep step with this true solar 
or, as it is called in navigation, apparent time. Its day be- 
gins at midnight. Greenwich (England) apparent time and 
local apparent time are designated respectively G. A. T. 
and L, A. T. 

Mean Time. An imaginary or "mean" sun is therefore 
utilized which crosses a given meridian at uniform intervals 
throughout the year. The interval is the average of the 
true sun days. The mean sun is thought of as in the plane 
of the earth's equator all the time. The total of days in 
the year is the same as for apparent time. Days are divided 
into 24 hours of 60 minutes each and each minute into 
60 seconds. This is the time our regular clocks keep. In 
navigation, the hours are numbered from to 24. Ship 
chronometers are set to keep Greenwich time and almanacs 
give data for this. It is called Greenwich civil (or Green- 
wich mean) time (G. C. T. or G. M. T.). The civil time 
day begins at midnight. Local civil time is designated 
L. C. T. 

The Equation of Time is the amount (up to 16 minutes)* 
varying from minute to minute of the day and through 
the year, which must be added to or subtracted from appar- 
ent time to give civil time and vice versa. The N. A. gives 
it for every 2 hours with plus or minus sign indicating 
procedure for converting civil to apparent time. 

Transit of a body occurs the instant its point in the celes- 
tial sphere is on the meridian of the observer or on the 
meridian 180 away. When the transit is over the meridian 
which contains the zenith, it is designated as upper; when 
over the meridian 180 away, as lower. 

26 Time 

Sidereal or Star Time. This is measured by the apparent 
daily motion of the Spring equinox or "First Point of 
Aries" (T) around the earth. As this point is not marked by 
any heavenly body, we are dependent on the astronomers 
to calculate its position and publish in almanacs the right 
ascensions of bodies measured eastward from it. 

"The equinox itself cannot be observed, being merely 
the intersection of two abstract lines upon the sky; but by a 
very long chain of observations, going back continuously 
to Hipparchus 150 B. c., and earlier, the relative spacing 
on the sky of all the lucid stars, and a great many more, 
has been determined with continually increasing accuracy, 
together with the 'proper motion 7 belonging to each. Rela- 
tive to this mass of material, the position of the equator and 
ecliptic are assigned, or what comes to the same thing, the 
coordinates of each star are given relative to the equinox 
and equator." (Encyclopedia Britannica, 14th ed., Vol. 22, 
p. 227, "Time Measurement/') 

A further difficulty comes from the constant slow west- 
ward shift or precession of the equinox. This is opposite 
to the movement of the earth in its orbit and amounts to 
50" of arc on the celestial sphere per year, and shortens the 
year by about 20 minutes. (For this we do not use the table 
given in Chapter 1 under Right Ascension which matches 
24 hours with 360 because here we are figuring on a whole 
year for 360.)* 

The year, then, from one equinox and back to the same 

* Now 360 = 1,296,000 seconds of arc And 365 days = 525,600 minutes 
of time. So to show the time which the year loses by this 50" shift: 
50" : 1,296,000" :: Xm : 525,600* 
1,296,000 X = 26,280,000 
X = 20m -K 



is 20 minutes shorter than the year from a certain position 
relative to a "fixed" star and back to the same. These two 
kinds of years are known respectively as Solar or Tropical 
(equinox to same) and Sidereal (star to same). 


FIG. 7. The Solar Year. 

The Solar Year equals an incomplete revolution of the Earth because of 
the westward shift of the Vernal Equinox (<][>). 

The solar year has: 365.2422 civil days, or 366.2422 side- 
real days. 

The sidereal year has: 365.25636 civil days, or 366.25636 
sidereal days. 

The solar year is the more satisfactory for calendar pur- 
poses since it keeps the seasons in a constant relation to 
dates. It is the one used in general and in the N. A. 

The .2422 fractional day of the solar year (just under 14) 

28 Time 

is what leads to our leap year system. To add a day to the 
calendar every 4 years gradually amounts to too much. So 
every year, whose number is divisible by 4, is a leap year, 
excepting the last year of each century (1900, for example) 
which is' a leap year only when the number of the century 
is divisible by 4. This keeps the calendar in almost perfect 

The matter of the two kinds of year is brought up to 
help prevent confusion which may arise from the fact that 
Bowditch (1938, p. 146) refers only to the sidereal year, and 
Button (7th ed. 1942, p. 248) only to the solar or tropical 

Remember, then, that in navigation, whether using civil 
or sidereal time, it is always part of a solar year not of a 
sidereal. (See FIG. 7.) 

The sidereal day is almost exactly the time of one com- 
plete rotation of the earth on its axis.* These days are 
practically uniform. The sidereal day at any meridian begins 
with the transit of the "First Point of Aries'* (T) over that 
meridian. The date is usually not used but sidereal time is 
spoken of as of a certain date and hour of G. C. T. 

A sidereal day is 23 h 56 m 4 s . 1 of civil time or 3 m 55 8 .9 
(civil) shorter than a civil day. 

*To find actual difference between the sidereal day and a complete 
rotation day: 

Precession of <> in 1 year = 50" of arc 

Precession of <p in 1 sidereal day = = .137" of arc 

Now 15" arc = 1 second time for earth's movement, so 

.137" : 15" : : Xs : Is 

15X = .137 

X = .009 seconds time = the amount by which 1 sidereal day, 
between transits of "^p, is shorter than 1 complete rotation day. 

Time 29 

A civil day is 24 h 3 m 56 S .6 of sidereal time or 3 m 56 S .6 
(sidereal) longer than a sidereal day. 

It will be seen that the difference between the shorter 
sidereal day and the longer civil day is 3 m 55 s -9 in civil or 
3 m 56 S .6 in siderealthe sidereal quantity being larger by 
.7 sec. 

The actual lapse of time of this difference is the same, 
however expressed. But the sidereal units days, hours, 
minutes, seconds are all shorter than similar civil units, 
so that for any time interval below the year, measurement 
in sidereal units will always show a larger quantity than 
measurement in civil units. 

Remember there are two senses in which both civil and 
sidereal quantities may be interpreted, as follows: 

Civil (a) Angular distance of mean sun from lower 
transit; and (b) A lapse of time since mean sun made lower 

Sidereal (a) Angular distance of T from upper transit; 
and (b) A lapse of time since T made upper transit. 

Now civil and sidereal time in the "a" sense are in similar 
units and can be combined without conversion. 

But civil and sidereal time in the "b" sense are expres- 
sions of duration in different units and cannot be combined 
to find sum or difference without converting one into terms 
of the other. (N. A. supplies tables to use in converting 
either way.) 

Of course an "a" sense of one kind cannot be combined 
with a "b" sense of the other. 

Apparent Time is found in navigation by sextant obser- 
vation of the sun and calculation therefrom of its hour 






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Time 31 

angle (to be explained later) or by conversion of civil time 
by the equation of time. 

Civil Time is not directly observable but is obtained by 
conversion of either apparent or sidereal. Practically it is 
taken from the ship's chronometer, corrected by the known 
rate of change of the chronometer or by radio time signals. 

Sidereal Time is found by sextant observation of moon, 
planet or star, and calculation therefrom of the body's hour 
angle which is then combined with its R. A. obtained from 
the N. A. It may also be obtained by conversion of civil and 
sometimes is taken from a sidereal chronometer. 

A rough estimate of local sidereal time (L. S. T.) may 
be made as follows: Imagine a line from the pole star drawn 
through fi Cassiopeiae (Caph) and a Andromedae (Alphe- 
ratz) to the celestial equator. (See Chap. 25.) It will cut the 
equator approximately at the Vernal equinox. When this 
line is on the meridian through zenith the L. S. T. is O h . 
An estimate of its swing around the pole star counterclock- 
wise may easily be made by dividing a circle around the 
pole star into quarters (6 hours each) and these quarters 
into thirds (2 hours each). 

We must now consider the definitions of some terms that 
are in very frequent use. 

A great circle of any sphere is one formed by the inter- 
section on the sphere's surface of a plane which passes 
through the sphere's center. The earth's equator, and any 
meridian (with its opposite one) which passes through both 
poles, are examples. The shortest distance, on the surface, 
between any two points on a sphere is always part of a 

32 Time 

great circle. The equinoctial is the great circle on the 
celestial sphere produced by intersection of the plane of the 
earth's equator. The ecliptic is another. 

A small circle of a sphere is one described by the inter- 
section on the sphere's surface of a plane which passes 
through the sphere but not through its center. All the 
parallels of latitude above and below the equator are small 
circles. Any circle on a sphere which includes less than half 
the sphere is a small circle. 

An hour circle is a great circle on the celestial sphere 
passing through the poles and some point in question such 
as the projection of Greenwich or the projection of a 
heavenly body, as seen from the earth's center. 

The hour angle of a body is the angle at the pole of the 
celestial sphere between the hour circle of the body and 
the celestial meridian of the observer. It is also measured 
by the arc of the celestial equator between the hour circle 
and the celestial meridian. It is either reckoned positively 
to the west all around to 360 or 24 hours, or, if over 180 
west, is subtracted from 360 and designated east. 

Greenwich hour angle is designated G. H. A. and local 
hour angle L. H. A. It is becoming customary to limit the 
use of L* H. A. to any westward measurement and to use t 
for an eastward or westward angle below 180, which is 
then known as a meridian angle. 

Table 4 may now be studied to review the three kinds 
of time. 

Diagrams are most useful in visualizing problems in time 
or in position finding. It is much easier to record apparent 
movements of heavenly bodies on the edge of a circle clock- 


Figure 8. A circle represents the celestial sphere in the plane of its equator 
and seen from above its north pole. A dot in the center represents earth. 
(We use a similar dot and circle to represent the earth and the plane of a 
given meridian of the C. S. when working latitude problems.) The circle, of 
course, stands for 360 or 24 hours. A line is drawn to the bottom part to 
indicate a projection of part of the meridian of the observer and always 
labeled M. 

Figure 9. Another line is drawn as a projection of the meridian of Green- 
wich. It is placed approximately according to the supposed longitude and 
labeled G. Here it shows observer at about longitude 60 W. 

FIG. 10 

Figure 10- Other lines represent hour circles of heavenly bodies as indicated 
with symbols. One of them is filled in after an observation has been made 
and the local hour angle has been calculated, as will be explained later. We 
may here recall once more that M, G, and the central dot are really in 
counterclockwise motion while any line for a heavenly body is in apparent 
clockwise motion. 


wise, with stationary earth in center, than to record the 
actual rotation of the earth at the center of the circle coun- 
terclockwise. A simple convention has therefore been 
worked out. It may be done free-hand and only approxi- 
mately but will show relationships clearly if we remember 
that the clockwise movement is only apparent. 

Examples of the chief uses of such diagrams will be seen 
in Figures 8 to 17. 

Figure II. The "First Point of Aries" (<f) or starting point of sidereal time 
may be similarly drawn in problems involving moon, planet or star. The 
Greenwich sidereal time would have to be first calculated. The N. A. gives 
it for Ok G, C. T. of any date. Adding to this the actual G. C. T., and also 
a small amount from a table representing excess of sidereal over civil 
during that civil interval, gives G. S. T. 

Figure 12 shows G. C. T., G. S. T., and R. A. of a star. Dotted line repre- 
sents opposite meridian needed for start of G. C, T. Curved lines with 
arrows show amounts approximately as follows: 

G. C, T. 
G. S. T. 

Long. W. 

(2 P. M.) 

60* (4 h.) 

FIG. 13 

Figure 13. Sidereal Time of 0& G. C. T. is the same as R. A. of the Mean 
Sun + 12* ( 24 if over 24) at 0* G. C. T. This latter expression is always 
abbreviated to R. A. M. Q + 12. The two cases of G. S. T. given here will 
show this identity. The one on the right being over 24 h. requires a sub- 
traction of 24. 

Long. = 90 W. 


18ii (day before) 
20^ (day before) 



+ corr. for 3 h 

(day before) 

+ corr. for 3 h 

FIG. 14 

14 shows two diagrams and the data they represent. The student 


Monday noon with 
L.C.T. 12 

Tuesday Sidereal noon 


L.C.T. lib 56m 4 .1 

Tuesday civil noon 
L.S.T. Oh 3*a 56s .6 
L.C.T. 12 

FK;. 15 

Figure 15, not to scale, is to illustrate the time difference between sidereal 
and civil days. 

The local hour angle (L. H. A. or f) 
is the starting point of the actual navi- 
gational calculations for position. (See 
FIG. 16.) Whatever system is used from 
t on, the calculation of t must first be 
done. Table 5 shows how the N. A.G 
method giving G. H. A. in arc has 
shortened it. 





G. C. T. 

G. C. T. 

G. C. T. 


R. A. M. -f 12 

G. H. A. (arc) 

G. A. T. 


Long. D. R. 

G. H. A. (time) 

G. S. T. 

L. H. A. or t 

G. H. A, (arc) 

R. A. 

Long. D. R. 

G. H. A. (time) 

L. H. A. or t 

G. H. A. (arc) 
Long. D. R. 

L. H. A. or t 

Time 37 

The arithmetic from W* through G. C. T. to t probably 
offers more inducements for errors than the actual calcula- 
tion which follows t. The student is cautioned to give this 
preliminary portion of all problems his most careful atten- 

Apparent time may be described as the hour angle of 
the true sun westward from the observer's meridian +12 
(24 if over 24). This statement may sound a bit obscure 
but its truth can be realized by an examination of the two 
cases shown in Figure 17. The one on the left is obvious. 
The one on the right requires a subtraction of 24 which 
leaves the arc MX. This is of course equal to the arc FO 
which is the required time. 

FIG. 17 

Civil time is found by a similar rule for the mean sun. 

Local apparent time is ordinarily found as follows: If 

body is east of observer's meridian, subtract local hour angle 

* Watch time. 

38 Time 

(in time) from 12. If body is west of observer's meridian, 
add 12 to local hour angle (in time). 

Today practical navigation can ignore sidereal time. 
The N. A. gives data in terms of G. H. A. in arc sufficient 
for all ordinary purposes and even for star identification. 
This was only begun in the 1934 issue and is not preferred 
by all as yet. But to try to grasp the scheme of things in 
Nautical Astronomy without understanding something of 
sidereal time, is unwise. 

For example, the new American Air Almanac makes use 
of two expressions of sidereal origin: G. H. A. T, which, 
of course, is the same as G. S. T., and S. H. A., which means 
Sidereal Hour Angle and is the body's angle westward from 
T. This is the same as 24 h. or 360 - R. A. of the body. The 
sum of G. H. A. T + a correction + S. H. A. * (-24 h, or 
360 if over 24 h. or 360) = G. H. A. *. 

Remember the following relationships: 

G.HA. combined with L.H.A. gives longitude in degrees 
G.A.T. " " L.A.T. " " 4< time 

G.C.T, " " L.CT, " " " " 

G.S.T. " " L.S.T. " " " " 

G.H.A, " " longitude in degrees gives L.H.A. 

G.A.T. " " " " time gives L.A.T. 

G.C.T. " " " " " " L.C.T. 

G.S.T. " " " " " " L.S.T. 

G.H-A. (in time) combined with R. A. gives G.S.T. 
G.S.T. combined with R.A. gives G.HA. (in time) 
L.H.A. (in time) combined with R. A. gives L.S.T. 
L,S.T. combined with R.A. gives L.H.A. (in time) 

Time 39 

Standard or Zone Time is based on the L. C. T. of me- 
ridians at 15 intervals from Greenwich. For convenience 
of railways and everyday affairs, a standard time zone ex- 
tends 7.5 each side of these meridians in which the time 
of the meridian is used. This system has been extended 
over the oceans and is used by the navies of the United 
States, Great Britain, France and Italy. The Greenwich 
zone is called Zero Zone. Each other zone is numbered from 
1 to 12 according to the hourly difference from Greenwich. 
East zones are called minus zones since in each of them 
the zone number must be subtracted from the standard 
time to obtain the G. C. T. Conversely, west zones are called 
plus. The twelfth zone is divided medially by the 180th 
meridian and the terms "minus** and "plus" are used in 
the halves of this zone which lie in east longitude and west 
longitude, respectively. (See Table 6.) These zone boun- 
daries are modified in the vicinity of land for special condi- 
tions and circumstances. Instead of adjusting the ship's 
time to apparent time at noon each day, the clock is adjusted 
to the standard time of the successive zones as they are 
entered, the change invariably being exactly one hour. 
When it is desired to obtain zone time from G. C. T., the 
sign of the zone in question must be reversed and the result 
applied to the G. C. T. Zone time has simplified the work 
of the navigator in many ways. His watch is usually kept 
on it. 

Certain relations between Z. T. and L. C. T. may confuse 
one at first. A study of Figures 1 8 and 1 9 should clear up this 



ZONE * 7 

ZONE. + 6 ZONL + 5 


9* 9 



T 9 

6' 9 

y 9 

f 9 

3' 9 

2* 9 

I* 9 

O r 8 

9' & 

8' 8 

r s 

6' 8 

y 8 

4' 6 

y Q 

Z' 8 


s \. 










1 I 

a i 

a i 

2 I 

Z 1 

a i 

2 1 

Z 1 

I I 

Z \ 

2 1 

2 1 

z \ 

2. \ 

Z 1 









52 11 

36 II 

40 II 

44 II 

48 II 

52 II 

56 12 

00 12 

04 12 

08 IE 

12 12 

16 12 

20 12 

Z* 12 

28 12 



FIG. 18. Relations between zone time and local civil time. 
The figure represents one instant only. 

Mean sun is over 90th meridian W. Longitude. 

Zone -f 6 time is 12 noon. 

All other meridians in Zone + 6 have zone time 12. 

Zone + 5 is one hour more. 

Zone + 7 is one hour less. 

L.C.T. of 90th meridian is 12 noon. 

L.C.T. of meridians to east increases 4 minutes for each. (Fast of Z.T.) 

L.C.T, of meridians to west decreases 4 minutes for each. (Slow of Z.T.) 

ZONE. * 7 ; ZONC * 6 

ZorsL + 5 


9- 9 





7" 9 


6' 9 


3" 9 


r 9 


6* 9 

t 9 

r 9 

0* 6 

9- 8 

a- e 


r e 


6' 8 


5' 6 


4' 8 




' 8 











Z& 12 

34 12 


16 e 

12. IZ 

06 12 

04 12 

00 11 


52 II 

46 11 

44 1) 

40 II 

36 It 

*i E 






oo le 

00 \l 

oo ia 

00 IZ 

oo is 

00 12 

00 12 

oo ia 

00 12 

00 12 

00 12 

00 12 

00 12 

00 12 

00 12 

00 12 




FIG. 19. Zone time of mean sun noons in zone. 

The figure represents 17 different instants. It shows at what zone time the 
mean sun will be over each meridian. 




Zero Zone: Long. 7^ W.-71/5 E. 







+ 1 

7i/ 2 - 221/2 

- 1 

+ 2 

221/2 37i4 

- 2 

+ 3 

371/2- 521/2 


+ 4 

521/2 6714 


+ 5 . 

671/2- szy z 

- 5 

+ 6 

821/2- 97i/> 

- 6 

+ 7 



+ 8 



+ 9 


- 9 





157i/ 2 _172i4 





It is possible through daily radio signals to keep a second- 
setting watch correct for G. C. T. However, many ships 
still have no radio and use only the standard chronometer 
which gains or loses at a certain rate. Most of the textbook 
problems in navigation start the data with the observer's 
watch time and proceed as follows: 

W (Watch) 

+C W (Chronometer minus Watch; add 12 h. to 
C. if necessary) 

C F (Chronometer Face) 

C C (Chronometer Correction) 

G C T (Greenwich Civil Time) 

42 Time 

A newer form of recording the data for G. C. T. is as 



(Watch Error) 

(Zone Time) 
(Zone Description) 

(Greenwich Civil Time) 


G. C. T. 

Sights had best be taken with a stop-watch unless an 
assistant is available to note time when observer calls for 
it. Taking a sight with a stop-watch in the left hand is simple 
and the stem may be punched at the instant desired. The 
stop-watch can then be taken to the chronometer and 
stopped when the latter reads some even minute. Subtract- 
ing the stop-watch total gives the chronometer reading at 
time of sight. This can be abbreviated as follows: 

C. 5. (Chronometer at stop) 

R. W. (Ran stop-watch) 

C. O. (Chronometer at observation) 

C. C. (Chronometer correction) 

G, C. T. (Greenwich Civil Time) 

Greenwich Date 

Because chronometers do not have 24-hour dials, there 
is always a problem, in applying a chronometer correction, 
whether to add 12 hours for a P. M. hour at Greenwich. 
Also the G. date may be that of the ship, one earlier, or one 
later. Button gives the following quick and easy mental 
method of G. C. T. and date determination: 

Time 43 

1-. Apply the zone description to the ship's approximate 
zone time, obtaining approximate G. C. T. 

2. If it is necessary to add 24 hours to ship's time in order 
to subtract a minus zone description, the G. date is one less 
than ship's. 

3. If after applying the zone description the total is over 
24 hours, the excess is the G. C. T. and the G. date is one 
more than ship's. 

4. Otherwise the G. date is same as ship's. 

One can diagram the problem by making a circle for 
equinoctial and a dot in the center for north pole. West is 
clockwise. Then draw projected local meridian to bottom 
and G. meridian according to longitude. Draw dotted lines 
for their opposite (or lower branch) meridians. Draw symbol 
for sun on circumference according to L. C. T. or Z. T. and 
connect it to center. 

G. C. T. will at once appear less or more than 12. If the 
latter, add 12 to chronometer. 

G. date is the same as local unless sun lies in the sector 
between lower branches. 

G. date is one more than local if sun is between lower 
branches and west of G. lower branch. 

G. date is one less than local if sun is between lower 
branches and east of G. lower branch. 

Change of Date 

Greenwich Civil Noon is a unique instant. It is then, and 
only then, that the same date prevails all over the earth. 

We might discuss this matter of date by referring to the 
sun's apparent motion around the earth, but it is felt that 



a sounder conception will be gained by sticking to the real 

situation and referring to 
the earth's rotation oppo- 
site a stationary sun. For 
convenience, we can disre- 
gard the progress of earth 
[18 in its orbit and the fact that 
a little more than one rota- 
tion occurs between sun 

Take it on faith, for the 
moment, that at Greenwich 
Civil Noon of July 4 this 
date prevails throughout 
the earth. 

The instant the Green- 
wich meridian has passed 
eastward under the mean 
sun, the 180 meridian (also 

FIG. 20. Time Frame. known as the International 

Earth seen from above North Pole Date line) will have come 

around closer to the mean 
20^. sun by an equal amount. 

Say these amounts are each 5 or i/% of an hour. G. C. T. 
will be 4 July 12 h 20 ffi and L. C. T. at 180 will be 5 July 
O h 20 m , that is, 20 minutes past midnight with a new date. 
(See FIG. 20.) Similarly: 

L. C. T. at Long 179 E. will be 5 July O b 16 m . 
178 " O h 12*. 

1 ^77 ** f)k ft**i 

176 " O h 4 m . 

175 " O h O m . 


Sun. G. c. T. = 

FIG. 21. "The Opposite-the-Sun" Meridian. 

FIG. 22. Time Frame and O. S. M. 

Time Frame and O. S. M. each retains its position relative to Sun while 
earth rotates east. Above = G. C. T. 15 h . 




Ss G 

x &> 

G 5 

V .3 

^ O qj ' 

<^ ^^ -H 
r\r -_ *-'' 



O g 

fr : 

s ^ 

Time 47 

The 175th E. meridian is now opposite the sun as was the 
180th at Greenwich noon. Naturally, any earth meridian 
opposite the sun is experiencing midnight or the start of a 
new day. 

Let us think now of a permanently "opposite-the-sun 
Meridian" (O. S. M.) not rotating with earth, but like a 
half hoop suspended at some distance above the earth's 
surface by attachment of its ends to the "poles." The earth 
rushes eastward under this O. S. M. (See FIGS. 21-22.) All 
the earth's surface from this O. S. M. eastward to the 180th 
meridian has the new date. As the 180th proceeds eastward 
with the earth's rotation, more and more area of earth's 
surface is brought between it and O. S. M. By the time 
the 180th comes under the sun at noon, Greenwich has 
reached its midnight and changes date to July 5. The half 
of earth east from Greenwich to the 180th is likewise July 
5, while the opposite half is still July 4. As the 180th com- 
pletes the remaining half rotation reaching midnight again 
and Greenwich reaches noon of July 5, the entire earth 
again has one date, July 5. Figure 23 will make this clear. 

Crossing the 180th Meridian 

A glance at Figure 23 will show that a ship sailing west- 
ward across the 180th must add one to the date, while one 
crossing eastward must subtract one from the date. In each 
instance, the name of the longitude (E. or W.) must be 

The student may wonder about a ship passing the O. S. 
M. This, of course, is not an earth mark, but one kept 
opposite the sun. A ship on earth is carried east under 

48 Time 

O. S. M. by the earth itself at the speed of its rotation, and 
passes midnight in consequence. No ship could travel fast 
enough (except near the poles) to pass under O. S. M. west- 
wardly as that would require a speed greater than the speed 
of earth's rotation. 

For practical convenience, separating continents, keep- 
ing certain groups of islands under one time, etc., the In- 
ternational Date Line is not just the same as the 180th 
meridian. A glance at a globe will show the several angles 
and curves that have been agreed upon. 


Before the invention of a timekeeper which would re- 
main close to correct in varying temperatures, navigators 
were never certain of their longitude. About 30 miles was 
as close as they could come by old methods of computa- 
tion, chief of which was through the measurement of 
"lunar distances/' This meant getting the angle between 
the edge of the moon and some other body by sextant, and 
noting the time, from which, by elaborate corrections and 
computations, the moon's right ascension could be found. 
As this changes about 30" of arc in every minute of time, it 
was necessary to observe within 30" of the correct dis- 
tance to be correct within 1 minute. Consulting the alma- 
nac showed at just what G. C. T. the moon had this 
R. A. Comparing this G. C. T. with the ship's clock at 
observation showed the error of the latter. Having thus 
obtained a fairly correct G. C. T., observations could later 
be made of the sun to get hour angle and hence Local 
Apparent Time. G. C. T. with the equation of time gave 

Time 49 

G. A. T. Then L. A. T. compared with G. A. T. gave 

John Harrison (1693-1776) was an English watchmaker. 
When he was twenty years old, the British Government 
offered a prize of 20,000 pounds for a method which would 
determine longitude within 30 miles. Fifteen years later, 
Harrison invented a compensating grid-iron pendulum 
which would maintain its length at all temperatures, and 
applied unsuccessfully for the prize. By 1761, Harrison had 
produced a better instrument. In 1764, this was taken to 
Jamaica and back on a voyage of over five months and 
showed an error of only l m 54.05 s . It depended on the un- 
equal expansion of two metals with change of temperature. 
The British Government awarded the prize to Harrison 
who, however, did not receive it until nine years had passed. 
Two years later, in 1776, he died at the age of 83. 

The usual ship's chronometer of today is a finely made 
instrument kept in a box and swinging on gimbals to keep 
it level, which is wound regularly but not set after it is 
once started correctly. The rate of change is noted and 
correct time is calculated by applying this rate. It is cus- 
tomary for a ship to carry three so that a serious error in 
one will be manifest by its disagreement with the other 
two. Chronometers are made with 12-hour dials which 
sometimes necessitates adding 12 hours to the face to get 
G. C. T. 

Recently there have appeared what are called "second- 
setting" watches. One model designed by Commander 
Weems and in wrist-watch size, makes second hand setting 
possible, without stopping the movement, by a device 
which rotates the dial under the second hand. Another 

50 Time 

model, which has the advantage of a 24-hour dial, has a lever 
which can stop the movement until the operator pushes 
the lever back. On a small boat where radio time signals can 
be had daily, this watch is probably an adequate substitute 
for a chronometer. 

3. The Nautical Almanac 

annually by the United States Naval Observatory and 
may be obtained for the current or coming year for 65 
cents in money order, from the United States Government 
Printing Office, Washington, D. C. 

The N. A., as it is called, is one of the four absolute 
essentials of equipment for doing celestial navigation. The 
other three are the sextant, the chronometer, and one of 
the many types of tables necessary for computation. 

The student should spend sufficient time in looking 
through the N. A. and reading the explanations printed in 
its last pages to become thoroughly familiar with it. 

As has been previously stated, navigational computation 
has been shortened and simplified by the recent inclusion 
of Greenwich Hour Angle for all bodies used. This and 
the declination are the principal items for which we use 
the N. A. The bodies, for which data are given are the sun, 
moon, Venus, Mars, Jupiter, Saturn, and 54 convenient and 
prominent stars. Special tables are provided for computing 
latitude from Polaris and several 'other useful reference 
tables are given. 

Data are given at the following intervals: 

Dec. G. H. A. 

Sun . 2 h 2 h 

Moon 1 h 1 h 

Planets 1 day 1 day 

Stars 1 month 1 day 


52 The Nautical Almanac 

Following the data for each of the bodies except the 
sun, there will be found tables for computing any values 
of declination and G. H. A. which lie between the values 
given in the main tables. These save a great deal of arith- 
metical interpolation. The sun's declination is easily in- 
terpolated by inspection and its G. H. A. is corrected by a 
table given on every third page. Star declinations hardly 
vary from one month to the next. 

While the use of sidereal time will not be advocated in 
this book, many professional navigators use it and so the 
tables, except those for the sun, give right ascension for 
all the bodies. The first table in the N. A. gives sidereal 
time of O h civil time at Greenwich for every day in the 
year, and another table (No. VI in 1943) gives the time 
which must be added to the above with the excess of 
G. C. T. over O h in order to give Greenwich Sidereal Time. 

The equation of time is given in the sun tables for 
conversion of civil to apparent time. We shall find little 
use for it. 

A two-page table of mean places of 110 additional stars 
is provided. It will be useful occasionally when a single 
star is observed and proves to be neither Polaris nor a 
planet nor one of the usual 54. In such case it may be 
among these 110 and the older method of sidereal time will 
have to be used to obtain t (the L. H. A.) since the R. A. 
values are given without the G. H. A. data. 

Tables are included for calculating times of sunrise, 
sunset, moonrise, moonset, and twilight. 

Commander Angas in An Introduction to Navigation 
(VoL XIII. of Motor Boating's Ideal Series, p. 40) suggests 
entering abbreviations for the month in the upper part 

The Nautical A Imanac 5 3 

of the star chart given in the N. A. as follows: At upper 
right corner put Nov. Then passing to the left and skip- 
ping one square, put Dec. Continue with Jan., etc., in every 
other square to the left until Oct. occupies the second 
square from upper left corner. "At nine p. m. local time 
in the middle of each month, the observer's meridian will 
about coincide with an imaginary vertical line on the 
chart drawn through the center of the space occupied by 
the month in question." 

The writer has found it a great convenience to make up 
a correction table booklet from several old almanacs. 
Bored with having to hunt through the almanac for the 
proper correction tables following the given body's data, 
he has cut out the necessary tables (which do not change 
from year to year), pasted them on loose-leaf sheets of reg- 
ular typewriter size paper using one side only and bound 
them in a 10 cent binder. Index markers make it easy to 
turn at once to all the correction tables needed for any 
body observed. Certain tables appear more than once, 
but this makes each division complete in itself, Altitude 
correction tables (to be discussed later) are included and, 
since these appear in still another part of the almanac, 
further saving of time is made possible, The arrangement 
is as follows: 



1. Height of Eye (Table C). 
Altitude (Table A, sun portion). 
Additional altitude for semidiameter (Table B). 

2. Greenwich Hour Angle. 

54 The Nautical Almanac 


3. Height of Eye (Table C). 

4-5. Altitude (Table D). 

6-7. Greenwich Hour Angle. 

8. Declination. 


9. Height of Eye (Table C). 
Altitude (Table A, star portion). 

1013. Greenwich Hour Angle. 
141 7. Declination. 

18. Height of Eye (Table C). 

Altitude (Table A, star portion). 
1921. Greenwich Hour Angle. 


22-23. Sidereal into Mean Solar (Table V). 

24-25. Mean Solar into Sidereal (Table VI). 

26-28. Proportional Parts (Table VII). 

29. Arc to Time (Table VIII). 

30. Star transit corrections. 

The American Air Almanac, published once In 1933 and then discontinued 
until 1941, is a simplified almanac issued in sections covering 4 months each. 
It is arranged so as to usually give the desired data from a single opening. 
All values are to the nearest minute of arc. Interpolation is unnecessary. 
Button says it "can be used for surface navigation in open waters without 
fear of introducing any serious error. Near land it should only be used 
with caution because errors resulting from its use are not the only errors to 
be expected in the observed position." 

4. Altitudes 

CELESTIAL HORIZON of an observer is the 
JL great circle of the celestial sphere that is everywhere 
90 from his zenith. At the extreme distance of the celes- 
tial sphere it makes no difference whether the observer is 
on the surface of the earth or is theoretically at the center 
of the earth the great circles in these cases will practically 
coincide. Altitudes of heavenly bodies are expressed in 
angular distance from the celestial horizon with the ob- 
server imagined to be at the earth's center. For the far 
distant stars, there will be no difference if the altitude is 
measured from the earth's surface, but for nearer bodies 
sun and moon there must be a correction to our actual 

True Altitude of a heavenly body, therefore, at any 
place on the earth's surface, is the altitude of its center, 
as it would be measured by an observer at the center o the 
earth, above the plane passed through the center of the 
earth perpendicular to the direction of the zenith. 

Sextant Altitude, as measured at sea, must be converted 
to the true altitude by application of corrections for certain 
items as follows: 

For sun and moon: 

Index correction Refraction 

Dip Parallax 


56 Altitudes 

For planets and stars: 

Index correction Dip 


Index Correction will be explained under the chapter 
on the sextant and is merely to correct a mechanical error 
of that instrument that may be present. 

Dip of the horizon, is the depression of the visible sea 
horizon below the celestial horizon due to the elevation 
of the eye of the observer above the level of the sea. Sex- 
tant altitudes taken from the bridge of a steamer or even 
the deck of a small yacht are enough larger than true alti- 
tudes to require correction by N. A. Table C (Height of 
Eye). The correction is, of course, subtracted (see Bow- 
ditch, pp. 1534, for the influence of unusual conditions 
of temperature and barometric pressure). 

Refraction means the bending which rays of light un- 
dergo when passing obliquely from one medium into 
another of different density. If the latter be more dense, 
the ray will be bent toward a perpendicular to the line 
separating the two media. The earth's atmosphere in- 
creases in density down to the earth's surface. Hence the 
path of an obliquely incoming ray of light, by passing 
from a rarer medium to one of increasing density, be- 
comes a curve concave toward the earth. The last direc- 
tion of the ray is that of a tangent to the curved path at 
the eye of the observer. The difference of this from the 
original direction of the ray is the refraction. Refraction 
therefore increases the apparent altitude of a heavenly 
body. It does not change its direction laterally. At the 
zenith, refraction is zero. At horizon it is greatest. The 



correction must always be subtracted. (See FIG. 24.) Table 
A for sun, planet or star, and Table D for moon, include 
correction for refraction. 




FIG. 24. Refraction. 

Parallax of a heavenly body in general is the angle be- 
tween two straight lines drawn to the body from different 
points. Geocentric parallax the only kind with which we 
are concerned is the angle subtended at a body by that 
radius of the earth which passes through the observer's 

FIG. 25. Parallax. 

<j = Moon (Lower Limb) Observed. 

JD = Parallax (N. A.) 

O = Observer. 

= Earth's Center. 

C = Observed Altitude. 

A = Altitude from Earth's Center = A' by Geometry. 

^' + jB = 180,and 

C + D -h B = 180, Therefore 

^ + J5=:C + JDH-J^and 

/4'=C -f A Therefore 

A=C +D. 

Altitude from Earth's Center = Observed Altitude 4- Parallax. 

Altitudes 59 

position. Horizontal parallax is the maximum value of this 
parallax for a particular body and is present when the body 
is on the observer's horizon. Parallax in altitude is the 
parallax when the body is at any point above the observer's 
horizon. It diminishes to zero at the zenith. Putting it 
the other way around, parallax is the difference in altitude 
of a body supposedly measured at the same instant from 
a point on the earth's surface and, with parallel horizon, 
from the earth's center. This is shown in Figure 25. 
Parallax is always additive. Table A for sun and Table D 
for moon include corrections for parallax. The N. A. gives 
horizontal parallax for the moon on each page with the 
other moon data and we must note it for use with Table D 
to correct sextant altitude. Parallax of planets may be 
neglected in practical navigation. 

Semidiameter of a heavenly body is half the angle sub- 
tended by the diameter of the visible disc at the eye of the 
observer. In cases of sun, moon and planets, whose dis- 
tances from earth vary at different times, the semidiam- 
eters will change. The moon is nearer an observer when 
at zenith than when at horizon by the length of the earth's 
radius and the ratio of this length to the total distance of 
the moon is large enough to cause measurable enlargement. 
(This shows that our ordinary impression that the moon 
looks larger when low is an illusion. This effect is prob- 
ably due to the nearness of earth landmarks, buildings and 
so forth, which make the moon seem large by comparison.) 
The increase in semidiameter due to increase in altitude 
is called augmentation. Semidiameter is to be added* to 
observed altitude in case the altitude of the lower limb of 
a body has been measured and to be subtracted in the case 

60 Altitudes 

of the upper limb. Tables A and B include corrections 
for semidiameter for the sun and Table D for the moon. 
Semidiameter of planets may be neglected and of stars is 
not measurable in navigation. 

For greater accuracy one should mentally make the sub- 
traction for dip and the correction for index error first, and 
use the result in entering the table containing corrections 
for refraction, finally using the algebraic sum of the dip 
and other corrections to correct the sextant altitude. 

5. The Sextant 

THE FIRST ELEMENT required in any problem of 
celestial navigation is the angular altitude of the body 
observed. This altitude is measured by means o a sextant 
or an instrument of the sextant family. Measuring the angle 
is done by bringing into coincidence at the eye rays of 
light received directly from the horizon and by reflection 
from the celestial body, the measure being afforded by the 
inclination of a movable mirror to a fixed one. The handle, 
the triangular frame with apex above and scaled arc below, 
the telescope, eye-shade glasses, and horizon mirror, whose 

Mt> X MRRO& 








FIG. 26. Sextant. 





Fxc. 27. Sextant Angles. 


The Sextant 63 

right half alone is silvered, are all rigid parts of a unit. 
The index mirror at top to catch the rays from the body 
is part of the movable arm which terminates below in the 
vernier scale with its screws and magnifying glass. This is 
moved along the arc, tipping the index mirror until the 
two images are brought together. It is then clamped by a 
screw on the right and the tangent screw on the left is 
used for finer adjustment. The scale is then read to give 
the angle of altitude. 

In measuring the altitude of a celestial body, it is neces- 
sary that the angle shall be measured to that point of the 
horizon which lies vertically beneath the body. To deter- 
mine this point the observer should swing the instrument 
slightly to the left and right of the vertical about the line 
of sight as an axis, taking care to keep the body in the 
middle of the field of view. The body will appear to de- 
scribe an arc of a circle, convex down. The lowest point 
of this arc marks the true vertical. 

When a ray of light is reflected from a plane surface, the 
angle of reflection is equal to the angle of incidence. From 
this it may be proved geometrically that, when a ray of 
light undergoes two reflections in the same plane, the angle 
between its first and its last direction is equal to twice the 
inclination of the reflecting surfaces. (See FIG. 27.) 

The vernier is an attachment for facilitating the exact 
reading of the arc scale of the sextant by which certain 
fractional parts of the smallest division of the scale are 
measured. A sextant vernier is a shorter scale usually con- 
taining one more division than an equal length of the arc 
scale. Both arc scale and vernier readings increase to the 
left. To read any sextant it is necessary to observe the arc 


The Sextant 


TO 150 


The numbers are for each 10. 
The next highest marks are for 1 . 
The next highest marks are for V2 = 30'. 
The lowest marks are for 1/6 = 10', 


TO 10 





\ \ 


1 1 

1 1 


\ \ 





FIG. 28. Arc and Vernier Scales. 

When of Vernier is at arc, 10 of vernier is at 19 50' of arc. This is a 
proportion of 50 to 119. The vernier has double spaces for clearer reading. 
Otherwise, the proportion would be 120 to 119. 

scale division next to the right from the vernier zero and 
add thereto the angle corresponding to that division of the 
vernier to the left which is most nearly in exact coinci- 
dence with a division of the arc scale. Figure 28 shows the 
arc scale and vernier on a typical sextant, 

Rule: The smallest measure to which a vernier reads 

length of 1 division of scale 10' 10' 

number of divisions of vernier as 60" = 10 " or l20 = 5 " etc * 

Hence, after observation read thus. (See FIG. 29): 

The Sextant 65 

Scale: right of vernier zero to degrees and 10' units. 
Find line on vernier to left of its zero closest to a scale 

Vernier number gives extra 1' units. 
Vernier marks give extra 10" units. 

Index Error exists when, with index and horizon mir- 
rors parallel, the zero of the vernier does not coincide with 
the zero of scale. Observe a star, or the sea horizon in day- 
light, directly through telescope and move index until 
reflected image coincides with direct. If now the vernier 
zero is to left of the scale zero, all readings will be too great 
by the amount of this divergence; if to right of scale zero, 
readings will be similarly too small. (See Chap. 24 for 

Index Correction (I. C.) is expressed as + or the 
amount of arc to be applied to the observed amount. 

Certain minute errors due to construction and not cor- 
rectable by adjustment are usually noted in certificates 
accompanying the instrument when purchased. 

Certain adjustments must occasionally be made. See 
Button, pp. 219-20, for detailed information as to: 

Index mirror Telescope Horizon mirror 

Properly speaking*, instruments of the sextant family 
should be designated as follows: 

Octant: 45 arc measures angles to 90 

Sextant: 60 arc measures angles to 120 

Quintant: 72 arc measures angles to 144 

Quadrant: 90 arc measures angles to 180 


66 The Sextant 

The author's instrument, therefore, with an arc of 75 
for angles to 150 would seem to qualify as a Super- 

AND 10' 5PAC.E6 





1 | 

lilt 11 



1 1 1 

| | 


111 II ARC 


1 1 

1 1 

1 1 






10" 6PACE.6 

FIG. 29. Reading the Sextant. 

Passing to left shows first coincidence at 75 50' of arc (Arrow). 
Reading Arc =69 10' 
Reading Vernier = 3' 20" 

Reading Total = 69 13' 20" 

The U. S. Navy now uses a sextant fitted with an end- 
less tangent screw which carries a micrometer drum from 
which the minutes and the tenths of a minute o arc are 
read. The tangent screw is thrown out of gear by pressing 
a lever so the index arm may be freely moved. Releasing 
the lever stops the arm at individual degrees and throws 
the tangent screw into gear again for finer adjustment. 
(See FIG. 26.) 

An artificial horizon may be provided by a glass-roofed 
dish of mercury or even a saucer of ink placed in an 
uncovered box with sides low enough to let the body's rays 
in but high enough to keep moving air from rippling the 
ink. The angle between the celestial body and its reflection 
in the artificial horizon is measured by sextant. Half of 

The Sextant 67 

this angle equals the altitude of the body. (See Chap. 24 
for details.) 

A bubble sextant is an instrument containing a mech- 
anism which supplies an artificial horizon. It is much more 
expensive and less accurate than the standard type of sex- 
tant but can be used when the natural horizon is invisible, 
as in hazy weather, or in polar regions, on the desert, or 
in an airplane high above the earth and clouds. 


The evolution of the modern sextant probably began 
with the astrolabe, used by early Greek and Arab astron- 
omers. It consisted of a graduated circle suspended in the 
vertical plane from a ring at the top. At its center a sight- 
ing bar was attached somewhat as a compass needle and, 
by looking along this at a body and noting the scale on the 
circle, altitude could be measured. Elaborate forms of the 
astrolabe were in use in the sixteenth century. 

Capt. John Davis, an English navigator, developed a 
quadrant in 1594 which had two arcs and required sight- 
ing in two different directions. A later form was used with 
the observer's back to the sun. Many subsequent instru- 
ments depended on a plumb line. 

In 1729 Pierre Bouguer invented an instrument which 
needed only a sight of the horizon while a beam from the 
sun was kept visible in line on a wooden peg. 

The cross-staff, something like a T square, was also in 
use at about this time and necessitated sighting sun and 
horizon separately. It too was used both facing and with 
back to the sun. 

68 - The Sextant 

The double reflecting mirror instrument was suggested 
in 1674 both by Robert Hooke, a professor of geometry in 
London, and Sir Isaac Newton, independently, but no 
models seem to have been made. 

In 1730 Thomas Godfrey of Philadelphia and John 
Hadley, an English astronomer, independently constructed 
double reflecting instruments much like our sextants of 
today. Hadley, who was Vice-President of the Royal Society 
of London, probably suppressed Newton's notes and cer- 
tainly ignored Godfrey's claim and obtained the credit for 
the invention. 

Capt. Campbell in 1757 enlarged the arc to make it a 
true sextant. Up to 1775 the instruments were of all-wood 
construction. Various improvements followed and the all- 
metal sextant appeared in the early nineteenth century. 
Verniers then came into use although described by Pierre 
Vernier long before, in 1631. Commander Hull of the 
U. S. S. Constitution in 1812 used a sextant which had 
shade glasses, telescope, vernier, ivory arc, brass fittings 
and ebony frame while, at this same period, a similar in- 
strument constructed of brass with a silver arc was in use 
by Nathaniel Bowditch. Relatively minor improvements 
followed but the sextant of today is not radically different 
from its ancestor of 100 years ago. (See The Evolution of 
the Sextant by Commodore E. S. Clark, U. S. N. I. P. Nov. 
1936, Navigational Antecedents by Commander H. D. 
McGuire, U. S. N. I. P. May 1933 and The American 
Inventor of the Reflecting Quadrant, also by McGuire, 
U, S. N. I, P. Aug. 1940.) 

6. The Compass 

ence on a piece of iron touched by it is of great 
antiquity. Its use in a form to indicate directions at sea was 
a subsequent development. The Chinese, Arabs, Greeks, 
Etruscans, Finns and Italians have all been credited as 
originators of the compass. Encyclopedia Britannica (14th 
ed., Vol. 6, p. 176, "Compass") says . . . "the earliest defi- 
nite mention as yet known of the use of the mariner's 
compass in the middle ages occurs in a treatise entitled 
De utensilibus, written by Alexander Neckam in the 12th 
century. He speaks there of a needle carried on board ship 
which, being placed on a pivot, and allowed to take its own 
position of repose, shows mariners their course when the 
polar star is hidden." 

The Magnetic Compass of today as used on ships is the 
Thomson (Lord Kelvin) instrument introduced in 1876 
with a few improvements. Kelvin used several magnetic 
needles in parallel attached under a circular card on which 
were printed the "points/* the whole supported on a pivot 
for easy rotation. A subsequent improvement was filling 
the compass bowl with an alcohol mixture and sealing it 
under the glass cover, a corrugated chamber being pro- 
vided for expansion of the liquid with increased tempera- 
tures. Instead of the flat glass top, a modern development 
is the "spherical" (really hemispherical) glass cover. Com- 
passes are swung on gimbals to keep them level when the 
ship rolls or pitches, and mounted in a pedestal called a 


70 The Compass 

binnacle. Through the bowl there is painted a thin black 
line from front to back. This is known as the lubber's line 
and the compass must be installed with this line exactly 
parallel to the ship's keel. The forward tip of the line, 
visible over the compass card, is the mark by which the 
helmsman notes the course, on the card, of the ship's 

Compass cards are marked in two principal ways: 
The old point system consists of 32 points around the 
circle, starting from north as follows: 

North Northeast by East 

North by East East Northeast 

North Northeast East by North 

Northeast by North East 

Northeast etc. 

Each point is 11 14 degrees from the next. A system of 
quarter points is added, each being equal to about 2.8 
degrees. The older (Merchant Marine) custom and the 
newer (Navy) custom of naming these quarter points are 
both somewhat difficult to memorize and of no real use to 
the student. 

The other method of marking is in 360 around the 
circle clockwise. This system has many advantages aside 
from the fact that steamers can be steered to 1 while the 
smallest unit of the point system is about 3. East is 90, 
South is 180, West is 270 and North is 360 or 0. This 
system is all one needs at sea for celestial navigation. 

Compass error results from two main causes, Variation 
arid Deviation, now to be discussed. 

Variation. The earth may be thought of as a great 

The Compass 71 

magnet whose poles, however, do not exactly correspond to 
the geographical poles. The north magnetic pole is at 
about Lat. 70 N., Long. 97 W. and the south magnetic 
pole at Lat. 73 S., Long. 155 E. (As unlike poles attract 
and likes repel, the "north" end of the compass needle is 
really "north-seeking" or south.) As the north magnetic 
pole is above Hudson Bay and about 1200 miles below the 
geographical north pole, it will be evident that, in the 


06 1 091 
FIG. 30. Compass Card showing three methods of marking. 

72 The Compass 

Atlantic Ocean, the compass will point west of true north 
and, in the Pacific, east of true north. The amount of this 
divergence from true is called variation. As long ago as 
1700, Halley constructed charts showing lines of equal 
variation over the earth. Today we obtain the variation 
from our charts for any locality. There is also a slow change 
which is noted on the chart as so much per year and, by 
figuring from the date of the chart, the present amount 
can be obtained. 

Deviation is a further compass error due to magnetic 
influences on the individual ship. The intricacies of the 
subject need not be gone into here. Hard and soft iron, 
horizontal and vertical iron, permanent, subpermanent 
and transient magnetism, are all terms used in the discus- 
sion and explanation of deviation. It will suffice to under- 
stand that, as a ship swings around, its metal and mag- 
netism will be brought into different positions relative to 
the north point of the compass (which retains its general 
position) and will exert varying pulls on it, sometimes to 
one side and sometimes to the other. Deviation varies 
with latitude. 

Heeling error is similar. It occurs when the ship heels 
or rolls to one side or the other. Iron which has been hori- 
zontal approaches the vertical and vice versa. The compass 
is influenced by this alteration of magnetic force, chiefly 
when on north and south courses. 

Local magnetic disturbance is due to magnetic material 
outside the ship in the neighborhood. In certain parts of 
the world (Australia, Labrador, Madagascar, Iceland, the 
Baltic, Lake Superior) this is a large source of error due to 
mineral deposits in the ocean or lake bed. Minor causes 

The Compass 73 

are present at docks due to other vessels, metal, etc. 

Correction of much of the error which develops with 
the ship on different headings or due to heeling is done 
by placing magnets and iron in certain positions near the 
compass. This is called compass adjusting and should only 
be done by a thoroughly trained worker. Nothing is done 
about local magnetic disturbance. 

Conversion of true, magnetic, or compass courses, one 
into another, is easy with the 360 card system. Between 
true and magnetic, we must know the variation. Between 
magnetic and compass, we must know the deviation. Bear 
in mind that a course, however described in these three 
ways, is the same direction on the earth. When a force pulls 
the N. point of the compass in a clockwise direction, it is 
called easterly and if counterclockwise, it is called westerly. 
If deviation is 5 W. and variation 7 E., the error is the 
algebraic sum or 2 E. If deviation is 6 W. and variation 
2 W., the error is, of course, 8 W. 

When thinking for the first time of these disturbance! 
of the compass card, it is a good scheme to think of the card 
as having just been rotated in a certain way to a certain 
extent and to imagine yourself pushing it back to its orig- 
inal position. Say you are heading on a compass course of 
20 and know that the deviation is 5 E. This means the 
card has been forced 5 clockwise. Mentally push it back 
5 counterclockwise. The heading will now be 25 or the 
magnetic course. Then suppose you also know the varia- 
tion is 6 E. This means the card has been forced 6 clock- 
wise. Mentally push it back 6 counterclockwise. The 
heading will now be 31 or the true course. In each case 
we have added. 

74 The Compass 

Only one rule need be memorized and it is this: From 
compass toward true add easterly errors. 

This means that from compass to magnetic we add east- 
erly deviation and from magnetic to true we add easterly 
variation. Similarly we subtract westerly errors. From true 
toward compass we do just the opposite: subtract easterly 
and add westerly. Any one of these can be quickly thought 
out if we start from the one rule and make the necessary 
reversals. Examples are shown in Table 7. 











1 W 



Methods of determining Deviation. 

1. By bearings (azimuth) of the sun. This is the usual 
method when at sea. Polaris and other stars may also be 

2. By comparison with a gyro compass. 

3. By reciprocal bearings. 

4. By bearings of a distant object. 

5. By ranges. 

The first will be discussed later on in this book under 
Azimuth. The student is referred to Button, Chapter II, 
for methods 2, 3, and 4: Method 5 which is probably the 
simplest and most convenient for small craft not at sea, 
may be briefly outlined as follows: 

The Compass 75 

a. Find some pair of objects in line, one nearer and one 
farther, easily visible and shown on the chart, and deter- 
mine the magnetic bearing of the line joining them, from 
the water where you are located, by means of the "com- 
pass rose" on the chart. 

b. Send your boat across this line heading first 0, then 
15 and every 15 around the circle, noting the bearing 
of the range at each separate heading. 

c. Make a table showing the above and add a column 
showing deviation on each heading, as in Table 8. 


Ship's Head 
by Compass 

Range by 



45 etc. 




1 W 

This table of deviations for each compass heading is 
not sufficient for our needs. It does not tell us, what we are 
more anxious to know, what compass course to steer in 
order to make a given magnetic course. We want a table 
arranged the other way around beginning with equal divi- 
sions of magnetic and showing the proper compass course 
for each. 

The Napier Diagram is the means by which the above 
is accomplished. Details will not be given here but the 
student is referred to Bowditch or Dutton for a full 

76 The Compass 

explanation. The diagram is basically a series of equilateral 
triangles each side of a base line. Deviation values for each 
15 compass course are plotted and a curve drawn, from 
which it is easy to obtain the proper compass course for any 
desired magnetic. 

Having constructed the curve as above, a table is made 
giving the desired compass course for each 15 of mag- 
netic, as in Table 9. 


For Magnetic 

Steer Compass 

90, etc. 


For values between these 15 magnetic intervals it is 
easy to interpolate. However, for more easily visualizing 
interpolation, there are two methods by which the table 
may be diagramed. 

First, a double vertical scale may be drawn like two 
parallel thermometer scales and lines then drawn across 
from the magnetic to the compass scale indicating equiv- 
alent values. 

Second, a double compass diagram may be had, with 
one scale outside the other, the inner representing mag- 
netic and the outer representing compass. Lines again con- 
necting equivalents make the process of conversion and 
interpolation especially easy. 

The Compass 77 

The Gyro Compass obtains its directive force from the 
force of the earth's rotation. A full description is given in 
Bowditch. The essential feature is an electrically driven 
wheel spinning at 6,000 r. p. m. whose axis seeks to remain 
in the plane of its meridian. The attachments and mechan- 
ical details result in a compass which has the following 
advantages over the magnetic compass: 

a. It is unaffected by the ship's magnetic field. (No devi- 

b. It seeks the true instead of the magnetic meridian. 
(No variation.) 

c. Its directive force is much greater. 

d. It may be located in a safe and central portion of the 
ship and repeater compasses directed electrically from it 
may be located anywhere. 

The disadvantages are: 

a. It is a complex and delicate mechanism. 

b. It requires a constant source of electrical power. 

c. It requires intelligent and expert care. 

d. It is expensive ($2,000 up). 

An Azimuth Circle is a ring formed to fit flat over a 
compass bowl and which can be turned to any desired 
position. It is graduated from to 360 clockwise. Sight- 
ing vanes permit the observer to take bearings of terres- 
trial objects by turning the circle until the object is in line 
with the vanes while a reflecting prism throws the compass 
card into view at the same time. An adjustable dark glass 
reflector brings celestial bodies into view and a concave 
mirror and reflecting prism make it easy to take the bearing 
of the sun. 

78 The Compass 

The Pelorus is a "dumb compass" or card without mag- 
netic needles which can be turned to any desired position, 
and set. A lubber's line as in the compass marks the direc- 
tion of the ship's head. Sighting vanes are provided. Pelo- 
ruses are placed so that views can be had in all directions 
which is seldom possible with the ship's compass. The card 
is set to correspond with the heading of the ship by compass 
and then bearings are taken with it which are the same as 
compass bearings. 



North to East 



N by E 









East to South 





SE by E%E 
SE by E^E 
SE by E34E 




S by El 
S by E 


South to West 


Sby W 
Sby W}^ 
S by WMA 
Sby WM^ 



West to North 




NWby W 



Nby W 
N by W 
Nby W 






D. M. S. 

0' 00* 
2 48' 45" 
5 37' 30" 
8 26' IS* 

11 15' 00* 
14 3' 45" 
16 52' 30" 
19 41' 15* 

22 30' 00" 
25 18' 45* 
28 7' 30* 
30 56' 15" 

33 45' 00* 
36 33' 45" 
39 22' 30* 
42 11' 15* 

45 00' 00* 
47 48' 45* 
50 37' 30* 
53 26' 15* 

56 15' 00* 
59 3' 45* 
61 52' 30* 
64 41' 15* 

67 30' 00* 

70 18' 45* 
73 7' 30* 
75 56' 15* 

78 45' 00* 
81 33' 45* 
84 22' 30* 
87 11' 15* 
90 00' 00* 

7* The Astronomical Triangle 

NAVIGATION USES an imaginary triangle on the 
celestial sphere whose three corners are the elevated 
pole (N. in N. latitude, etc.), the zenith/and the projection 
of the observed heavenly body. Sometimes we can easily 
determine the three sides and only need to compute one 
of the angles. Again we can easily determine two of the 
sides and one angle and only need to compute the remain- 
ing side. These computations are done by spherical trigo- 
nometry which will be briefly outlined in the next chapter. 

(V4 of Celestial Sphere seen from West) 

Earth DM Declination 

? N. Celestial Pole MP Polar Distance 

Z Zenith AM Altitude 

M Body Observed MZ Zenith Distance 

' Celestial Equator * Hour Angle 

QjL Latitude Z* Azimuth 

ZP Co-Latitude 

(V4 o Celestial Sphere seen from North) 

.of body. 

Q Sun 

LM Local Meridian 

CM Greenwich Meridian 

GHA Greenwich Hour Angle 

to Longitude 

t Local Hour Angle 

FIG. 3L The Astronomical Triangle. 

80 The Astronomical Triangle 

For the present, we will consider what the principal rela- 
tions are and leave solutions until Part II. 

Remember that the celestial triangle is merely a magni- 
fication of a terrestrial triangle. Each part, side or angle, 
or relationship, on the celestial sphere corresponds to the 
similar unit on the earth. Co-latitude (and hence, of course, 
latitude), for instance, on the celestial sphere is the same in 
degrees as that which we may be seeking on earth. 

Figure 31 shows on the left a typical astronomical tri- 
angle with the parts named. Remember that the angular 
distance of a quarter circle, no matter of what size, is 90 
so that if we know a portion of a quadrant we can find the 
remaining portion by subtracting the first from 90. The 
circular figure on the right shows the celestial sphere as 
seen from above with projections of the body and two 
earth meridians. 

Table 11 needs little explanation. The first column 
shows what one starts with; the second shows what is 





Portion of 
Triangle Found 


Altitude (ti) 


Zenith Distance (z) 

G.C.T. \ 

Naut. Aim. J 

/Declination (d) 


Polar Distance (p)* 

Previous Observation 
Previous Observation 
(As above) 

Latitude (L) 
Longitude (Lo) 

G.H.A. cx> Lo 


Co-Latitude (Co-X) 
Local Hour Angle (t) 
Azimuth (Z) 

* If L and d have opposite names (one N and one S) then p= 90+ d. 

The Astronomical Triangle 81 

thereby obtained; the third shows how it is used; and the 
fourth gives the parts of the astronomical triangle thus 

Thus 3 sides and 2 angles of the astronomical triangle 
are sometimes obtainable without the use of formulas. 
The remaining angle at the observed body is never needed 
in ordinary navigation. 

As the -fundamental relationships of celestial navigation 
are here set forth, this table should be studied with great 
care and thoroughly understood. 

The Celestial Co-ordinator is an ingenious device to give rapid approxi- 
mate solutions to problems of the astronomical triangle. It consists of a 
rotating disc on a larger background. The type using the orthographic pro- 
jection has its under portion printed in black. It is a circle representing 
observer's meridian with the upper half ruled with horizontal lines for 
altitude circles every 5. These are crossed by elliptical arcs from the zenith 
for azimuth circles every 5. Over this and fastened to it at the center is a 
transparent circular disc ruled similarly but in red and in both halves. Its 
border represents observer's meridian. A straight line across it is for the 
equinoctial. Crossing this at right angles is the polar axis. Lines parallel to 
the equinoctial are declination circles every 5. Elliptical arcs from the poles 
represent hour circles at 334 (15 minutes of time) intervals (in the type 
produced by the N. Y. Power Squadron). In using the co-ordinator the 
upper part is set according to observer's latitude. Then various combina- 
tions of problems involving h, t, d, and Z may be solved. The greatest use- 
fulness appears in star work. Given t and d, one gets the approximate h 
and Z which shows where to look for the star. Given h and Z of an unknown 
observed star, one gets d and t. From t and known longitude, one gets 
G.H.A. With d and G.H.A. and the Nautical Almanac, the star can be 

8. Trigonometry 

NAUTICAL ASTRONOMY makes constant use of 
trigonometry. The commonest example of this is the 
need for computing altitude for comparison with a sextant 
observation. This and various other needs will be ex- 
plained in Part II. A knowledge of this branch of mathe- 
matics is not essential to the navigator but some familiarity 
with its fundamental concepts is highly desirable. There- 
fore, both for the benefit of the novice and as a mind 
refresher for those who at some distant past date have 
studied trigonometry, the following pages are provided. 
They contain enough to indicate the general features of 
the subject and to suggest how various formulas, which 
will appear later, may have been derived. Remember that 
in plane trigonometry, only angles are measured in de- 
grees. In spherical trigonometry not only angles but also 
sides are so measured. The reader is referred for further 
details to any modern text on the subject. An excellent 
one is Palmer and Leigh: Plane and Spherical Trigonom- 
etry, 4th edition (McGraw Hill Book Co., Inc., New York 
and London, 1934). 

Plane Trigonometry 

This branch of trigonometry investigates the relations 
that exist -between the parts of triangles which lie in a 




(A and B = Acute Angles) 



Sin (sine) A 
Cos (cosine) A 
Tan (tangent) A 

: side opposite = A = cosB (s!n A = always < 1) 
hypotenuse c 

side adjacent 

: = sin B (cos A = always < 1) 

side opposite = _0 
side adjacent b 

( tan A under 45= always < 1 

side adjacent & j cot A under 45= always > 1 ? 

Cot(cotangentU - gide oosite - 7 " tan ^ | C ot A over 45-always < 1 J 

hypotenuse _ __ 
side opposite a 


1-cos ^l is called versed sin A (vers A) 
% (1-cos 4) is called haversine A (hav 

By formula (#22 in Palmer & Leigh) : 


1-cos A 

Sin H ^ - y ;1 -T which, being squared, becomes: 

SinaJ^^L = J^ (1-cos A). 

Haversine of an angle = square of the sine of Y% the angle. 


+ 8 



w a 


















o ta 

>-s ft 

^ 2 

8 -I 






Trigonometry 85 

The sum of the angles of a plane triangle = 180. 

The complement of an angle = 90 minus the angle. 

The supplement of an angle =180 minus the angle. 

When one quantity so depends on another that for every 
value of the first there are one or more values of the sec- 
ond, the second is said to be a function of the first. 

Connected with any angle there are six ratios that are of 
fundamental importance, as upon them is founded the 
whole subject of trigonometry. They are called the trigono- 
metric functions of the angle. To each and every angle 
there corresponds but one value of each trigonometric 

Tables 12, 13, 14, 15 will show the nomenclature used 
and some of the fundamental relationships of the func- 
tions of, first, acute angles and, second, angles of any size. 

Spherical Trigonometry 

This branch of trigonometry investigates the relations 
that exist between the parts of a spherical triangle. 

A spherical triangle is the figure on the surface of a 
sphere bounded by three arcs of great circles. The three 
arcs are the sides of the triangle, and the angles formed 
by the arcs at the points where they meet are the angles 
of the triangle. The angle between two intersecting arcs 
is measured by the angle between the tangents drawn to 
the arcs at the point of intersection. The sum of the sides 
of a spherical triangle is less than 360. The sum of the 
angles of a spherical triangle is greater than 180 and less 
than 540. In a spherical triangle there are six parts, three 
sides and three angles, besides the radius of the sphere 



(The sign of the hypotenuse or distance is always +) 





. x 

bin = - 



rt~\ *v 

Ian = - 


Cot = - 




Csc = - 


NOTE: The angles here and in Figure 33 have been drawn 
increasing clockwise (in contrast to Pigure 137 in Bowditch) so 
as to conform to the 360 compass and its quadrants. 


Trigonometry 87 

which is supposed known. In general, if three of these parts 
are given, the other parts can be found. 

A right spherical triangle is one which has an angle 
equal to 90. In such a triangle, two given parts in addi- 
tion to the right angle are sufficient to solve the triangle. 

Napier's Rules of Circular Parts. In a right spherical 
triangle, omitting the right angle, consider the two sides 
(a and b) adjacent to the right angle, the complements of 
the two other angles (co-^4 and co-) and the complement 

FIG. 32. For Napier's Rules. 

of the remaining side (co-c). Arrange these in a circle as 
in Figure 32. Any one of these five parts may be selected 
and called a middle part; then the two parts next to it are 
called adjacent parts and the other two parts, opposite 
parts. Napier's rules are: 

L The sine of a middle part equals the product of the 
tangents of the adjacent parts. 

2. The sine of a middle part equals the product of the 
cosines of the opposite parts* 

The ten formulas for solution of a right spherical trian- 
gle can be derived from these two rules. 

38 Trigonometry 



(A = Acute) 
(See Fig. 33) 




sin A. 
cos (90-A) 

cos A 
sin (90-A) 

tan A 
cot (90-A) 

cot A 
tan (90 -A) 

sec A 
esc (90-4) 


sec (90~A) 


-cos (90 -hi) 
sin (180-A) 

sin (90+A) 

-cot (90+A) 
-tan (180- A} 

-tan (90-M) 
-cot (180--A) 

esc (90-M) 
-sec (180 ~A) 

-sec (90+A) 
esc (180-A) 


-sin (180+A) 
-cos (270-A) 

-cos (180+ J.) 
-sin (270-A) 


cot (180+^) 
tan (270- A) 

-esc (270-1) 

-esc (180+A) 
-sec (270-A) 


cos (270-M) 
-sin (360-A) 

-sin (270+A) 
cos (360-A) 



-esc (270*4- A) 
sec (360-A) 

sec (270+A) 
~csc (360-A) 

Table 15 leads to the following rules which are useful in 
certain "Sailings" problems which will be found in Part 

1. The value of any function of any angle in the three 
higher quadrants is always equal to the value of the same 
function of some angle in the first quadrant, as: 

Given Angle in Quadrant 



Corresponding Angle in Quadrant I 

180 given angle 
given angle -180 
360 given angle 

2. To find an angle, known to be in a higher quadrant, 
for a log in Table 33, Bowditch, take out the angle in 
Quadrant I and treat as follows: 

(360 OR o 8 ) 


(TO) -X 

(To be used with Table 15) 

FIG. 33. Angles in the Different Quadrants. 

The A angles at center are constructed equal. Each A angle near circum- 
ference equals A since its horizontal side is parallel, by construction, to the 
X axis and, by geometry, a straight line (here a radius) connecting two- 
parallel lines makes an angle on one side with the first line which is equal 
to the angle it makes on the other side with the second line. 

90 Trigonometry 

For Quadrant Use 

II 180-angleinl 

III 180+angte in I 

IV 360- angle in T 

Oblique Spherical Triangles are solved by formulas de- 
rived from the following theorems: 

1. Sine theorem. In any spherical triangle, the sines of 
the angles are proportional to the sines of the opposite 

2. Cosine side theorem. In any spherical triangle, the 
cosine of any side is equal to the product of the cosines 
of the other two sides, increased by the product of the sines 
of these sides times the cosine of their included angle. 

3. Cosine angle theorem. The cosine of any angle of a 
spherical triangle is equal to the product of the sines of 
the two other angles multiplied by the cosine of their in- 
cluded side, diminished by the product of the cosines of 
the two other angles. 


(1) sin A _sin B _sin C 
sin a ~sin b ""sin c 

(2) cos a = cos b cos c -f sin b sin c cos A 

(3) cos A sin B sin C cos a cos B cos C 

The following are useful haversine formulas for any 
spherical triangle: 

For an angle when the sides are given: 

hav a hav (b ^ c] 

hav A = = r =-^ ' 

sin b sin c 

Trigonometry 91 

For a side when other sides and included angle are 
given : 

hav a = hav (b -^ c) 4- hav 6 
where hav hav A sin b sin c 

9* Logarithms 

multiplication of various long numbers. To multiply 
one cosine by another and this perhaps by a haversine 
would be a tedious process with old style arithmetic. By the 
use of logarithms, the processes of multiplication, division, 
raising to a power, and extracting a root of arithmetical 
numbers are usually much simplified. 

ItN= 6 X , then x = the logarithm of N to the base 6. 

The logarithm of a number to a given base is the expo- 
nent by which the base must be affected to produce that 

As 100 = 10 2 so logic 100 = 2, or the logarithm of 100 to 
the base 10 is 2. This is abbreviated to log 100 = 2. 

Logarithms were invented by Lord Napier of Scotland 
(1550-1617) and described by him in 1614. He used the 
base 2.7182818 called "e." This system is used in advanced 
and theoretical work. 

Prof. Briggs of London (1556-1631) modified the above 
by using the base 10. This system is called the common or 
Briggs system and is the one usually used in computing 
and is given in the 1938 Bowditch, Table 32 for logs of 
numbers and Table 33 for logs of trigonometric functions 
of angles. 

The word "natural" is used to distinguish from the log- 


Logarithms 93 

arithm. Table 31 gives natural, that is, actual values of, 
sines, cosines, tangents and cotangents for all angles to 90. 
Table 33 is where one will find the log sines and log 
cosines, as well as the logs of the other four trigonometric 
functions, of angles to 180. Table 34 gives both natural 
haversines and log haversines of angles to 360. (Although 
omitted to save space, -10 belongs after every log in Tables 
33 and 34.) 

Multiplication of two numbers is accomplished by add- 
ing their logs and then finding the number of which this 
sum is the log. 

Division is accomplished by subtracting the log of the 
divisor from the log of the dividend and then finding the 
number of which this result is the log. 

Raising to a power is done by multiplying the log of the 
given number by the index of the power and then finding 
the number of which this product is the log. 

Extracting a root is done by dividing the log of the given 
number by the index of the root and then finding the num- 
ber of which this quotient is the log. 

The student should consult Bowditch, pp. 324-26, for 

While rules are given for obtaining logs of numbers and 
numbers for logs, I believe the rules on the next two pages 
will be found somewhat simpler. 

Table 16 is to show examples of logs with various char- 
acteristics (numbers to left of decimal point) for one man- 
tissa (number to right of decimal point). 

94 Logarithms 

To Find Log of a Number 


For number > 1 = 1 < the number of figures including 
zeros to left of number's decimal point. 

For number < 1 = 9 - the number of zeros directly to 
right of number's decimal point and -10 to right of 
mantissa. ^ 

Manti&sa: * 

Disregard decimal point of number when looking it up 
in Table 32 Bowditch. F 

For number of 1 figure: Add 2 zeros to right and treat as 
3 figures. 

For number of 2 figures: Add 1 zero to right and treat as 
3 figures. 

For number of 3 figures: Find number in left column and 
mantissa opposite it in column headed 0. 

For number of 4 figures: Find first 3 in left column and 
mantissa opposite them in column headed by 4th figure. 

For number of 5 or more figures: Find mantissa for first 4. 
Then add to it the following: figure in column d x remain- 
ing figure or figures, first pointing off as many places as there 
are remaining figures and disregarding fraction or, if it 
exceeds .5, raising total to next number. 

Logarithms 95 

To Find Number for a Log 

May find exact mantissa in Table 32 Bowditch and take 
out first 3 figures from left column and 4th from top. 

If mantissa lies between 2 given in table: 

Take 4 figure number of next lower mantissa. 

Note difference between next lower and given mantissa 

Note difference between next lower and next higher man- 
tissa (col. d). 

Note P. P. table under figure found for d. 

Note 1st diff. figure in right column, 

Note figure on same line in left column. This is 5th figure 
of number. 

If over 5 figures in number (as when characteristic is 5 and 
number therefore has 6 figures to left of decimal point), get 
4 figure number for next lower mantissa and then use 


= 5th and succeeding figures. (Always < 1.) 


Disregard decimal point but retain any zeros that may come 
between it and other figures and add 1 or more zeros on right 
if characteristic calls for more figures in number. 

If number is > 1 : Place point to right of 1 more figures 
than number of characteristic. 

If number is < 1 (log ending in 10 with characteristic 9 
or less) : Subtract characteristic from 9 and add that many 
zeros before figures already found and put point to left. 


} Logarithms 







10.09152 or 




19.09152 10 



18.09152 10 









15.09152 10 



14.09152 10 



13.09152 10 



12.09152 10 



11.09152 10 



10.09152 10 



9.09152 10 



8.09152 10 


- 3.09152 




6.09152 10 





- 6.09152 

4.09152 10 


- 7.09152 

3.09152 10 



2.09152 10 


- 9.09152 

1.09152 10 





- 11,09152 

9.09152 20 

I? art II: I?:roceelTJLxres 

10. Introduction to 
Position Finding 

SOLUTIONS FOR LATITUDE, as such, are not so 
important in the modern practice of navigation as they 
once were. The same can be said of longitude. This is be- 
cause, as we shall soon see, the newer navigation qbtains 
from one observation a "line of position," on which the 
ship is situated, and crosses this with another line from an- 
other observation thus obtaining a "fix" on the chart, from 
which the latitude and longitude can then be read off giv- 
ing the exact position. Discussion and argument still go on 
between navy and merchant marine on the merits of the 
newer methods. Inasmuch as the whole art and science of 
position finding has evolved through first obtaining lati- 
tude and longitude and because many officers of the mer- 
chant marine still depend on these older methods and must 
use them in their examinations for promotion, it seems well 
to briefly explain them to the beginner. 

One of the first things to understand is the nautical 
mile. It is defined in the U. S. A. as being 6,080.27 feet in 
length, equal to 1^ part of a degree, or 1 minute of arc, 
of a great circle of a sphere whose surface is equal in area 
to the area of the surface of the earth. The earth is some- 
what flattened at the poles which slightly alters the length 
of 1 minute high on a meridian. This, however, is disre- 
garded in navigation and a change of 1 minute of latitude 
always is taken to mean a change of 1 nautical mile north 


100 Position Finding 

or south. Since the meridians converge toward the poles, the 
difference of longitude produced by a change of position 
of 1 mile to the east or west will increase with the latitude. 
For instance, 1 mile on the equator will cause a change of 
longitude of 1 minute while at latitude 60 it will cause a 
change of 2 minutes. 

Before doing any actual work, the amateur navigator 
must understand the principles of the Mercator chart. 
(Gerardus Mercator, Flemish cartographer, 1512-1594.) The 
transfer of a spherical surface, such as a globe map of the 
world, onto a flat surface, presents many difficulties. The 
portion of the globe between Lat. 60 N. and Lat. 60 S. 
may, however, be transferred by placing a cylinder of 
transparent paper around the globe, tangent at the equa- 
tor, and projecting onto it the features of the globe as seen 
from its center. Cutting this cylinder vertically at some 
point and laying it out flat will show the meridians of 
longitude not converging but as parallel vertical lines. 
The parallels of latitude will be horizontal parallel lines 
but farther apart the farther away they are from the equa- 
tor. As the distance between meridians becomes more and 
more in excess of the true proportional distance, the in- 
creasing distance between parallels makes up for it and 
proportion in a given region is maintained. (This is not 
literally accomplished as described but is done mathemati- 
cally. See Button, Chap. I.) Of course, there is great dis- 
tortion of areas in high latitudes. The great advantage of 
a chart on this basis is that any course which cuts successive 
meridians at the same angle becomes a straight line. If it 
were cutting meridians at some angle other than 90 and 
drawn on a globe, it would have to be a curve of a spiral 

Position Finding 101 

form. Such a curve is called a Rhumb Line. Scales of miles 
as on ordinary maps are not possible here but, since a minute 
of latitude equals a nautical mile, one uses the latitude scale 
marked at the side of the chart at the level of the region 

Position plotting sheets are blank charts made on the 
Mercator system and important for the navigator in the 
graphic solution of problems. One series issued by the Hy- 
drographic Office is of 12 sheets covering latitudes from 
to 60 and can be used for north or south of the equator. 
The price is 20 cents a sheet, sold singly for a 5 area. Their 
use saves the mutilation of charts, and they are large enough 
(about 4x3 ft.) to permit the recording of sufficient data 
for a good record. The parallels of latitude are, of course, 
numbered, but the meridians of longitude are not. The 
user numbers the latter according to his location. A smaller 
and more convenient size (26 x 19 inches) is issued in 16 
sheets covering latitudes 0-49 and sells for 10 cents a sheet. 

Tables of logarithms of trigonometric functions are nec- 
essary for the solution of various equations which will 
appear in the descriptions of the several procedures. Bow- 
ditch (H. O. 9) contains all the essential tables. Among 
the many newer systems which have been devised for posi- 
tion finding, H. O. 211 by Comdr. Ageton is the one pre- 
ferred by the present writer. Reasons for this will be given 
in the chapter on Short-Cut Systems. Meanwhile, equa- 
tions will generally be presented in two forms. Both are 
solvable by the Bowditch tables but the second in each case 
is especially designed for solution by H. O, 211. 

The several procedures will first be briefly outlined with 
the idea that the student should get a rapid survey of the 

102 Position Finding 

general principles before getting" into the details of indi- 
vidual problems. In Part III will be found examples worked 
out for each of the more important procedures. 

The expression "dead reckoning'' which will frequently 
be used comes from deduced (ded.) reckoning and is ab- 
breviated to D. R, It is the method of finding a ship's posi- 
tion by keeping track of courses steered and distances run 
from the last well-known position. This is explained in 
Chapter 21. 

Throughout the following discussions of procedures, 
preference will be given to the new methods in which 
G. H. A. is taken from N. A, 

11. Latitude 

LATITUDE of a place is the arc of the meridian 
X of the place subtended between the equator and the 
place. It is labeled north or south in relation to the equa- 
tor. It may also be described as the angular distance on the 
celestial sphere along the hour circle of the place between 
the equinoctial and the projection of the place, or its 
zenith. By geometry, it also equals the altitude of the ele- 
vated pole. 

Figure 34 shows the four cases which cover all latitude 
calculations from observations of a body on the meridian. 
It is important to study these and understand the equa- 
tions, which may be summarized as follows: 

1 . L & d opposite names: L = z d 

2- L 8c d same name ScL> d: L = z + d 

3. L & d same name $cd> L: L = d z 

4. L & d same name, lower transit: L = 180 - (d + z) 

Regardless of what body is used, these principles apply. 
The next problem is to determine just when the body is 
on the meridian. There are three ways: 

L Measure with sextant the altitude of a body about to 
make an upper transit, that is, crossing the meridian from 
east to west, and continue to measure it at short intervals 
noting the time of each observation, until the altitude 


L = d (90 h) = d z L = 180 [d + (90 7z)] 

= h + (90 - d) = h + p 

Each Big Circle = Projection of Celestial Sphere on Plane of the Meridian 
E = Earth M Projection of Body 

P = Elevated Pole h - - - - 

Z = Zenith z 

QQ' = Equinoctial d 

NS = Horizon " 

$pj = Latitude 

FIG. 34. The Four Cases of Latitude from Meridian Altitude Observation. 


= Altitude 
= Zenith Distance 
= Declination 
= Polar Distance 

Latitude 105 

begins to decrease. Then take the greatest altitude as the 
meridian altitude. This is not especially accurate, but is 
often used for the sun. 

2. Observe the true bearing of the body and measure 
its altitude when it is directly south or north as the case 
may be. With a good compass, steady ship and not too high 
altitude, this gives fair results. 

3. Calculate in advance the time of transit. This is the 
most dependable method. It necessitates knowing the cor- 
rect longitude if ship is stationary or moving true north 
or south. If ship is making any progress east or west, the 
rate of longitude change must be known in addition to 
the longitude and time at the start of the calculation. The 
usual method of making this calculation for apparent 
noon is known as Todd's and was devised by him when a 
midshipman at Annapolis (see Bowditch or Button). 
Tables based on his equations have been published as 
H. O. 202 (Noon-Interval Tables). 

Latitude by Noon Sun 

The G. C. T. of apparent noon may be found by the 
following modification of Todd's method, using G. H. A. 
without looking up Eq. T. No mention is made of watch 
time or its difference from chronometer as this becomes very 
confusing to the beginner and can easily be dispensed with 
on smaller boats either by taking observations with chro- 
nometer nearby or by setting a stop-watch exactly with 
chronometer. The true sun is here assumed to be "mov- 
ing" at the same speed as the mean (civil time) sun. The 
error of combining a mean time interval with a civil time 
interval is immaterial (about 1 second per hour). 

106 Latitude 

Interval to Noon and G. G. T. of 
Local Apparent Noon 

1. G. C. T. at an instant in A. M. when longitude is 

2. G. H. A. of sun in arc from N. A. for this instant. 

3. Combine G. H. A. with longitude obtaining t in arc, 
always E. 

4. Convert this arc into time. This is interval to apparent 
noon at known longitude. 

5. If ship is not changing longitude, add t in time (#4) to 
G, C. T. (#1), obtaining G. C. T. of Local Apparent Noon. 

6. If ship is changing longitude, subtract t in time (#4) 
from 12, obtaining!,. A.T.of beginning of interval. 

7. Bowditch, Table 3: Enter with course in degrees (top 
or bottom of page) and speed in knots (col. labelled Dist.), 
obtaining miles made E. or W in 1 h. (col. labelled Dep.). 

8. Bowditch, Table 3: Enter with latitude of ship (figure 
for course in degrees, top or bottom of page) and miles made 
E. or W, in 1 h. (col, labelled Lat.), obtaining change of 
longitude . or W* in minutes of arc per hour (col. labelled 

9. H, 0. 202 (Noon-Interval Tables): Enter with L. A. T. 
o beginning of interval (#6) and change of longitude E. or 
W. in minutes of arc per hour (#8), obtaining interval to 
apparent noon with ship maintaining course and speed. 

10. Add this interval to G. C. T. (#1), obtaining G. C. T. 
of L. A. N. 

Latitude 107 

Another and simpler method but which requires charting 
is that of Commander Weems, Put in my own words it is as 

Weems' Method for Interval to Noon 

1. At some time in morning determine sun's t east. 

2. Convert this arc to time and fraction of hour to 
decimal. (This is time sun needs to reach meridian of # 1 

3. On chart from position of #1 run line for course as far 
as speed for time of #2 would take ship. 

4. Find DLo in minutes for length of this line. If not 
due E. or W., drop perpendiculars from each end to middle 
latitude and measure on that. 

5. Since sun moves W, 15' per minute of time, divide 
DLo of #4 by 15' to get time in minutes that sun would 
require to cover this DLo. 

6. If course is easterly, this time is saved, so subtract it 
from time of #2 for interval to noon (sun on ship's me- 

7. If course is westerly, this time is lost., so add it to time 
of #2 for interval to noon. 

8. Apply this interval time to Z. T. or G, C T, of #1 to 
get Z. T. or G. C. T. for noon sight at L. A. M 

108 Latitude 

The Meridian Sight 

Whatever method of finding the proper time for the 
meridian observation is used, the following steps must 
then be carried out: 

1. Take sextant altitude of body. 

2. Note G. C. T. 

3. Make usual altitude corrections. 

4. Subtract from 90 to get z. 

5. Find decimation in N. A. for G. C. T. of observation. 

6. Combine d and z according to which of the four cases 
was present and obtain latitude. 

Reduction to the Meridian 

A small cloud may spoil the actual meridian sight. Ob- 
servations are therefore often taken any time within 28 
minutes of noon, before or after, because such can be 
''reduced" to the meridian by Tables 29 and 30 in Bow- 
ditch. This procedure is based on the following equation: 

Meridian altitude = corrected altitude at observation + 
(change of altitude in 1 minute of time from meridian X 
square of time interval from meridian passage) or H = h 
+ a t 2 . 

Entering Table 29 with D.R. latitude and declination, 
obtain a. 

Entering Table 30 with this a and the time of observa- 
tion from meridian passage (given for i/ 2 minute), obtain 
at 2 . 

This a t 2 is to be added to a corrected altitude observed 
near upper transit (or subtracted from a corrected altitude 

Latitude 109 

observed near lower transit) to obtain the corrected alti- 
tude at meridian passage. 

Having thus found the meridian altitude, it is treated 
as explained under Meridian Sight to obtain latitude. This 
latitude is that of the vessel at the instant of observation. 
The latitude at noon will depend on the run between 
noon and sight. 

Latitude by Star, Planet and Moon Transits 

Latitude may also be found by measuring the altitude of 
some body, other than the sun, at meridian transit. This 
procedure i$ not generally recommended. The time for 
observation at twilight is quite limited and, while waiting 
for a given transit, the horizon or body may fade. So, 
only the one co-ordinate, latitude, is obtainable. Hence, 
the usual practice is to measure two or more stars whose 
bearings differ by about 90, from which, as will be ex- 
plained later, the exact position of the ship can be found. 

The general principle of finding G. C. T. of transit is 
this: Longitude must be known. When body is crossing 
local meridian, its G. H. A. = the longitude if west, or 
360 minus the longitude if east. Hence find from N. A. 
at what G. C. T. the body will have G. H. A. equal to the 
longitude, or to 360 the longitude. This will be G. C. T. 
of transit. Details of a new uniform procedure will be 
found in Part III, Chapter 30, Problems, Transit. 

Latitude by Polaris 
The altitude of the elevated pole of the celestial sphere 

110 Latitude 

equals the latitude. So in the northern hemisphere, it is 
convenient to measure the altitude of the north star or 
Polaris which is close to the celestial pole and, by means of 
certain tables in the N. A., "reduce" this altitude to that 
of the actual celestial pole. Polaris has an apparent motion 
counterclockwise in a circle with radius of about 1 around 
the actual pole. The steps in the process to thus find latitude 
are as follows (references are to 1943 N. A.): 

1. Take sextant altitude of Polaris. 

2. Note G. C, T. and longitude. 

3. Make usual altitude corrections. 

4. N. A. table for G. H. A. of Polaris (W) (p. 280) at 
O h G. C. T. of date. 

5. N. A. Table "Correction to be added to tabulated 
G. H. A. of stars" (pp. 214-16) using G. C. T. and obtain- 

ing correction. 

6. Add results of #4 and #5, subtracting 360 if over 
360, obtaining G. H. A. at G. C. T. of observation. 

7. Apply longitude to this G. H. A. obtaining L. H, A. 
of Polaris (W). 

8. N. A Table III (p. 284) entering with L. H. A. and 
obtaining correction for L. H. A, to be applied to true 

9. Add this to or subtract it from true altitude obtain- 
ing approximate latitude. 

Latitude by Phi Prime, Phi Second 

This old method for determining latitude fron! (1) a 
single altitude of a body not on the meridian, (2) G. C. T. 



of the observation, and (3) known longitude, is now seldom 
used, but is given here as an example of one of the phases 
of navigation which preceded the Saint-Hilaire method. 

The method is best restricted to conditions where the 
body is within 3 hours of meridian passage, of declination 
at least 3, and not over 45 from the meridian in azimuth. 
This last means that a line from body to zenith should not 
make an angle of over 45 with meridian. (See FIG. 35.) 

Projection of the Celestial Sphere on the Plane of the Horizon. 

h = Altitude 
Mm Perpendicular from Body to 

Meridian = I 

$>=mZ - Zenith distance of m 
= mQ = Declination of m 
QZ = Latitude to be found 

FIG. 35. Latitude by Phi Prime, Phi Second. 

WQE = Equator 
2 = Zenith 
P = Elevated Pole 
M = Body 
d = Declination 

112 Latitude 

One of the two sets of equations given in Bowditch (1933) 
for the solution is as follows: 

sin / = cos d sin t 
sin <" = sin d sec I 
cos $ = sin h sec / 

Give <j>" same name as d. 

Mark </>' North if body bears north and east or north 
and west. 

Mark <j>' South if body bears south and east or south 
and west. 

Combine f and <" by adding, if different names or sub- 
tracting, if same. 

The result will be latitude, except in the case of bodies 
nearer lower transit when 180 <" must be substituted 
for <". (The rules for marking <' and for combining are 
Bowditch's reversed, in order to make the process conform 
to other procedures in general use.) 

As the author has pointed out (U. S. N. L P. Sept. 1935), 
these Bowditch equations can be converted for satisfactory 
use with H. 0. 211 substituting R for / as follows: 

esc R = esc t sec d 


esc d>" = 


Latitude 113 


1 . Take sextant altitude of body. 

2. Note G. C. T. and longitude. 

3. Make usual altitude corrections. 

4. N. A. for declination and G. H. A. 

5. Combine G. H. A. and longitude for t. 

6. Solve by 211 method for latitude. 

12 . Longitude and 
Chronometer Error 

THE LONGITUDE o a place is the arc of the equator 
intercepted between the prime meridian (Greenwich, 
England) and the meridian of the place, measured from 
the prime meridian toward the east or west through 180. 


Projection of the Celestial Sphere on the Plane of the Horizon. 

W Qf, = Equator p = Polar distance 

Z = Zenith h = Altitude 

P = Elevated Pole QZ - Latitude 

M = Body t = Local hour angle to be 

d = Declination found 

FIG. 36. Longitude by Time Sight. 

Longitude 115 

The "Time Sight" method for longitude dates back to 
1763 and is still much in use in the merchant service 
though not in the navy. It calls for (1) a single altitude of a 
body preferably near the prime vertical (bearing east or 
west), (2) G. C. T. of the observation and (3) known lati- 
tude. Practically, the body should be between 3 and 5 hours 
of meridian passage, (See FIG. 36.) 

The long-used equation for this problem is: 

hav t = secL esc p cos s sin (s h) 
where s = Vz (h + L + p) 

Having thus found t, it is combined with the G. H. A. 
from N. A. to give longitude. 

Hulbert Hinkel, Jr. (U. S. N. I. P. April 1935) presented 
the following substitute equation for use with H. O. 211: 

sec s esc (s h) 


1. Take sextant altitude of body. 

2. Note G. C. T. and latitude. 

3. Make usual altitude corrections. 

4. N. A- for declination and G. H. A. 

5. Solve by 211 method for t. 

6. Combine t with G. H. A. for longitude. 

NOTE: A method for finding the time at which the sun will be on the prime 
vertical will be found in Part III, Chap. 30, Problems. 

116 Chronometer Error 

Chronometer Error 

Radio time signals now furnish the best means of deter- 
mining chronometer error at sea. 

In some circumstances, however, the radio receiver may 
fail, or there may be no radio. In such case, if correct 
longitude is known, the process is as follows: 

L Take sextant altitude of body as near prime vertical 
as possible. 

2. Note chronometer (approx. G. C. T.) and longitude. 

3. Make usual altitude corrections. 

4. N. A. for declination. 

5. Solve by 211 method for L 

6. Combine t with longitude for G. H. A. 

7. N. A. to find the G. C. T. of this G. H. A. 

8. Difference of this G. C. T. from chronometer equals 
part or all of error. 

9. If body was sun, moon, or planet, go back to #4 and 
use the revised G. C. T. found in #7 to pick out more 
exact declination. Use it in repeating #5 and repeat re- 
maining steps through #7. Difference of this second 
revised G. C. T. from chronometer will show practically 
all error. 

NOTE: The method for #7 of finding from N, A. what the G. C. T. is for 
a certain G, H. A. is the same as the method of finding G. C. T. of transit. 
(See Part HI, Chap. 30, Problems, Transit.) 

13* Azimuth and 
Compass Error 

A ZIMUTH of a celestial body is the angle at the zenith 
,/\between the meridian of observer and the vertical 
great circle passing through zenith and the body. It is 
usually measured from the north in north latitudes, east 
or west through 180, and similarly from the south in 
south latitudes and designated Z. It is also measured from 
the north point clockwise through 360 and is then iden- 
tical with true bearing and labeled Z n . (See FIG. 37.) 

Azimuth of a body is needed for two main purposes: (1) 
for drawing a line of position, as will be explained in the 
next chapter, and (2) to determine the error of a magnetic 

Azimuth is found by the following means: 

L Gyro compass (with azimuth circle). 

2. Table. 

3. Formula (with and without sextant observation). 

4. Diagram (with and without sextant observation). 

5. Celestial Co-ordinator. 

6. N. A. (after obtaining latitude by Polaris). 

The first is possible because of the fact that gyros show 
true directions with no deviation or variation. 

The second is much used. H. O. 71 ("Red" Azimuth 
Tables for declinations - 23 and to Lat. 70, usually 
for the sun) and H. 0. 120 ("Blue" Azimuth Tables for 




declinations 24 70 and to Lat. 70, for stars) are the 
standard books. Entering with L, d, and t (the first two in 
even degrees and the last in I0-minute time intervals) 
gives Z. A rather long and tedious calculation is necessary 
in most instances because the quantities usually lie between 


Projection of the Celestial Sphere on the Plane of the Horizon. 

WQE Equator M = Body 

Z'= Zenith Z = Azimuth direct from Ele- 

p = Elevated Pole vated Pole 

Z n = Azimuth by 360 system 
FIG. 37. Azimuth. 

these entering values and interpolation is necessary. 
"Cugle's Two-Minute Azimuths" recently published in 
two volumes are more expensive, but also save some time. 
They are arranged on the same basis as the H. O. volumes 
except for having 2-minute time intervals. They cover lati- 

Azimuth 119 

tudes to 65 and declinations to 23. H. O. 66, Arctic 
Azimuth Tables, is a small volume, also on the same basis, 
for latitudes 70 - 88 and declinations to 23. It is lim- 
ited to the hours 4 to 7 A. M. and 5 to 8 P. M. at 10-min- 
ute intervals. H. O. 200 includes a short table which is 
entered with h cy d and t and requires some interpolation. 
It covers declinations to 89 30'. 

The new series of volumes, H. O. 214 (see Chap. 16), 
provides probably the most satisfactory means of similarly 
finding azimuth. L is in whole degrees, d in whole and usu- 
ally half degrees and t in whole degrees which corresponds 
to 4-minute time intervals. The volumes already published 
carry latitudes to 79 and decimations to 74 30'. (A prob- 
lem showing method of interpolation will be found in 
Chapter 30.) 

There are many formulas for azimuth, but they all fall 
into the following three classes: 

Time Azimuth (Z obtained from t, p 3 and L). The equa- 
tions and method are lengthy and will not be given here 
but may be found in Bowditch. 

Altitude Azimuth (Z obtained from h, p, and L). Collins' 
equation (U. S. N. I. P. July 1934) is recommended: 

hav Z = sec h sec L sin (s h) sin (s L) 
where s = Yz (h + L + p) 

Hinkel (U. S. N. I. P. June 1936) gives the following for 
use with H. O.211: 


sec h sec L 

In either case it is measured from N. in N. lat. and from 

120 Azimuth 

S. in S. lat. The quantity s p may have a negative value 
but this does not matter since the secant of a negative angle 
less than 90 is positive. (See Table 14.) 

Time and Altitude Azimuth (Z obtained from t 9 h, and 
d). The Bowditch equation is: 

sin Z = sin t sec h cos d 

There is a defect in this method in that nothing indi- 
cates whether the azimuth is measured from north or south. 
However, as the approximate azimuth is always known, the 
solution will almost always be evident. When in doubt, with 
sun almost E. or W. and L and d same name, find altitude 
when sun is E. or W. as follows: sin h = esc L sin d. If ob- 
served altitude was less, sun was on side toward elevated 
pole, etc. 

Using H. 0.211: 

esc t sec d 

esc Z = 

sec h 

An azimuth diagram is often accurate enough for prac- 
tical purposes and saves much of the time ordinarily spent 
in calculating formulas or interpolating tables. A copy of 
Rust's azimuth diagram comes with one of the short-cut 
systems, Weems' "Line of Position Book." Entering with 
t, h, and d one arrives at Z. Another diagram by Capt. Weir 
is mentioned by Button. Weir's uses t, d, and L to obtain Z. 

A Celestial Co-ordinator (see Chap. 7) set for L, t and d 
will give approximate values for Z and 7z. 

The azimuth of Polaris may be found from the Nautical 
Almanac as follows: 

1. Find approximate latitude by procedure given in 
Chapter 1 1 under Latitude by Polaris. 

Azimuth 121 

2. Enter Table IV (p. 285 in 1943 N. A.) with L. H. A. 
and approximate latitude and take out azimuth. 


Time Azimuth 

1. G. C. T. and L. and Lo. (or t). 

2. N. A. for G. H. A. and d(p = 90 d). 

3. G. H. A. and Lo. for t (unless given). 

4. Table (H. O. 66, 71, 120, 214 or Cugle's) or formula or 
diagram (Weir). 

Altitude Azimuth 

1. Sextant altitude. 

2. G. C. T. of observation, and L. 

3. Make usual altitude corrections* 

4. = 90d). 

5. Formula, 

Time and Altitude Azimuth 

1. Sextant altitude. 

2- G. C. T. of observation, 2md Lo. 

3. Make usual altitude corrections. 

4. N. A. for G. H c A. and d. 

5. G. H. A. and Lo. for t. 

6. Formula, or Table (H. O. 200), or diagram (Rust). 

Azimuth of Polaris 

1. Latitude by Polaris. 

2. N. A. Table IV. 

122 Compass Error 

Compass Error 

The compass azimuth, taken with an azimuth circle, o 
a body, usually the sun and preferably at low altitude, is 
compared with the true azimuth of the body, which is de- 
termined by one of the methods described. The difference 
is the error: variation plus deviation. The difference be- 
tween this total error and the variation, obtained from the 
chart, is the deviation on the particular heading. 

14. Sumner Lines of Position 

CAPT. THOMAS H. SUMNER, an American ship- 
master, on December 18, 1837, near the end of his 
ship's voyage from Charleston, S. C., to Greenock, Scotland, 
was in need of data as to his position. About 10 A. M. an 
altitude of the sun was obtained and chronometer time 
noted. As the D. R. latitude was unreliable, two additional 
latitudes 10' and 20' farther north were assumed and the 
three possible longitudes were worked out. When the three 
positions were plotted on the chart, they were found to be 
in a straight line. "It then at once appeared that the ob- 
served altitude must have happened at all the three points 
. . . and at the ship at the same instant." The conclusion 
was that, although the absolute position of the ship was 
uncertain, she was necessarily somewhere on that line. Capt. 
Sumner published his discovery in 1843 and it is considered, 
with the previous invention of the chronometer and the 
subsequent development of the Saint-Hilaire method, as 
one of the three greatest contributions to the science of 

Circles of Equal Altitude 

If we look up at the top of a vertical flagpole from a 
certain distance away from, its base, we will be gazing 
upward at a certain angle. We may walk all around the 
pole keeping at the same distance, but this angle will not 


124 Sumner Lines of Position 

It is obvious that a heavenly body directly over the nonh 
pole would show the same altitude from every point on a 
given parallel of north latitude. 

It is somewhat less obvious but none the less true that, if 
a certain heavenly body is in the zenith at a point anywhere 
on the earth at a given time, and if circles in the same 
hemisphere are imagined around this point as a center, 
then the altitude of such body from every point of any one 
circle at the given time will be the same. Measuring the 
radius of such a circle in degrees on the earth's surface, 
the largest possible circle, dividing the earth into hemi- 
spheres, would have a radius of 90, From any point on 
this circle of 90 radius, the body would be in the horizon 
and have an altitude of 0. From the center with radius 0, 
the body would be in the zenith and have an altitude of 90. 
On a circle with radius of 10 the body's altitude would be 
80, etc. The radius is, therefore, the complement of the 
altitude and so equals the zenith distance. Expressing it in 
minutes of arc, the zenith distance equals the number of 
nautical miles the observer is away from the center of the 
circle, where the body is in the zenith. (See FIG. 38.) 

In an astronomical sight, the following is learned about 
the point on the earth in whose zenith the body is: 

1. Longitude-from G. C. T. and G. H. A. 

2. Latitude from d. 

3. Distance from observer from h. 

There is nothing to show at what point on the circle of 
equal altitude the observation was made. But an observa- 
tion of another body preferably in a direction at right angles 

Sumner Lines of Position 



* = Extremely distant fixed star 
Circle = Earth in plane of vertical circle of body observed 

Z = Zenith of observer 
OB = Line from observer toward body 
h = Altitude 
z = Zenith distance 

' = Place on earth where body is in zenith (geographical position) 
ZO is prolonged to center of earth at O' 
O'B' is parallel to OB considering distance of body 

Therefore, by geometry: Angle OO'B' = angle ZOB, or angular distance on 
earth from observer to body's geographical position = zenith distance of 
body at place of observer (for nearer bodies, parallax correction makes them 
equal anyway). 

FIG. 38. Zenith distance and the Radius of the Circle of Equal Altitude. 

1 26 Sumner Lines of Position 

to the first (or the same body several hours later) can be 
made in order to get a new circle. 

This will intersect the first circle at two places, one of 
them being the ship's position. But as the ship's position is 
always approximately known within about 30 miles, and as 
these two intersections may be thousands of miles apart, 
there is no question as to which is the correct one. 

The Line of Position 

It is never necessary to determine the whole of a circle of 
equal altitude. A very small portion of it is sufficient, and 
such an arc may be considered as a straight line for the 
length needed to cover the probable limits of the position 
of the observer. Such a line is known as a Sumner line or 
line of position. It gives a knowledge of all the probable 
positions, while a sight worked with a single assumed lati- 
tude or longitude gives only one probable position. It always 
lies at right angles to the direction of the body from the 
observer, as a tangent to a circle through a point is perpen- 
dicular to the radius at that point. 

In the two earlier line of position methods, sights were 
worked for latitude when the body was nearer north or 
south and for longitude when the body was nearer east or 
west. The three methods of determining a line of position 
are as follows: 

1. Chord method: for one sight assume two values of 
latitude and determine longitudes or assume two values of 
longitude and determine latitudes. Two points are thus 
fixed on the chart and the line joining them is the line of 

Sumner Lines of Position 127 

2. Tangent method: for one sight assume one latitude or 
one longitude and determine the other co-ordinate. One 
point on the line is thus obtained. The azimuth of the body 
must now be found either by formula, table, or diagram, 
and a line drawn from the one point at this angle in the 
direction of the body. Then perpendicular to this line and 
through the point is drawn the line of position. When using 
the time-sight formula for longitude, it is convenient to 
also use the altitude azimuth formula. When using the Phi 
Prime, Phi Second formula for latitude, one should use 
the time and altitude formula for azimuth. 

3. Saint-Hilaire method: assume both latitude and longi- 
tude using either the D. R. position or one nearby and 
calculate what the altitude and azimuth of a body would 
be, there, at the time an actual altitude was taken on the 
ship. The difference between the actual and the calculated 
altitude shows how far to move along the azimuth line from 
the assumed position before drawing a perpendicular. This 
is the line of position. (This method will be discussed in 
more detail in the next chapter.) 

15. Saint-Hilaire Method 

THE FRENCHMAN Adolphe Laurent Anatole Marcq 
de Blond de Saint-Hilaire was born at Crcy-sur-Serre 
(Aisne) July 29, 1832 and entered the French Navy in -1847. 
He was made a Commander of the Legion of Honor in 
1881 and reached the rank of Contre-Amiral in 1884, fol- 
lowing distinguished service in the Tunis expedition. He 
died in Paris, December 30, 1889. Little information is 
available about him. His method of the calculated altitude 
was published as "Calcul du point observe. Methode des 
hauteurs estim^es/' in "Revue Maritime et Coloniale" Vol. 
XLVI, pp. 341-376, August 1875. He split the astronomical 
triangle by dropping a perpendicular from the body onto 
the meridian and solved the resulting right spherical trian- 
gles by logarithms, using the D. R. position. 

Lord Kelvin before the Royal Society, February 6, 1871, 
had announced a method of comparing calculated with 
actual altitude for locating the line of position. He used an 
assumed position and computed tables for solving the tri- 
angle. These were published as "Tables for Facilitating 
Sumner's Method at Sea" in 1876. 

It would be interesting to know if Saint-Hilaire knew of 
Kelvin's announcement, and when Saint-Hilaire first devel- 
oped his method. 

Everything so far in this Primer has been leading up to 
the Saint-Hilaire method. It is a most important develop- 
ment in the science of position finding and is the basis of 
practically all modern navigational systems. 


Saint-Hilaire Method 129 

Details of Procedure 

1. Take an altitude and note G. C. T. 

2. Make usual altitude corrections. 

3. Note D. R. latitude and D. R. longitude. 

4. N. A. for G. H. A. and d. 

5. Combine G. H. A. and D. R. longitude for t (for D. R. 

6. With t, d, and L using one of several formulas to be 
given and perhaps tables or a diagram, calculate altitude 
(h c ) and azimuth (Z). These are the values the body would 
have had if the observation had been made at exactly the 
D. R. position. 

7. The difference between your observed altitude cor- 
rected as in #2 and the calculated altitude of #6, in min- 
utes of arc, represents the difference in miles between your 
actual distance from the place where the body is in the 
zenith, called its geographical position, and the distance of 
the D. R. position from the body's geographical position. 
(We could compare the observed zenith distance with a 
calculated zenith distance but the altitude way is more 
convenient.) This difference is called the altitude difference 
or intercept and is given the abbreviation a. If the observed 
altitude is greater than the calculated, it means your actual 
position was nearer the body's geographical position than 
was the D. R. position. (You were more "under" the body.) 
But if your observed altitude is less than the calculated, 
then you were really farther from the body's geographical 
position than was the D. R. position. (You were less "under" 
the body.) The a is, therefore, labeled "toward" or "away." 

130 Saint-Hilaire Method 

8. Through the D. R. position on the chart draw a line 
at 'the calculated azimuth angle in the direction of the body's 
geographical position. 

9. Lay off on this line from the D. R. position a distance 
in miles equal to the intercept, either toward or away from 
the body's geographical position, as the case may be. 

10. Through the end of this intercept draw a line per- 
pendicular to the azimuth line. This is the line of position. 
(See FIG. 39.) 

One of the best formulas for calculating the altitude 
called for in #6 is known as the cosine-haversine and is 
described by Button as "the most widely used formula of 
trigonometry applied to nautical astronomy. It is univer- 
sally applicable to all combinations of the values of t, d, and 
L." Here it is: 

hav z = hav (L ~ d) + hav 
where hav & = have t cos L cos d 
from which h c - 90 - z 

(L and d: add opposite names, subtract likes) 

Convenient to use with this is the time and altitude azimuth 

sin Z = sin t cos d sec h c 

Other formulas or tables or a diagram may also be used. 

The cosine-haversine formula and a formula for azimuth 
should be memorized once and for all by every student 
navigator. Knowing these one could be independent of all 
short-cut systems and could function even when only tables 
of a foreign publication were available. 

Saint-Hilaire Method 







D,R. posmon 


FIG. 39. The Saint-Hilaire Method. 

132 Saint-Hilaire Method 

The Fix 

We have seen that one line of position does not fix the 
exact location of the ship. Another line must be obtained 
which will intersect the first giving what is called a fix, and 
so show the real position. The second line preferably should 
be at about 90 to the first although a smaller angle down 
to 45 may be effective. The ideal fix is probably from three 
stars at about 120 intervals around the observer, giving 3 
lines which should intersect at a point or at least with 
formation of a small triangle. When the same or another 
body is used later in the day to get the second line, the first 
must be moved forward parallel to itself according to the 
course and distance made good. These latter factors are 
estimates from compass and log readings or engine revolu- 
tions and open to error through current, wind, etc. Hence 
a line which is over five hours old in a moving ship is not 
trusted. A fix obtained after moving forward a previous 
line is known as a running fix. 

The best method for a series of two or more star sights is 
to use the D. R. position at the time of the last of the series 
in working each sight. Then plot the line for the last sight 
in the usual way but advance the lines of the previous sights 
according to time difference, speed and course of ship to 
get fix at time of last sight. 

When chart work is impossible, a method for computing 
the position of the intersection of position lines (Bowditch, 
pp. 190-3) may be followed. This requires knowledge of the 
position of one point on each line and so the lines had best 
be obtained by the tangent method. (See Chap. 14.) 

Saint-Hilaire Method 1 33 

The various practicable combinations of observations for 
a fix are as follows: 


Sun and Sun 
Simultaneous or close 

Sun and Moon 

Sun and Venus 

Venus and Moon 
Simultaneous or close 

Star and Star 

Star and Planet 

Star and Moon 

Planet and Planet 

Planet and Moon 

The Computed Point 

This is the point where a line from the D. R. position 
dropped perpendicular to the line of position intersects 
the latter. It is the mean of the possible positions of the 
ship on the line. It, therefore, is the best point to use as the 
ship's probable position in the absence of an intersecting 
line and without contrary evidence from weather or 

16. Short- Cut Systems 

ular form by Soule and Collins, copyrighted by the 
U. S. Naval Institute, was published by the U. S. Hydro- 
graphic Office as "Supplement to the Pilot Chart of the 
North Atlantic Ocean" in 1934. It gives a ready compari- 
son of the salient features of twenty-nine different methods 
for determining the elements of the astronomical triangle 
in order to plot the line of position. There are seventeen 
systems working from an assumed position and three from 
the D. R. position, making twenty for the Saint-Hilaire 
method. The remaining nine are longitude methods for 
chord or tangent lines. Since this publication, I know of 
five more methods which have appeared: Aquino's ' 'Tan- 
gent Secant Tables," "Hughes* Tables for Sea and Air 
Navigation/' Ageton's "Manual of Celestial Navigation" 
(for D. R. or A. P.) and H. O. 214 (all of which use the 
Saint-Hilaire method) and Aquino's method for finding 
latitude and longitude directly. 

Assumed vs. D. R. Position 

The advantages of using an assumed position are that 
fractions of degrees of latitude or hour angle may be 
avoided and arithmetic simplified. Dutton (7th ed., p. 204) 
states that any position correct to within about 40' may be 
used. "No matter what position is assumed within the 40' 
limit, the line of position will plot in the same place on the 

Short-Cut Systems 135 

chart, the altitude difference, a, changing with each change 
of assumed position." The chief disadvantage is that chart 
work is complicated by using a different assumed position 
for each of several sights taken in a short interval. This 
means drawing each azimuth line through a different point. 
Another bad feature is that long altitude intercepts may 

The advantages of working from the D. R. position are: 
it is run forward on the chart from the last fix; only the one 
position is plotted for any group of sights; errors are more 
apparent; much smaller altitude intercepts result which 
makes for accuracy; and (theoretically) the azimuth should 
be more accurate. The disadvantages are that a little more 
arithmetic and time are required. 

In view of the foregoing and after using both, the writer 
has chosen the D. R. position system. Of the various meth- 
ods and tables available, he recommends the use of "HL O. 
211" by Comdr. A. A. Ageton, U. S. N. Its full title is "Dead 
Reckoning Altitude and Azimuth Table." 

H. 0.211 

Advantages of H. O. 211 are as follows: It gives sufficient 
accuracy without interpolation; except for H. O. 214, it is 
the shortest D. R. method; it follows a uniform procedure 
for sun, moon, star or planet; it consists of only one table of 
36 pages running in two columns of almost all whole num- 
bers; the table is indexed and very simple and easy for rapid 
use; azimuth is determined without question as to origin; 
and there are only two special rules. (See FIG. 34.) 

The only disadvantage is that when the hour angle lies 


Rule 1: 

Short-Cut Systems 

1 ft is > 90 

K is always < 90 I ^ J Take iC from bottom of 

Take from top of table! when j table 

Z is always > 900 f . _ _ . , 

Take Z from hot- L cxce P t J A 1S same name as & > latltude 
torn of table J when 1 Take Z from top of table 

Rule 2: 

Of course, if ^ = 

^ takes same name as declination (N. or S.) 

FIG. 40. Rules for H.O. 211. 

(Projection of celestial sphere on plane of horizon. R is dropped from 
body M, perpendicular to meridian.) 

Short-Cut Systems 137 

within about 3 of 90 the solution requires such close 
interpolation that Ageton recommends discarding those 

The table consists of double columns of log cosecants and 
log secants each X 10 5 which eliminates decimal points in 
all but a few pages. Without interpolating between the 
values, which are given for each half-minute of arc, accuracy 
(excluding, of course, sextant errors) is within i/% mile. 

The formulas used, derived from Napier's second rule, 
are as follows: 

esc R = esc t sec d 
esc d 

esc K = 

sec R 

esc h c = sec R sec (K - ' L) 

esc R 
esc Z = 

sec h c 

Give K same name as d. 

K ' ' L: subtract likes, add opposites. 

NOTE: See Part III, Chap. 29, for convenient forms to use 
with H. O. 211. See Ageton, A. A.: "The Secant-Cosecant 
Method for Determining the Altitude and Azimuth of a 
Heavenly Body." U. S. Naval Institute Proceedings. Oct. 
1931, Vol. 57, No. 344, pp. 1375-1385. 

H. O. 214 

This recent development in navigational tables is en- 
titled "Tables of Computed Altitude and Azimuth/' The 
complete set will consist of nine volumes. Eight have already 

138 Short-Cut Systems 

been published covering latitudes to 79 (10 to a volume). 
Either the D. R. or an assumed position may be used. The 
procedure in either case is the shortest of its kind possible 
at present. The azimuth from the table may be interpolated 
for the D. R. position if extreme accuracy is desired. Tables 
for star identification are included at the end of the data for 
each degree of latitude. The fact that the whole set is of 
nine volumes need not discourage the average cruising 
yachtsman, since each volume covers 600 miles in latitude. 
A few of the volumes will probably be sufficient in most 

In Chapter 30 will be found a fix from two stars worked 
out by H. O. 214 with both the D. R. position and the as- 
sumed position methods. 

It will be seen that the method which uses the D. R. 
position for each sight exceeds in length the method using 
a different assumed position for each by two short altitude 
corrections (L and t) and a combination of the three cor- 
rections, for each sight. 

However, the method using a different assumed position 
for each sight requires the calculation of the assumed longi- 
tude, sometimes by a subtraction, and the plotting of each 
azimuth line from a different point. The intercepts are 
usually much longer from assumed positions and the azi- 
muths not quite so accurate. 

The following summaries will show each step. 

H. O. 214: Method Using D. R. Position 

1. Take an altitude by sextant and note G. C T. 

2. Make usual altitude corrections for h . 

Short-Cut Systems 139 

3. Note D. R. Latitude and D. R. Longitude. 

4. Find d and G. H. A. in N. A. and t from G. H. A. & 
D. R. LQ. 

5. Enter tables with nearest whole degree of latitude and 
local hour angle and nearest whole or half degree of declina- 

6. Copy the 4 values found which will be h, Z, A d and A 
t. These last two represent the change in altitude due to a 
change of 1' of arc of declination and of local hour angle. 

7. Calculate by inspection the differences between fig- 
ures used to enter tables and actual values of d, t (D. R.) 
and (>.#.) 

8. Use table on back cover pages to multiply d and t 
differences (found in #7) by A d and A t respectively. 
Give each result a + or a sign according as the tables show 
the altitude to be increasing or decreasing in passing from 
the chosen tabulated value toward the value to be used. 
These are altitude corrections for d and t. 

9. Use another table in back of book entering with L 
difference (found in #7) and azimuth (found in #6) and 
take out altitude correction for L. With Z > 90: if D. R. 
latitude exceeds chosen tabulated value, give L correction 
a sign; otherwise mark it +. With Z < 90: if D. R. lati- 
tude exceeds chosen tabulated value, give this L correction 
a + sign; otherwise mark it . 

10. Combine these three altitude corrections algebra- 

1 1 . Combine the resulting total with altitude from table 
(found in #6) to get h . 

12. Combine h c and h for intercept. 

13. Plot from D. R. position. 

140 Short-Cut Systems 

H. 0. 214: Method Using Assumed Position 

1. Take an altitude by sextant and note G. C. T. 

2. Make usual altitude corrections for h . 

3. Note D. R. Latitude and D. R. Longitude. 

4. FinddandG.H.A.inN.A. 

5. Assume the whole degree Latitude nearest D. R. 

6. Assume Longitude nearest D. R. which when com- 
bined with G, H. A. will give a whole degree for t. 

7. Enter tables with assumed latitude and local hour 
angle and nearest whole or half degree of declination. 

8. Copy the values found for h, Z and A d. This last rep- 
resents the change in altitude due to a change of 1' of arc of 

9. Calculate by inspection the difference between decli- 
nation used to enter tables and the actual declination. 

10. Use table in back cover pages to multiply d difference 
(found in #9) by A d. Give result a + or - sign according 
as the tables show the altitude to be increasing or decreasing 
in passing from the chosen tabulated value toward the 
value to be used This is the altitude correction for d. 

11. Combine this correction with altitude from table 
(found in #8) to get h c . 

12. Combine h c and h for intercept 

13. Plot from the assumed position. 

17. Special Fixes 

A>ART FROM THE USUAL TYPE of fix described 
in Chapter 15, there are several methods for certain 
circumstances, some of which will now be described. 

The Zenith Fix with Sumner Arcs 

When the altitude of a body is over 89 its zenith distance 
is, of course, a matter of less than 60'. It will be recalled 
that the zenith distance of a body in minutes of arc repre- 
sents the distance in nautical miles from observer's position 
to the geographical position of the body, namely, that point 
on earth which has the body in its zenith. (See FIG. 38.) 
Ordinary lines of position under these circumstances are 
not accurate because a straight line here will only coincide 
with a very small part of the circle of equal altitude. It is 
better to draw the actual circle or an arc of it. This can be 
done as follows: 

Take a sextant altitude 3 or 4 minutes before transit, 
note G. C. T. and make usual corrections. The G. H. A. of 
the body at this G. C. T., obtained from N. A., represents 
the west longitude of its geographical position. The decli- 
nation represents the latitude of its geographical position. 
This point can, therefore, be plotted on the chart or plotting 
sheet. With the point as a center and with dividers set by 
the latitude scale for the number of miles equal to minutes 
of zenith distance, an arc can be swept in the general direc- 
tion of the ship. Following a similar observation from the 


142 ' Special Fixes 

same position about 3 or 4 minutes after transit, another 
arc from a new center can be similarly swept which will 
intersect the first and give a fix. In case the ship has mean- 
while moved, the first arc can be advanced by shifting its 
center according to course and distance covered to give a 
fix at second observation. Or the second arc can be pushed 
back similarly to give a fix at first observation. For accuracy, 
a third arc from an observation at transit may be used, but 
is not essential. 

The azimuth in this situation is rapidly changing from 
east of the meridian to west of it. If body is to transit be- 
tween zenith and elevated pole, azimuth (on the 180 
system) will diminish to and then increase. If body is to 
transit with zenith between it and elevated pole, azimuth 
will increase to 180 and then diminish. 

Theoretically, this method should apply to any body 
passing near zenith but it is unlikely that a star would be 
found and satisfactorily observed in this way at twilight. 
With the usual method available of a fix from two stars 
there would be no need for it. 

Practically, it is of occasional use in daylight with the sun 
at noon or Venus at transit. These bodies will only pass 
near zenith if the ship is in low latitudes and when decli- 
nation and latitude are about equal. (See Lecky, 22nd ed., 
p. 500; Button, 6th ed., p. 212; Bugger, U. S. N. I. P. Jan. 

The Noon Fix with Equal Altitudes 

This is described by Lecky (22nd ed., pp. 482-4). Noon 
latitude is obtained by a meridian altitude in the usual way. 
If the ship is practically stationary and the sun's altitude 

Special Fixes 143 

not less than 75 the longitude may be found as follows. 
Use another sextant. From 10 to 15 minutes before local 
apparent noon observe sun's altitude and note G. C. T. 
Clamp the sextant. Observe sun with this clamped sextant 
after noon until sun is seen on horizon as before and again 
note G. C. T. The half-sum of these two times will be G. C, 
T. of apparent noon at ship. G. H. A. obtained from N. A. 
for this G. C. T. will equal longitude. 

The Fix on Equator at Equinox 
On the two equinox dates of each year the sun rises prac- 
tically due east and sets practically due west throughout the 
earth except in extreme polar regions. At the equator it 
will pass through zenith at noon, the azimuth remaining 
90 all day from either pole, E. in A. M. and W. in P. M., 
except for the instant when it is at noon. Set the sextant 
at 90 X (see Note) and observe sun eastward in A. M. till 
its lower limb just meets horizon and note G. C. T. (A quick 
look with sextant to N. or S. should show no change if you 
are actually on equator.) G. H. A. obtained from N. A. for 
this G. C. T. will equal long.; lat. = 0. 

NOTE: X represents the corrections for height of eye and semidiameter 
which must be allowed for in order to observe the sun at a true altitude 
of 90. (Parallax and refraction do not operate when body is in zenith.) 
If the sextant were set for 90, the subsequent application of these cor- 
rections would show a true altitude in excess of 90 when observation is 
made from the usual height of a ship's bridge. 

For example: 

Table A. Corr. for sun (S. D.) alt. 90 = + 16' 
" B. Corr. for sun (S. D.) Mar. 21 = + O'.l 
" C. Corr. for 39 feet (H. E.) = - 6'.1 

Total + 10'. 

144 Special Fixes 

Reversing the sign and applying to 90 
90 00' 

89 50 7 = sextant setting. 

At any other time of day on equator at equinox, if a 
sextant altitude is taken, and corrected, then t = 90 h 
and t combined with G. H. A. for time of observation = 
longitude; latitude = 0. 

The Fix on Equator Not at Equinox 

Hinkel (U. S. N. I. P. June 1936) gives the following 
equation for local hour angle, to be worked by H. O. 211, 
when on equator, not at equinox, but with a definite value 
for the sun's declination. 

esc h 

sec t = - 
sec d 

Combining t with G. H. A. gives longitude; latitude = 0. 

Aquino's Fix from Altitude and Azimuth 

Capt. Radler de Aquino of the Brazilian Navy in his 
article "A Fix from Altitude and Azimuth at Sea and in 
the Air" (U. S. N. I. P. Dec. 1936) presents a method of 
obtaining a fix from simultaneous sextant and compass 
observations of a single body. The azimuth must be taken 
with a gyro compass in perfect order or with a magnetic 
compass whose error is definitely known. The student is 
referred to the original article for details, equations used 

Special Fixes 145 

and their derivation but it is sufficient to state here that 
latitude and longitude are obtained directly from the rea- 
sonably short calculation. A form for the procedure is 
given in Chapter 29 and a sample problem in Chapter 30. 
Any tables containing log tan, log sec and log esc for every 
T of arc may be used. Aquino has prepared an especially 
convenient table of these functions which is contained in 
his "A Navega^ao Hodierna com Logaritrnos de 1633!" 
This can be purchased from R. de Aquino, 133 Rua Raul 
Pompeia, Copacabana, Rio de Janeiro, Brazil. The tables 
occupy only 18 pages. The practical difficulty in this other- 
wise ideal method is, of course, in obtaining a sufficiently 
accurate azimuth. 

The Fix with Weems' Star Altitude Curves 

Star Altitude Curves, published by the Weems system of 
Navigation, Annapolis, Maryland, represent a new develop- 
ment in position finding and under favorable conditions 
offer the means for obtaining a fix in the amazingly short 
time of two minutes. 

The following items are dispensed with: assumed posi- 
tion, Nautical Almanac, right ascension, hour angle, azi- 
muth and the plotting of position lines. 

The student is referred to the publisher for details or can 
find a good description in Button, 7th ed., pp. 352-5. 

Very briefly, each page is a "grid" formed by the respec- 
tive equal altitude curves, or lines of position, of three stars 
plotted on a Mercator chart at 10' intervals against latitude 
(left and right edges) and local sidereal time (top and bot- 
tom). One of the stairs is usually Polaris. 

146 Special Fixes 

At the top are shown the names of the stars used on that 
page and the half or whole hour of local sidereal time which 
is covered. Each page covers about 10 of latitude and the 
collection covers latitudes from to 70 30' and local side- 
real time from O h to 24 h . 

No correction is needed when the altitudes are observed 
with a bubble sextant. With a standard sextant, only the 
correction for dip of the horizon is used. 

For any place and sidereal time, a circle of equal altitude 
of a star remains nearly the same from year to year. An 
annual correction is indicated under the name of each star 
at the top. 

A second-setting watch on G. S. T. is used, or else G. C. T. 
is converted to G. S. T. 

The procedure in simplest form, for northern hemisphere, 
is as follows: 

1. Apply approximate longitude (in time) to G. S. T. to 
obtain approximate L. S. T. 

2. Turn to the corresponding page for this time and for 
the approximate latitude. 

3. Observe altitude of one of the "longitude stars" and 
note G. S. T. 

4. Observe altitude of Polaris. (Time of this is not nec- 
essary, as altitude of Polaris changes so slowly.) 

5. Find the proper altitude line and fraction on the 
Curves for the longitude star and do the same for Polaris. 
Mark the intersection of these two lines. 

6. Project this intersection horizontally to the scale to 
find latitude. 

7. Project this same point vertically to the scale to find 
L. S. T. 

Special Fixes 147 

8. Take the difference between this and G. S. T. of ob- 
servation to find longitude in time. 

9. Convert above to arc to find longitude. 

This remarkable procedure is at present most suitable 
for high-flying fast airplanes. The limitation to the three 
stars given on each page without curves for sun, moon or 
planets, make it a supplement to, rather than a substitute 
for, the usual methods of marine celestial navigation. 

18. Polar Position Finding 

THE AMATEUR NAVIGATOR will rarely be con- 
cerned with lines of position in arctic regions. How- 
ever, two reasons have led me to include something of 
these matters in this Primer: first, we will be completing 
the story of position finding for the whole earth instead 
of the usual limit of 65 latitude N. or S.; second, there is 
practically nothing printed about polar position finding 
in any of the texts or manuals in common use. 

Let us consider for a moment certain special conditions 
which exist at, say, the exact geographical north pole. After 
understanding these, we can more easily understand the 
conditions in the neighborhood of the pole which are simi- 
lar but which become less so, the farther away from the 
pole we go, until they gradually become those with which 
we are already familiar in latitudes below 66. 

At the North Pole 

At the north pole, then, every direction radiating out 
from an observer is south. 

Circles drawn with observer as center will be leading 
west clockwise or east counterclockwise. 

The horizon of an observer at the north pole marks the 
equinoctial, or celestial equator, above which all bodies 
are in northern declination. A north declination as given 
in N. A. will then be the true altitude, Sextant altitudes, 


Polar Position Finding 149 

of course, would differ by the amount of the usual cor- 

The sun at Spring equinox, March 21, appears above the 
horizon and skims around it in about 24 hours. It remains 
visible and rises gradually to an altitude of about 23 27' 
at summer solstice, June 21. It then gradually sinks to dis- 
appear at Fall equinox, Sept. 23, and remains invisible for 
the six months of arctic night. 

The moon's declination is north about one half of each 
month and at such times, of course, this body will be above 
the horizon. It sometimes attains an altitude of over 21. 
The presence of the sun does not make it invisible and so 
when the two are in evidence they can often be used for 
lines of position which intersect and give a fix. 

All stars of northern declination are theoretically visible 
during the arctic night from Fall to Spring equinox. As 
their declinations change so little from year to year, this 
upper half of the "inner surface of the celestial sphere" 
presents an unvarying picture except for the moon and 

The four navigational planets appear at times during 
arctic night. Naturally their different orbits and varying 
declinations will change the program from year to year. 
In 1938, for instance, Venus was in northern declination 
from March 1 6 to August 9, Mars from February 1 to Octo- 
ber 29, Jupiter at no time, and Saturn from March 6 to 
the end of the year but never as much as 5, which is too 
low an altitude for navigational use. Conditions favoring 
visibility of a planet in northern declination during the six 
months of arctic daylight would be: more altitude, more 
angular distance from the sun, and less bright sunlight. 

150 Polar Position Finding 

There is no azimuth at the pole, since true north is in 
the zenith. 

Polaris when visible will be found within 1 1'.6 of the 

To find direction of Greenwich: choose a heavenly body, 
find its G. H. A. in N. A. for a given instant, set a pelorus 
with this G. H. A. on the body at the time chosen, and 
on the pelorus will point toward Greenwich. The direction 
of any other place will be its longitude west, read in de- 
grees on pelorus, as 74 for New York City. 

The north point of the magnetic compass will point south 
toward the magnetic north pole or approximately along 
the meridian of longitude 97 W. (The latitude of the north 
magnetic pole is about 70 N.) 

A gyro compass, started with N. point toward Green- 
wich, for instance, theoretically should hold its direction 
steady in relation to the universe and so appear to make 
one clockwise revolution in one sidereal day, as the earth 
turns under it counterclockwise. "Actually, however, this 
result would not be mechanically possible due to friction 
of the supports. This would produce a slight tilt of the gyro 
axis, which in turn would result in a continuous precession 
about the vertical axis due to pendulousness of the compass." 
(Sperry Gyroscope Co., Inc. communication.) 

Owing to ice floes making the horizon rough, a bubble 
sextant is necessary. This supplies its own horizon. 

Being on top of the spinning world, the rule that "day 
and night are equal at equinox" is not quite true at or near 
the pole. One has to get away far enough to be able to have 
some of the earth come between oneself and the sun to 
make any night possible at all. 

Polar Position Finding 151 

In Polar Regions 

The behavior of the magnetic compass in polar regions 
deserves some mention. At the north magnetic pole it be- 
comes useless inasmuch as the directional force is to pull 
the north end downward. A dipping needle supported on 
a horizontal axis proves this. Getting away from the mag- 
netic pole the compass gains in directional force, but this 
never becomes as strong as it is in our latitudes. Suppose 
the compass is carried around the geographical north pole 
on the parallel of latitude 80 N. Starting north of the 
magnetic north pole, the north point of the compass will 
point south and variation ~ 180. Continuing east on the 
80th parallel, this variation will be west and will steadily 
decrease till a point is reached about halfway around the 
parallel when compass will be pointing true north and 
variation = 0. Still continuing east, the compass will be- 
gin to show east variation which will steadily increase till 
our starting point has been reached when variation again 
= 180. 

The nature of the Mercator chart makes it useless for 
high latitudes and so a polar great circle chart is used. (See 
Chap. 21.) A straight line drawn on this represents a por- 
tion of a great circle. This is usually sufficiently close for a 
true line of position although the latter is always part of 
a "small circle/' (A straight line on a Mercator chart simi- 
larly is a rhumb line and not truly a part of a small circle.) 
A method for allowing for curvature in long position lines 
will be explained. 

The astronomical triangles in polar problems are pe- 
culiar in that the side between elevated pole and zenith is 









si s j3 ss 5: si 


CM CO *^h *O VO 




S 2 g| ^ S 



S S S S 2 i2 



OX CO OO **d^ i> 
. i i i CM CO 



oq o- vO vo r-^ 




ijO t^-* OO to CO 

r-C. 7 to <^3 to 

t i CM CM 


\O *^3 co CO co 
T ^ CM CO ^ tr> 





^^ Z2 "0^1 



* < CM CO co -rH 




OO CM VO g~*> jj^ 



CM co co CM oq r^ 

^ ^ CM co co d^ 




f to OO CM OO 
CM CO "^i^ VO t^- 



ox -*=H oq co ox 

^ * CM CM 





~-* CM CM CO -^ 


VO OX c^ v^ Ox 




co -*H Jr-. oq ox 

~ ^ ^ ^ S 



* H ^ I <M CM CO 


Polar Position Finding 153 

relatively much shorter. However, this does not prevent, 
except in extreme cases, using H. O. 211 for solutions or 
the cosine-haversine formula for calculated altitude. H. O. 
66 "Arctic Azimuth Tables" may be used in certain lim- 
ited conditions. (See Chap. 13.) Weems' "Line of Position 
Book" Polar edition 1928, is a system using an assumed 
position for which the tables have been extended to include 
all latitudes. 

Weems, in an article "Polar Celestial Navigation" (U. S. 
N. I. P. Nov. 1933), suggested a method preferred for 
position finding within 5 or 10 of the pole. Briefly it is 
as follows: 

Note D. R. position on polar chart, Greenwich meridian 

Assume observer is at pole, in which case h c = d and G. 
H. A. is what corresponds to Z, measured clockwise for N. 
pole, counterclockwise for S. pole. 

Take sextant altitude of sun, note time and make usual 
corrections = h . 

Find G. H. A. and d from N. A. for this time. 

Plot line from pole toward sun's geographical position 
at the proper G. H. A. and extend this line through pole. 

Find difference in minutes of arc between d (represent- 
ing h c ) and h . If latter is greater, then position line is to- 
ward sun from the pole and vice versa. Call this differ- 
ence a. 

Lay off a from pole on the line drawn through pole, 
and draw through its outer extremity a preliminary posi- 
tion line perpendicular to sun G. H. A. line. 

Measure distance on this position line between sun's 

154 Polar Position Finding 

G. H. A. line and observer's approximate position on the 
position line. 

Use Table 17, from Weems' article, entering with the 
above distance and the observed altitude to get corrections 
for h and Z. These are needed because the position line 
really curves if the above distance is considerable. The cor- 
rection for Z will be added or subtracted according to ob- 
vious circumstances. 

Plot new sun bearing line from observer's approximate 
position on first position line using corrected Z, measuring 
from a zero line parallel to Greenwich meridian. 

Lay off from starting point toward sun on this new sun 
bearing line the correction for h obtained above, and draw 
through its extremity a perpendicular to the new sun bear- 
ing line. This perpendicular will be the corrected line of 

Repeat this whole process for moon when it bears about 
90 from sun. 

Intersection of sun's and moon's corrected lines = fix. 

Any other pair of available bodies at suitable angles may 
be used. 

19* Identification 

AT NIGHT, after the horizon has faded, with a clear 
sky it is easy to identify most of the bright stars because 
various groups are easily recognized and relationships to 
such groups are obvious. 

At evening twilight, however, before the horizon has 
blurred, there will often be only a very few stars visible and 
these widely separated with no groups to give clues of iden- 
tity. One may have only a few minutes to take altitudes of 
such stars before the horizon becomes useless. Identification 
may be necessary and even a vital matter in a case where an 
observation is long overdue and perhaps only one star is 
seen for a moment between clouds. The approximate bear- 
ing of the star should be noted as well as the sextant altitude 
and G. C. T. If the latitude is fairly well known, identifica- 
tion is made as follows: 

1. H. O. 127 is first used. With latitude, altitude and 
azimuth (from N. or S. according to latitude) as arguments, 
take out declination and hour angle (local, in time). 

2. Convert L. H. A. from time to arc. 

3. Combine L. H. A. with longitude to get G. H. A. o 
star at G. C. T. of observation. 

4. You must now reduce this to the G. H. A. at O h same 
date in order later to find the star in the N. A. So find the 
table in N. A. entitled "Correction to be Added to Tabu- 
lated Greenwich Hour Angle of Stars." Use the G. C. T. 
in this table and take out the corresponding angle. 

5. Subtract this angle from G. H. A. at observation (add- 


156 Identification 

ing 360 to G. H. A., if necessary) and get G. H. A. at O h . 

6. Look in N. A. star tables of the month of the ob- 
servation and under the proper date find star which has 
approximately this G. H. A., and the declination already 
found in H. O. 127. This will be the star observed. 

In case no star is found whose G. H. A. and declination 
approximate those you have determined, a search should be 
made through the data of the four navigational planets and 
one will probably be found which will meet requirements 
and so settle the identification. 

In the rare instance where the body is neither one of the 
54 usual stars nor one of the 4 navigational planets, it will 
probably prove to be one of the 110 additional stars whose 
mean places are given in two pages of N. A. Only R. A. and 
d without G. H. A. values are provided for these and so the 
older method of sidereal time must be employed. Use H. O. 
127 as above to get d and L. H. A. (in time). Finding G. S. T. 
by the usual calculation and then applying D. R. longitude 
(in time) to get L. S. T., the latter is combined with the 
hour angle taken out of H. O. 127 to get the star's R. A. 
Using this R. A. and the d obtained from H. O. 127, search 
the list of 110 additional stars and make the identification. 
(NOTE: This and the fix by Weems* star altitude curves are 
the only procedures given in this Primer where the use of 
sidereal time is essential.) 

If the H. O. 214 identification tables are used instead of 
H. O. 127, the local hour angle will be obtained in arc 
direct. See the N. A. for practical suggestions on identifi- 


I s i 

Sextant, corre 
Special watoh 
Sextant, corre 

Sextant, corre 
Aziawth Circle 
DR or obaervat 
DR or observat 



Sextant, corre 
DR or observat 



ft- J. o 


P to 

Calculation fo 
noon or tra 
Sextant within 
of above , c 
NA. for time of 

!** < 


P 4 



O O t* 



a P.^ 


o ^ to CD 

s*s ss 

o> o S V 


Hj CK P 


< CD a -b *-i 

P n 


S o 

P P.C 0> 

c- c* o a 

M- CD c*- c* 


00 g- 







O >-C 



o o 






a t-3 


J^To" . w ? q 




03 O 5 


C* H- C-0 O 






*< M H" 

sap." H* 
<< ro H- 
VA/\ J g 

hj P.P P 1^ . 
CD H-C+ c*- P 

< P toccta o o 





^"a ^ 

H" O 

rf*. oa 
* g 

a M- CD * <B co 


i 1 


S- "^ 

*1 c* ^ P 




5" cj M. ^ "^ 

30 h-- CO 

8Se* TS 

4 j 

P> M> CD 

CD l-l 


^^_ (D 

- o 

p.. to ^.~~. 

A , 

c a P g 



1 PI 


O c? t-t 

*p^ 1-t, 



i l( 1^ 



a P 

*? -t 





i-t S> *4H>&$p. 

S^ cf 


S 1 


t ^P-i-fci ** 

* "^JTf? 







- ! 





c* O 




* a" c?2- 

** * 2 



S 1 

J K^^* >l 






P ca a 

** ** -a 

< P 




^8 g** 


1 ; 








'!>*' ,n 

5 ^p ^ 

^ ^ 


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20. Tabular Summary 

TABLE EIGHTEEN has been prepared to give at a 
glance the outlines of fourteen types of calculation 
which have been under consideration. Several others might 
have been included but it is felt that this list is sufficient for 
our purposes. Even after months of study the student will 
find it hard to definitely remember the exact steps of a 
process unless he is using it constantly. This table, it is 
hoped, will serve as a refresher and will surely save time that 
would be otherwise wasted in repeatedly looking back 
through the text. The arrangement in four columns is self- 
explanatory and horizontally traces a calculation from start 
to finish under the headings of Given, Obtain, Procedure, 
and Result. 

See Table 18 inserted. 


Part III: Supplernentary 

21. Sailings; D. R.; Current 

ELESTIAL NAVIGATION, strictly speaking, does 
not include the subject of sailings, dead reckoning, or 
current. However, the methods we have recommended for 
position finding at sea all require a D. R. position to start 
from and so it is proper to include something on the means 
by which such a D. R. position is ascertained. 

We saw in Chapter 10 that the D. R. position was ob- 
tained by keeping track of courses steered and distances run 
from the last well known position. We will see presently 
how such data is translated into a new latitude and longi- 

We also saw in Chapter 10 that a rhumb line was a curved 
line on the earth's surface which intersected all meridians 
at the same angle. 

The term sailings is used for various means of solving 
problems involving calculation of a ship's position in rela- 
tion to a place left or to a place one is approaching. The 
following items come into consideration: 

Course: the constant angle a rhumb line makes with 

Distance: the length of the rhumb line in nautical miles 
between starting point and destination. This usually refers 
to a day's run more or less and not to a whole voyage. 

Difference of latitude: the number of degrees, etc., the 
latitude is altered by the run made. Expressed in minutes 
it equals nautical miles. 

Difference of longitude: the number of degrees, etc., the 
longitude is altered by the run made. 


162 The Sailings 

Departure: the number of nautical miles of easting or 
westing made good by the run. It is also the length in 
nautical miles of an arc of a parallel. (Do not confuse these 
uses of the word departure with its use in the expression 
"take a departure" which consists in fixing the position of 
the ship, before losing sight of land, by some landmarks and 
using this position as the starting point for dead reckoning.) 

The matter of departure requires still further comment. 
For a small area of the earth's surface, where its spherical 
form may be neglected, the departure will be the same, 
whether measured on the parallel of the point left or on 
that of the point reached. (See FIG. 41.) 

The formula Dep. = Dist. sin C may be shown (see 
Button) to hold for any rhumb line and any distance. As 
so computed for a rhumb line, no matter how long, Dep. 
is not the easting or westing measured on the parallel of the 
point left or on that of the point arrived at. It is the distance 
steamed east or west made up of the sum of the Deps. of any 
number of small triangles each constructed like those in 
Figure 42. This sum proves to be practically equal to the 
Dep. between the places measured on a parallel in the mid- 
dle latitude of the two places. 

The Sailings 

I. Plane (earth's surface assumed to be flat) considers: 
Course, Distance, Diff. Lat., Departure. 

Types: Single, Traverse. 

Solved by: Logs, Traverse Tables, Construction on 
plotting sheet or chart. 



Plane Sailings 


C= Course 

Dist Distance 

DL= Difference of latitude 

Dep = Departure 

1"= Starting point 
T'= Destination 

h Intersection of line from T dropped 
perpendicular to meridian of T 

By definition: 







Dist=-^;=Dep esc C 

sm C * 

FIG. 41. Plane Sailing. 

164 Plane Sailings 

II. Spherical (earth's true shape used) considers: Course, 
Distance, Diff. Lat., Departure, Diff. Long. 

Types: Parallel, Middle Latitude, Mercator, Great 
Circle, Composite. 

Solved by: Methods similar to those for Plane. 

Plane Sailings 

Plane sailings can be solved by plane trigonometry and 
logarithms as indicated in Figure 41, using Bowditch Table 
32 for logs o numbers and Table 33 for logs of functions 
of angles. It will be seen that there are two or more formulas 
for the solution of each side of the triangle to provide for 
the various combinations of data given and to be found. 
The two commonest problems are: first, given course and 
distance, to find difference of latitude and departure; sec- 
ond, given difference of latitude and departure, to find 
course and distance. In the latter problem, the formulas are 
not suitable for long distances. 

A shorter way is to use the traverse tables. Bowditch Table 
3 gives Diff. Lat. and Dep. for each unit of distance from 
1-600 under each degree of Course. 

The simplest way, and one sufficiently accurate for 
yachtsmen, consists in laying down the various given terms 
by scale upon a position plotting sheet and then measuring 
the required terms. 

A Single calculation is all that is usually called for. 

The Traverse problem arises when a ship has made an 
irregular track sailing on several different courses. It con- 
sists in finding the difference of latitude and departure for 
each course and distance by the traverse tables and reducing 

Parallel Sailing 165 

all to a single equivalent course and distance. This is done 
by tabulating the amount of miles made good to north or 
south and to east or west on each course; adding up each 
of the four direction columns; taking the algebraic sum of 
the north and south totals for Diff. Lat. and of the east and 
west totals for Dep.; and finding in the traverse table the 
course and distance corresponding to this Diff. Lat. and 
Dep. This resulting course and distance can then be drawn 
on the plotting sheet from the starting point and will show 
the D. R. position at the end of the irregular track. 

Spherical Sailings 

Parallel sailing problems come up when a ship's course 
is due east or west. There is no change in latitude and all 
the distance covered is departure. The problem is to find 
the Diff. Long, corresponding to the Dep. This may be done 
by logs using the formula (see Dutton, Chap. 1): 

Diff. Long. = Dep. sec L 

Or, if it is desired to find the amount of departure necessary 
to reach a certain longitude: 

Dep. = Diff. Long, cos L 

It may be more convenient to use Bowditch Table 3. Diff. 
Long, will be found in minutes of arc or should be con- 
verted thereto as the case may be. The labels of the table 
should now be imagined changed as follows: 

Course Lat. 

Dist. * Diff. Long. 

Lat. Dep. 

166 Middle Latitude Sailing 

(See Bowditch for explanation of how this is possible.) 

Pick out the two known values and read off the one desired. 

Middle latitude sailing is founded on the assumption 

that the departure between two places with different lati- 

Departure between T & T' = JK 

(Proof depends on calculus and the difference in latitude should not be great 
and latitudes must not exceed 50.) 

PIG. 42. Middle Latitude Sailing. 

tudes equals the length in miles of that parallel of latitude 
which lies midway between the parallels of the places and 
between the meridians of the places. This is sufficiently 
accurate for a day's run under latitude 50. (See FIG. 42.) 

Given latitude and longitude of starting point, course, 
and distance made, to find latitude and longitude arrived at. 

Bowditch Table 3 is used entering with course and dis- 
tance to find Diff. Lat. and Dep. as in Plane Sailing. Con- 

Mercator Sailing 167 

verting minutes to degrees, etc., this Diff. Lat. is applied to 
latitude left to give latitude arrived at. 

The two lats. are now averaged to find Mid. Lat. 

The problem thus becomes one in parallel sailing and 
Table 3 is used with the same changes of labels as described 
for parallel sailing, as follows: 

Entering with Mid. Lat. (Course) and Dep. (Lat.) we 
find Diff. Long. (Dist.). Convert minutes to degrees, etc. 
Applying to longitude left gives longitude arrived at. 

Given latitude and longitude of both starting point and 
point reached, to find course and distance made good. 

Subtract lesser latitude from greater to get DifL Lat. and 
convert to minutes. 

Average the two latitudes to find Mid. Lat. 

Combine the two longitudes to find Diff. Long and 
convert to minutes. 

Entering Table 3 with Mid. Lat. (Course) and Diff. Long. 
(Dist.) we find Dep. (Lat.). 

Entering Table 3 with Diff. Lat. (Lat.) and Dep. we find 
course and distance. 

When the places are on opposite sides of the equator, 
this method is not applicable. 

Mercator sailing is used when the distance involved is 
greater than can be accurately handled with middle latitude. 

As previously described, a Mercator chart has parallel 
meridians. The result is that the farther one goes from the 
equator, the more unnaturally extended' are the parallels 
of latitude between any two meridians. Proportions are 
kept right because the meridians between parallels of lati- 
tude become more and more extended. Expressing it in 
another way: 

168 Mercator Sailing 

Globe map: 

Long, degrees become shorter toward pole. 

Lat. degrees stay same. 
Mercator chart: 

Long, degrees stay same toward pole. 

Lat. degrees become longer. 

The length of the meridian as thus increased on a Mer- 
cator chart between equator and any given latitude, ex- 
pressed in minutes as measured on equator, constitutes the 
number of Meridional Parts corresponding to that latitude. 
Bowditch Table 5 gives meridional parts or increased lati- 
tudes for every minute of latitude between and 80. 

Given course and distance from a known position we 
wish to know latitude and longitude arrived at. This is 
accomplished as follows: 

1. Use Table 3 entering with course and distance and 
find figure in Lat. column, which is difference of latitude 
expressed in nautical miles or minutes of arc. (Or use logs 
and equation Diff. Lat. Dist. cos C.) 

2. Reduce above to degrees and minutes. 

3. Apply above to latitude left (adding if course was 
northerly, subtracting if southerly) and obtain latitude 
arrived at. 

4. When both the place left and place reached are on 
the same side of the equator, look up in Table 5 the me- 
ridional parts for each latitude and subtract the smaller 
quantity from the greater. When the places are on opposite 
sides of the equator, similarly look up the two quantities 
but add them. In each case the result is the meridional 
difference of latitude or ra. 

Great Circle Sailing 169 

5. Table 3 is now used with the following two substitu- 
tions of title: 

Lat. m. 

Dep. Diff. Long. 

Enter table with course and m (Lat. column) and take out 
Difference of Longitude (Dep. column) expressed in min- 
utes. (Or use logs and equation: Diff. Long. = m tan C, if 
C is not near E. or W.) 

6. Reduce above to degrees and minutes. 

7. Apply above to longitude left (adding if course was 
away from Greenwich, subtracting if toward) and obtain 
longitude arrived at. (If over 180, subtract from 360 and 
change name.) 

In case we require course and distance by rhumb line 
between two given positions which are far apart: 

1. By subtraction find difference of latitude. 

2. By subtraction find difference of longitude. 

3. Use Table 5 and find m as described in 4 above. 

4. Find course by using logs and equation: 

Diff. Lone, 
tan C = 


5. Find distance by using logs and equation: 

Dist. = Diff. Lat. sec. C. 

Great Circle Sailing 

The shortest distance over the earth's surface between 
two places is that measured along the great circle which 
passes through them. Generally speaking, the economy of 

170 Great Circle Sailing 

the great circle over the rhumb line is greatest for long 
distances at high latitudes when the course lies more E. or 
W. than N. or S. 

Every great circle, excluding the equator and meridians, 
cuts successive meridians at a different angle. To keep a 
ship on a great circle course, therefore, would require con- 
stant change of heading. As this is impracticable, the course 
is changed at regular intervals such as every 150 or 300 miles. 
Thus the ship really follows a series of rhumb lines. 

Every great circle track, if extended around the earth, 
will lie half in the northern and half in the southern 

The Vertex of a great circle track in one such hemisphere 
is that point which is farthest from the equator, or has the 
highest latitude. 

A great circle track between two places in, say, the north- 
ern hemisphere will everywhere be north of the rhumb 
line between these places. This is because a rhumb line is 
a spiral, concave toward the pole, which approaches but 
never reaches the pole. On a Mercator chart, then, a great 
circle track between two places in the northern hemisphere 
appears as a curve above the straight rhumb line joining 
the places. 

Great Circle or Gnomonic Charts are charts on which 
great circle courses will appear as straight lines. They are 
constructed for a given area of the globe by passing a plane 
tangent to the center of the area and then projecting by 
rays from the globe's center all features of the area onto 
the plane. As the plane of every great circle passes through 
earth's center, and as one plane always intersects another 
in a straight line, the projected great circles will be straight 

Great Circle Sailing 171 

The Polar Chart is a great circle chart for use in high 
latitudes where a Mercator is too distorted and spread out. 
The meridians are straight lines radiating from the pole 
and the parallels are circles with the pole as center, increas- 
ingly separated from the pole outward. (See Button, Chap. 

Hydrographic Office Publications 1280-1284 are great 
circle charts of the principal oceans. They are spoken of as 
on the Gnomonic Projection. The meridians appear as 
straight lines converging toward the poles, and the parallels 
appear as non-parallel curves, concave toward the poles. 
No compass rose can be applied to the whole of such a 
chart and latitude and longitude at a particular point must 
be determined by reference to nearby meridians and par- 
allels. The chart becomes distorted in the longitudes more 
removed from the center and so is not satisfactory for navi- 
gation. However, it provides the most convenient means for 
determining great circle track and distance between points. 
The track is always a straight line. An explanation is printed 
on such charts of how to determine the length of the track 
and the course at any point of the track. After selecting a 
number of points of the track on the great circle chart, their 
latitude and longitude are determined and such points are 
then plotted on a Mercator chart. A fair curve is then drawn 
through these points and gives the great circle on the 

In the absence of a great circle chart, the track, courses, 
and distance may be determined by computation. This 
rather long process will not be gone into here. A good ex- 
planation and method will be found in H. O. 211. A very 
easy and rapid method of finding initial course and distance 
is provided for in H. O. 214. 

172 Composite Sailing 

Composite Sailing 

This is a combination of great circle and parallel sail- 
ing. It is used when the great circle course between two 
points passes through higher latitudes than it is thought 
wise to enter. This may be from considerations of ice or 
cold or wind. There are three main ways of determining the 
track; by gnomonic charts, by computation, or by graphic 
methods. (See Bowditch.) An approximation to the shortest 
track between the points without exceeding the given lati- 
tudes is had by following a great circle between the points 
until the limiting parallel is reached, following this parallel 
until the great circle is again met, and finally following the 
great circle to destination. 

The track may be laid on a Mercator chart as follows: 

1. Draw the track on a great circle chart. 

2. Determine the latitude and longitude of a number of 
points of the track. 

3. Transfer these points to the Mercator. 

4. Draw a smooth curve between them from point of 
departure to destination. 

5. Discard the portion of the curve which lies north (or 
south) of the limiting parallel, retaining the two remaining 
portions of the great circle and that part of the limiting par- 
allel included between their northern (or southern) ends. 

Other methods of composite sailing are the following: 

1. One great circle to a predetermined place; a Mercator 
to another predetermined place; and another great circle 
to destination. 

2. One great circle to the limiting parallel at a given 
longitude; another great circle from there to destination. 

Dead Reckoning 173 

3. One great circle from departure tangent to the limiting 
parallel; another great circle from destination tangent to 
the limiting parallel; course along parallel between the two 
points of tangency. This is the shortest possible composite 

W. S. N. Plotting Charts 

A new series of charts or plotting sheets is featured by the 
Weems System of Navigation. The charts are designed to 
eliminate the sailings calculations. Mercators and great 
circles are measured directly. There are four charts in the 
series covering all areas from equator to pole. 

Dead Reckoning 

The working of dead reckoning involves a combination 
of the methods of traverse and middle latitude sailing. See 
Chap. 29 for appropriate form of table to be kept and the 
following summary from Bowditch: 

"When the position of the vessel at any moment is re- 
quired, add up all the differences of latitude and departure, 
and write in the column of the greater, the difference 
between the northing and southing, and the easting and 
westing. Apply the difference of latitude to the latitude of 
the last determined position, which will give the latitude 
by D. R., and from which may be found the middle latitude; 
with the middle latitude find the difference of longitude 
corresponding to the departure; apply this to the longitude 
of last position, and the result will be the longitude by 
'D. R." 

Button says, p. 95: "In modern practice nearly all navi- 
gators do their dead reckoning work graphically." 

174 Current 


Current is a broad term covering anything and every- 
thing that causes a discrepancy between the D. R. position 
for a given instant and a fix from celestial observation for 
the same instant. Button summarizes as follows: 

1. Foul bottom of ship. 

2. Unusual condition of trim. 

3. Error of patent log or revolution curve. 

4. Inaccurately known compass error. 

5. Poor steering. 

6. State of wind and sea. 

7. Observation errors. 

8. Real ocean currents or streams. 

9. Tidal currents found along the coast. 

The Estimated Position 

Either a Computed Point (Chap. 15) or a D. R. position 
which has been revised for Current is called an Estimated 

Real Ocean Currents 

The Set of a current is the direction toward which it is 

The Drift of a current is its velocity in knots (nautical 
miles per hour). 

The two usual problems which arise may be solved by 
traverse tables or trigonometry but simple graphic solu- 
tions on the chart with a protractor are satisfactory, as 
shown (after Button) on pages 176-179, following. 

Current 175 

Current does not affect the ship's speed through the water. 
Wind, weather, waves, etc., however, do so. 

In case the current is directly against or directly with the 
ship's course, the algebraic sum of speed and current gives 
speed over ocean floor. 

r~. . , Distance in miles 

Time required = 

bpeed over bottom in miles per hour 

-, , . , Distance in miles 

Speed over bottom m miles per hour = 

r Time consumed 

FIG. 43. First Current Problem. 

Current 177 

First Problem 


Course and speed through water. 

Set and drift. 
To find: 

Course and speed over ocean floor. 
Example: (See FIG. 43.) 

Ship was at A. 

Steamed on course 211 at 12 knots. 

Through current of set 75 and drift 3 knots. 

To find course and speed over ocean floor: 

Draw N S meridian through A . 

Lay off clockwise angle N A B 211, for course. 

Using scale of chart make A C 12, miles, for speed. 

Draw D E through C parallel to AT S. 

Lay off clockwise angle D C F = 75 for set. 

Lay off C G = 3 miles by same scale, for drift. 

Draw A G. 

Then clockwise angle N A G = course made good over 
ocean floor =199. 

And A G by same scale = 10 miles = speed of 10 knots 
over ocean floor. 

Fio. 44. Second Current Problem. 
178 - 

Current 179 

Second Problem 


Set and drift. 

Speed of ship 

Course you desire to make good. 
To find: 

Course you must steer. 

Speed you will make on it. 
Example: (See FIG. 44.) 

Ship is at A. 

Current has set 75 and drift 3 knots. 

Ship's speed = 12 knots. 

Course to be made good =195. 

To find course to steer and speed attainable on it: 

Draw N S meridian through A. 

Lay off clockwise angle N A B = 195, for course to be 
made good. 

Lay off clockwise angle N A C = 75, for set. 

Using scale of chart make A D = 3 miles, for drift. 

Erect a meridian D N f . 

With D as a center and a radius of 12 miles by same scale, 
sweep an arc cutting A B at E. 

Draw D E. 

Then clockwise angle N' D E = course to steer = 206. 

And A E by same scale = 10.2 miles = 10.2 knots speed 
attainable on course to be made good. 

22. The Day's Work 

BEGINNING A VOYAGE, a good departure should 
be taken while landmarks are still in view. This be- 
comes the start of the dead reckoning which with new de- 
partures from fixes by celestial navigation is kept up till 
the voyage is completed. The patent log, having been put 
over, is read and recorded on taking departure. If an en- 
gine revolution counter is used it is read and time noted. 
The following summarizes the required daily work at sea 
(see Button): 

Dead Reckoning is carried forward from one well- 
determined position to the time of next observation or 
fix. Comparison of D. R. and fix gives "current" since last 
fix. Log or engine revolution counter is read at each obser- 

Compass Error is determined at morning and afternoon 
observations of the sun and, if possible, whenever the course 
is changed. 

Sun Observations should be made in forenoon and after- 
noon when on the prime vertical providing the altitude is 
then over 10; otherwise, as near prime vertical as possible 
with altitude exceeding 10. Sun should also be observed 
at local apparent noon or for ex-meridian. 

Fixes from at least two stars should be made at morning 
and evening twilight. Other combinations as star and planet, 
planet and moon, etc., can be used. 

Additional work., in case of failure to obtain A. M. fix or 
other of the above observations, consists in daylight sights 


The Day's Work 181 

of sun and moon, or sun and Venus, for a fix; latitude by 
Polaris may be done at dawn; or a running fix from two sun 
sights may be all that can be obtained. 

Chronometers are wound at a certain time each day and 
note made of any changes between them. 

Radio time signals are received daily, if possible, and 
chronometer correction noted. 

Reports of work are made to the captain at 8 A. M., noon, 
and 8 P. M. These include: 

Position by D. R. 

Position by observation. 

Set and drift of current. 

Deviation of compass. 

Course and distance made good since last report. 

Course and distance to destination. 

Any necessary change in course. 

Ship's run (at noon since last noon). 

23- Essential Equipment 

(For Celestial Navigation and Dead Reckoning) 

Absolute minimum 

Magnetic Compass 
Azimuth Circle or Pelorus 
Patent Log 
Nautical Almanac 

Dead Reckoning Altitude and Azimuth Table (H. 0. 21 1) 
Bowditch (H. O. 9) 
Plotting sheets and charts 

Dividers, parallel rulers, triangle, magnifying glass 
Notebook, pencils, eraser, pencil sharpener 

Also important 

Gyro Compass 
Engine revolution counter 
Telescope or binoculars 

Second-setting 24 h. face watch 
Course protractor 

Tables of Computed Altitude and Azimuth (H. O. 214) 
Noon-Interval Tables (H. O. 202) 
Star Identifier (H. O. 2102-B) 
Celestial Globe (preferred to above) 
Collected Correction Tables (see Chap. 3) 


24. Practical Points 


ON BEGINNING a voyage, the navigator must pro- 
vide himself with the latest ocean charts of the regions 
to be sailed, and note the date of the last corrections. All 
weekly "Notices to Mariners" should be obtained which 
have appeared subsequent to this date and any applicable 
data therein should be entered on the charts. The material 
necessary for coastwise navigation and piloting such as tide 
and current tables, coast pilots, light lists and large scale 
charts of coasts, harbors, etc., are not part of celestial navi- 
gation and are mentioned here only in passing. The essential 
equipment for taking and working sights should be checked 
over carefully and put in order. 

The navigator's watch (pocket or wrist) should be kept 
on zone time for ease of translation to G. C. T. 

The best standard of accuracy attainable in position find- 
ing may be expressed as a circle with a radius of one mile 
around the fix point on the chart. This is practically perfect 
work. It may occasionally be bettered, but is often not at- 
tained. Refraction, faulty observation, rolling ship, errors 
in time, etc., all make the actual practice at times more of 
an approximation than an exact procedure. Experience 
brings knowledge as to how accurate a given fix will prob- 
ably prove to be. 

The height of eye for the place on the ship which will 
usually be used by the navigator in taking sights should be 
accurately determined after the ship is loaded. 

1 84 Practical Points 

Sextant (General) 

The minimum altitude for reliable observations is 10 
and 15 is still better. 

On a rolling vessel, stand on the center line to avoid 
changes in height of eye. 

When a distant haze makes the horizon poor, it may be 
found better by lowering the height of eye. 

In clear weather, increasing the height of eye makes for 
greater accuracy. 

Index error of the sextant should be checked each time 
it is used. 

"Bringing a body down to the horizon" is especially con- 
venient for a star or planet sight but may also be well used 
for sun or moon. Set sextant at zero. Look through it di- 
rectly at body. Loosen index arm and slowly push it forward. 
A second image of body will appear to drop from the origi- 
nal. Lower the aim of sextant, abandoning original image 
and following this second one which is in the mirror side 
of the horizon glass. Continue pushing index arm forward 
and lowering aim to keep the mirror image of body in view 
until horizon appears in clear half of glass. Clamp sextant 
at an approximate contact and adjust to exact contact by 
vernier screw. 

After one preliminary observation, a sextant may be set 
ahead of a body's altitude on even 5' or 10' intervals of arc 
and several times of contact noted. Then use the average of 
these altitudes and times. 

The author heartily endorses the following, written by 
J. T. Rowland in Yachting, March 1941: "I would rather 
use one good shot, in which I feel confidence, than introduce 

Practical Points 185 

the error which may be present in a doubtful one. My sys- 
tem, therefore, is to take a series of sights until I get one in 
which I have faith and then use that. True this calls for 
some discernment on the part of the navigator but it is sur- 
prising how quickly one can develop an aptitude for 'calling 
one's shots/ 

It is more important to be exact with time than with the 
altitude. Ten seconds of time error can throw a position off 
2i/ miles whereas 10" of arc error will alter the result only 
about 1/5 mile. 

When reading sextant, have the eye directly over the 
marks to avoid error of parallax between vernier and limb. 

Focus the telescope on some distant object before screw- 
ing it in to sextant frame. 

Eye-strain is less if both eyes are open when taking sights. 

Between sights at twilight it may be helpful to close the 
eye used on sextant when going into the light to consult 
chart or read sextant. Returning to take another observation 
the sextant eye will then be in better shape for work. 

Start with tangent screw about halfway from its two ex- 
treme positions so as to allow plenty of room for finer 

Sextant (Care of) 

On small boats it is well to have a short lanyard attached, 
with an eye splice to fit over the wrist, in order to avoid 
dropping the sextant overboard. 

The sextant is a delicate instrument and must not be 
dropped, bumped or jarred. It should be kept from sudden 
changes of temperature or moisture as far as is possible* It 

1 86 Practical Points 

should be in its case with index arm clamped when not 
working and the case should be attached to some safe place 
in the boat. 

After use in wet weather the mirrors should be dried 
with chamois, linen or lens paper, not with silk. 

When dirty, the arc and vernier may be cleaned with 
ammonia or sperm oil. 

When the graduations of the sextant arc become dim, a 
paste of gritless lamp black and light oil may be smeared 
over arc and wiped off. This will clean the marks and also 
make them more easily read. 

Sextant (Index Error) 

Graduations of arc and vernier are continued somewhat 
to right of zero of arc and of zero (index) of vernier. 

When a star's reflected image is brought into coincidence 
with the star seen directly, or when in daylight the reflected 
sea horizon is matched with the direct view of it, the sextant 
should read zero. If, however, zero of vernier lies to left of 
zero of arc (called "on"), the error is + and its correction 
will have to be subtracted. Or if zero of vernier lies to right 
of zero of arc (called "off"), the error is and its correction 
will have to be added. 

In the latter case, the reading of how much it is "off" is 
done in a special way. Read the vernier as usual, looking 
to left till a vernier line is found coinciding with an arc 
line, and count the minutes and seconds to this vernier line. 
Then subtract this amount from the maximum reading of 
the vernier. The result will be the amount of error. 

Another method of measuring index error is by measur- 

Practical Points 1 87 

ing the apparent diameter of the sun. This is done by first 
bringing the upper limb of reflected image tangent to lower 
limb of direct image and reading the sextant; then bringing 
lower limb of reflected image tangent to upper limb of di- 
rect image and again reading the sextant. Mark reading 
when on the arc as and when off as +. The algebraic sum 
of the two readings will be Index Correction. 

If a series of such observations is made, add up the read- 
ings of each pair without signs and divide by 2 to get sun's 
diameter. Look in N. A. for sun's semidiameter for the date 
and multiply by 2 for true diameter. Pick from the series 
that pair which most closely approximates this true value. 
Use this pair for Index Correction. 

Artificial Horizon 

One may use a pie plate of ink set in a box 18" x 18" x 8" 
without a cover. 

Sit back of and above plate in line with sun and its image 
in ink. 

With sextant, bring real sun image down to ink image. 
Former will be in right-half of horizon mirror, latter in left. 
Though hard to find at first, lateral swinging will locate it. 

In forenoon bring real sun image, lower limb, tangent 
to upper limb of ink image. Former will rise till transit, 
separating the images unless pulled down by tangent screw. 

In afternoon, bring real sun image, lower limb, tangent 
to upper limb of ink image. Former will fall, causing over- 
lap, unless raised by tangent screw. 

Read the (double) sextant altitude of observation; apply 
I. C. to this; divide by 2. 

188 Practical Points 

Correct for refraction, parallax and semidiameter but not 
for height of eye. 

Result is true altitude. 


Swing in the strongest shade glass before observing sun. 
If this cuts off too much light use the medium shade glass, 
etc. If sun is looked at directly with too little protection it 
will make observation impossible for several minutes. 

Test for index error after shade glasses are in place since 
they may be responsible for the error. 

Hold sextant vertically, aimed toward the horizon be- 
neath the sun. Move index arm from zero position slowly 
forward till sun's image appears in mirror of horizon glass 
and continue till lower edge of image is about tangent to 
horizon seen through clear half of horizon glass. Set the 
clamp screw (or release lever). 

Determine vertical position of sextant by swinging it in 
an arc around line of sight axis and watching sun image 
swing in a curve convex to horizon. When image is lowest 
is when sextant is vertical. At this point adjust the tangent 
screw to bring sun's lower edge just tangent to horizon. At 
this instant call out "Mark," which signals your assistant 
to take the time. If alone, punch the starter of a stop-watch 
and proceed as described in Chapter IL 

Owing to refraction when the sun's center is actually in 
horizon, the lower limb appears to be about one semi- 
diameter above horizon. 

The usual correction tables include correction for sun's 
semidiameter only when lower limb is used. If unusual 

Practical Points 1 89 

circumstances should permit only an observation of upper 
limb, as when sun is partly obscured by clouds or even in 
an eclipse, proceed as follows: 

Apply I. C. to sextant altitude. Then subtract 2 X sun's 
semidiameter for closest date found in N. A. (This gives 
sextant altitude of lower limb.) Finally, apply usual cor- 
rections from Tables A, B, and C for refraction, parallax, 
semidiameter and dip, to obtain corrected altitude. 


A bright moon may illuminate an otherwise dim hori- 
zon so that an altitude of the moon or a star nearby may be 

At times with the moon about half-full, care must be 
taken to choose the limb whose upper or lower edge ex- 
tends through what would be the vertical diameter of the 
moon if full. 


Since navigational planets are brighter than stars they 
usually appear first at evening twilight and disappear last 
in the morning. Hence they can be observed when horizon 
is clearest and should be used when available. 

When Venus is over 2 hours of R. A. distant from the 
sun, it may be seen in daylight. Calculate its approximate 
altitude and azimuth in advance. Set the sextant for this 
altitude and look through it with the medium telescope 
attachment in the direction of the calculated azimuth. 
Lenses and mirrors must be clean. A distinct white disc will 

190 Practical Points 

be found and exact altitude can then be obtained. Later, 
knowing where to look, Venus can be seen with the naked 


A way for timing star sights is as follows: 

1. Start stop-watch at first observation. 

2. Read stop-watch at 2nd. observation and record. 

3. Read stop-watch at 3rd. observation and record. 

4. Stop stop-watch on an even minute of chronometer. 

5. Subtract total stop-watch reading from chronometer 
as in #4 for time of first observation. 

6. Add stop-watch reading of #2 to above for time of 
second observation. 

7. Add stop-watch reading of #3 to time of first ob- 
servation for time of third observation. 

Some use the following method for observing a star: 
Invert sextant set at zero, aim at star, push index arm for- 
ward till horizon is brought up approximately to star, clamp 
sextant, invert again and make final fine adjustment. This 
prevents starting to observe one star and ultimately making 
contact with another. 

When observing a star, it is well to get its approximate 
bearing to aid in subsequent identification if doubt should 

The writer obtained a perfectly satisfactory altitude of 
Capella an hour before midnight 29 October, 1936, when 
a full moon and clear air made the horizon as distinct as 
could be desired. 

Practical Points 191 

Morning twilight is a better time for star sights, other 
things being equal, than evening. This is because the navi- 
gator may, by getting up a little earlier, pick out what stars 
he intends to use from the total display with no doubt of 
their identity, and then keep them in sight until the horizon 
has become sufficiently sharp for measuring their altitudes. 
Twilight is shortest in the tropics and increases with lati- 

When working two or three star sights it is best to put 
the data in parallel columns and work across the page. By 
this is meant figuring one item for each before going on to 
the next item, such as altitude, declination, hour angle, logs, 
etc. This is easier and errors show up more readily. 

Avoid the use of the confusing inverting telescope. In 
poor light it is better to work without any telescope. 


In using sextant for meridia:n altitudes remember: Image 
goes up as body goes up. 

Turning vernier screw down on left side increases alti- 
tude reading, and pulls body's image down. 

Hence, while watching for body to transit: So long as 
you have to keep turning vernier screw down on left to 
keep body on horizon, the body is rising, and first dip of 
body below horizon without having turned vernier screw 
is start of body's fall after transit. 

As declination and latitude get farther apart, conditions 
become more favorable for finding latitude and less so for 

In working the sun for reduction to meridian, t should 

1 92 Practical Points 

not exceed in minutes the number of degrees in sun's 
meridian zenith distance. 

Reduction to the meridian is not nearly so reliable as 
the meridian sight. Any error in D. R. longitude makes t 
in error and as the formula uses t 2 the error becomes much 
greater. When observed altitude is over 60 the procedure 
is also less reliable. 

A rough observation of Polaris may be made as stars begin 
to fade at morning twilight and this can be made accurate 
when horizon has become clearer. 

Polaris has been observed for latitude during an electrical 
storm at night. 


When Mizar of the Big Dipper or Ruchbah of Cassiopeia 
are either above or below Polaris, the latter bears true N. 
and can be used for compass error. 

"When it is desired to swing ship using sun azimuths, 
it is convenient to use a graph laid out on coordinated (cross- 
section) paper. The interval 5 P. M. to 5:40 P. M., for in- 
stance, may be laid off on a horizontal line, a square for 
each minute, numbering every 5-minute square. The verti- 
cal scale, centered on the left-hand end of the horizontal 
scale, is graduated to the degrees of azimuth, with two 
squares for each degree. The azimuths taken from the tables 
for 5 P. M, and 5:40 P. M. (using, as an example, lat. 49 N., 
and declination 17 SO 7 N.) are then plotted and a line 
drawn connecting them. The azimuth for "any intermediate 
time is now readily available. Suppose the azimuth for 
5: 1 1 P. M. is desired: Run up the vertical line of 5: 1 1 until 

Practical Points 1 93 

it intersects the azimuth line, then run horizontally to the 
left to the intersection with the vertical scale, where the 
azimuth is indicated as N. 87 15' W." (Chapman, p. 225.) 
See Button, p. 347, for method of obtaining a curve of 
magnetic azimuths by H. O. 214. 

Line of Position 

It will sometimes be possible to get a line of position 
which is parallel to the course from a body abeam. When 
this is plotted it will at once show which way the ship is 
being set from the course and how far. 

A line of position can sometimes be obtained which is 
perpendicular to the course from a body dead ahead or 
astern. This gives a good check on the run. 

If a body is observed on the prime vertical (due E. or 
W.) the line obtained runs N. and S. and is unaffected by 
any error in the D. R. latitude. 

The meridian altitude gives a line running E. and W. 
which is independent of any longitude error. 

If only a single Summer line is obtainable, it may be of 
great value combined with a terrestrial bearing to give a 
fix; or the position on a line near shore may be told from 
soundings; or if the line is parallel to shore it will tell 
distances from land; or, finally, if it is perpendicular to 
shore the captain will know what to look for. 


Draw lines of position for a fix before labeling them so 
letters will not obscure intersection. 

Above a line of position put the time it was obtained, 

194 Practical Points 

using four figures for hours and minutes only, as: 0009, 
0017, 0231, 1604. This is usually Zone Time. 

Below the line, print name of body observed. 

When a line has been advanced in a running fix, put 
above it both the time it was originally obtained and the 
time of the new line it is to cross. Below, print the name of 
body observed. 

Lines over 5 hours old are not dependable. 

Lines under 30 minutes old can be used without calling 
the resulting fix a running one. (See Chap. 15.) 

Carry forward position from a fix by simple dead reckon- 
ing. Disagreement between this and the next fix is attrib- 
uted to "current" and the fix starts a new D. R. The D. R. 
track or course line is drawn according to true course and 
labeled above with C followed by course in degrees and 
below with S followed by speed in knots. 

Given a D. R. position and a line of position not running 
through it and known current: 

A perpendicular from D. R. to L. of P. determines Com- 
puted Point on the L. of P. 

After plotting the current line from D. R., a perpendic- 
ular to L. of P. determines Estimated Position on L. of P. 

The principles of chart construction, Mercator or polar, 
should be studied in Chapters I and VI of Button. 

The Weems Universal Plotting Charts in notebook form 
are convenient. A compass rose, three unnumbered latitude 
lines making equal spaces, and a scale for longitude are 
on each left-hand page. The latitude lines are numbered 
by the navigator according to the area under consideration 
and two or more longitude lines are drawn in and num- 
bered, the spaces varying with latitude. Opposite each chart 

Practical Points 1 95 

is a blank page for observational data. Thus all the record 
is kept easily available and the marking up of regular charts 
is avoided. 

If plotting sheets are not available, the following sug- 
gestion is made by Chapman in Piloting, Seamanship and 
Small Boat Handling (1940, p. 220), "A simple method 
of laying off intercepts in a work book is to employ the 
rulings of the page as meridians of 1' of longitude each, 
crossing them with the parallel of D. R. latitude. From any 
intersection of a meridian with this parallel draw a line at 
an angle equal to the degree of latitude. This line becomes 
a distance scale on which each space between rulings is 1 
mile or F of latitude available for use in laying off the inter- 
cepts and in measuring the latitude difference between 
the fix and D. R. latitude. In effect we have a miniature 
Mercator chart." 

25* Navigator's Stars 
and Planets 

THERE ARE MANY good books by which the student 
can become familiar with stars and with the constel- 
lations or groups of stars whose names date back many 
centuries. A somewhat different presentation has been 
worked out here for the student navigator. The fifty-four 
stars of the regular list in the N. A, are taken up in the 
order given there, that is in order of increasing right 
ascension from W 5 m 25 s .7 to 23 h l m 54 s .2 (1943). An 
arbitrary starting time is chosen but the list may be used 
at any later month providing one identification is made. 
For each subsequent star is found more eastward and de- 
scribed in relation to one or more close predecessors. 

It must be remembered that directions are those on the 
concavity of the celestial sphere. Curved lines must be 
imagined in following them. For instance, all directions 
of arcs of great circles radiating out from the North star 
are south. A small circle around the pole star followed 
clockwise as we observe it is always leading east, counter- 
clockwise west. 

As each star to be described is farther eastward on the 
celestial sphere, it is more elevated either later in the same 
evening or at the same time later in the year. The whole 
sphere appears to be revolving counterclockwise around 
the North star as a center (due north and as high in degrees 


Navigator's Stars and Planets 197 

as your latitude in northern hemisphere) rising in the east 
and setting in the west (or coming back under the North 
star in north latitudes) and showing more of itself in the 
east each night as the earth journeys around the sun. 

The easiest way to become familiar with these stars is 
to study and use a small celestial globe. Set for latitude, date 
and time, it shows better than any chart or table what stars 
are spread out for the observer. 

In the following list, stars marked with * are of too low 
declination to be visible from mid-north latitudes as Chi- 
cago or New York. Figures following star names are appar- 
ent magnitudes. 


1. a Andromedae (Alpheratz) 2.2 

Looking N. E. about 9 P. M. in mid-August in Lat. 42 
N. you see four bright stars in a row slightly sloping up to 
right, about equal distances apart, with a slight concavity 
in the row on upper side. The 4th or southmost star is the 
one named above. The angular distance between these 
stars is about 15 of the celestial sphere and this is the 
distance hereafter designated as 1 unit. Alpheratz is the 
N. E. corner of the "Square of Pegasus," a very easily recog- 
nized group with sides each about 15 long and at this time 
with its S. E. corner pointing down toward the horizon. 

2. ft Cassiopeiae (Caph) 2.4 

About 1/2 way from #1 to the pole star is #2. It is the 
westernmost of 5 stars forming the letter "W" whose upper 
surface is toward the pole. 

198 , Navigator's Stars and Planets 

3. Ceti (Deneb Kaitos) 2.2 
About 3 units S. by E. of # 1 . 

4. 8 Cassiopeiae (Ruchbah) 2.8 

4th. star to the E. in the "W" mentioned in #2. 

*5. a Eridani (Achernar) 0.6 
About 2^ units S. by E. of #3. 

6. a Arietis (Hamal) 2.2 

li/ units S. of 2nd. star from N. in row of 4 mentioned 

*7. Eridani (Acamar) 3.4 
units N.E. of #5. 

8. aPersei(Marfak)1.9 

The first or northmost of row of 4 mentioned in #L 

9. a Tauri (Aldebaran) 1.1 

About 2 units S. E. of 2nd. in row of 4 mentioned in #1 
is small dim cluster: the Pleiades. Proceed 1 unit more to 
#9. Red tinged. 

10. ft Orionis (Rigel) 0.3 

2 units ( S. S. E. of #9. The S. W. corner of a rough 
rectangle-Orionwhose length is N. and S. 

11. a Aurigae (Capella) 0.2 
li^ units E. by S. of #8. 

12. y Orionis (Bellatrix) L7 

1 unit S. E. of #9. The N. W. corner of Orion. 

13. c Orionis (Alnilam) 1.8 

The middle of 3 stars in center of Orion which are 
slanting S. E. and N. W. 

Navigator's Stars and Planets 199 

14. a Orionis (Betelgeux) 0.5 to 1.1 
The N. E. corner of Orion. 

*15. a Argus (Canopus) -0.9 
3 units S. S. E. of #10. 

16. a Cants Majoris (Sirius) 1.6 

units S. E. of #13. Our brightest star. 

17. e Canis Majoris (Adhara) 1.6 

1 unit S. by E. of #16. Westernmost of 3 close stars. 

18. a Canis Minoris (Procyon) 0.5 
units E. of #14. 

19. /? Geminorum (Pollux) 1.2 

li/ units N. of #18. Stars 14-18-19 make a right-angled 
triangle whose hypotenuse faces N. W. 

*20. e Argus 1.7 

li^ units S. E. of #15. Westernmost of 3 making equi- 
lateral triangle. 

*21. X Argus (Al Suhail al Wazn) 2.2 
2 units E. N. E. of #15. 

*22. /3 Argus (Miaplacidus) 1.8 

ls/4 units S. of #21. Stars 15-21-22 form an equilateral 

23. aHydrae(Alphard)22 
2 units E. S. E. of #18. 

24. aLeonis(Regulus)l.3 

I]/2 units N. N. E. of #23. Stars 18-23-24 make a right- 
angled triangle whose hypotenuse faces N. by W. 

200 Navigator's Stars and Planets 

25. a Ursae Majoris (Dubhe) 2. 

3^ units N. by E. of #24. The upper extremity of "big 
dipper" away from handle. It and star i/ unit S. constitute 
the "pointers" which point to the pole star. Seven stars in 
whole dipper. Handle curves down. 

26. ft Leonis (Denebola) 2.2 

units E. byN. of #24. 

*27. a' Cruets (Acrux) 1.6 

114 unit E. N. E. of #22. Southmost star in "Southern 

*28. y Cruets 1.6 

y z unit N. of #27. Top star in "Southern Cross/' 

*29. ft Crucis 1.5 

\/$ unit S. E. of #28. East star in "Southern Cross." 

30. e Ursae Majoris (Alioth) 1.7 

3rd. in handle of "big dipper" counting tip of handle 
as 1st. 

31. Ursae Majoris (Mizar) 2.4 

2nd. in handle of "big dipper" next to tip. 

32. a Virginis (Spica) 1.2 
214 units S. E. of #26. 

*33. Centauri 2.3 

units S. S. E. of #32 or 2 units N. E. of #27. 

34. a Bootis (Arcturus) 0.2 

2 units S. by E. of tip of handle of "big dipper." Stars 
26-32-34 make an equilateral triangle with sides 2i/ 3 units 

Navigator's Stars and Planets 201 

*35. a Centauri (Rigel Kentaurus) 0.3 
1 unit E. by N. of #27. 

36. p Ursae Minoris (Kochab) 2.2 

\y z units N. by E. of tip of handle of "big dipper." The 
upper extremity of "little dipper" away from handle. Seven 
stars in whole dipper, the tip of handle being pole star 1 
unit away. Handle curves up, 

37. a Coronae Borealis (Alphecca) 2.3 
li/ s units E. N. E. of #34. 

38. 8 Scorpii (Dschubba) 2.5 

. 2i/ 2 units E. S. E. of #32. Center one of 3 similar in row 
1/2 unit long going N. and S. 

39. a Scorpii (Antares) 1,2 

3 units E. S. E. of #32. Reddish. 

*40. a Trianguli Austmlis 1.9 
1 unit S. E. of #35. 

41. -7 Ophiuchi m (Sabik) 2.6 
1 unit N. E. of #39. 

*42. \ Scorpii (Shaula) 1.7 
unitS. E. of #39. 

43. a Ophiuchi (Rasalague) 2.1 
2 units S. E. by E. of #37. 

44. y Draconis (Etamin) 2.4 

units E. of tip of handle of "big dipper. 

*45. c Sagittarii (Kaus Australis) 2. 
unit E. S. E.of#39. 

202 Navigator's Stars and Planets 

46. a Lyrae (Vega) 0.1 

1 unit S. E. by S. of #44. In conditions named in #1 this 
is the bright star almost in zenith. 

47. o- Sagittarii (Nunki) 2.1 

2 units E. of #39. 

48. a Aquilae (Altair) 0.9 

2i/ 2 units S. E. by S. from #46. 

*49. aPavonis 2.1 
2 units S. E. of #45. 

50. a Gygni(Deneb) 1.3 

H/ units E. N. E. of #46. Stars 46-48-50 make a right- 
angled triangle with hypotenuse facing E. by S. 

51. tPegasi (Enif)2.5 

2 units E. of #48. 

*52. aGruis(AlNa'ir)2.2 
li/ s units N: E. by E. of #49. 

53. a Piscis Australis (Fomalhaut) 1.3 

3 units S. of S. W. star of "Square" (see #1). 

54. a Pegasi (Markab) 2.6 
S. W. star of "Square." 


In contrast to stars which usually twinkle, planets shine 
with a steady light. Instead of being a mere point of light, 
a navigational planet shows a distinct disc when viewed 
with binoculars. This disc can sometimes be seen with the 
unaided eye. The planets are always within a belt 8 each 

Navigator's Stars and Planets 203 

side of the ecliptic (projection of plane of the earth's orbit 
or apparent path of the sun). Venus, Mars and Jupiter are 
brighter than any star. Certain individual characteristics 
of the four navigational planets are as follows: 

Venus is the brightest heavenly body after the sun and 
moon. It is 12 times as bright as Sirius, the brightest star. 
Its orbit lies nearer the sun than our earth's and so it is 
never seen more than about 47 of angular distance from 
the sun. For the same reason it shows phases, like the moon, 
which can be easily detected with binoculars and which 
vary its magnitude or brightness from 4 to 3. It ap- 
pears as either a "morning" or "evening star." It gets around 
the sun in 225 of our days. When its R. A. differs by over 
2 hours from that of the sun, Venus may be seen in daylight 
if above horizon. 

Mars shines with a reddish light and varies in magnitude 
from 3 to +2. It must be distinguished from Aldebaran 
(#9) and Antares (#39) which are both reddish stars of 
magnitude 1 and near the ecliptic. Its orbit lies next outside 
the earth's and its year is 687 of our days. Much speculation 
has been given to the question of whether Mars is inhab- 
ited by intelligent beings. 

Jupiter is a brilliant white planet of 2 magnitude. Its 
orbit lies next outside those of the asteroids and its year is 
almost 12 of our years. Its diameter is over 12 times that of 
the earth and it has 1 1 moons. Usually 4 of these can be 
seen with binoculars. Roemer in 1675 made a remarkably 
close estimate of the speed of light from observations of the 
eclipses of these moons. 

Saturn is yellowish white and of magnitude to +L 
Only two stars are brighter than this planet: Sirius (#16) 

204 Navigator's Stars and Planets 

and Canopus (#15). It is more easily mistaken for a star 
than are the other three we have described. But when seen 
in a good telescope, it is a unique heavenly body because 
of its "rings." These are layers of billions of whirling par- 
ticles extending out from its equator. When seen on edge 
they are a thin line but as the position changes they are 
seen to be broad bands. There are also 10 moons. Saturn's 
orbit lies next outside Jupiter's and its year is 29J/2 of our 
years. Consequently, we do not notice much shift of its 
position among the stars between one year and the next. 
It is so much less dense than the earth that it would float 
on water. 

26. Reference Rules 
for Book Problems 

ON THE PAGE FOLLOWING is Table 19, given for 
occasional reference only. Learning its contents with- 
out understanding each step would be the very sort of thing 
in the study of navigation which this Primer attempts to 
prevent. When doing actual work at sea the labeling of 
angles east or west is usually self-evident or easily deter- 
mined from a rough diagram of the sort described in Chap- 
ter 2. The interested student, however, will want to work 
out problems for practice which he finds in textbooks. Then 
it is a more difficult matter to imagine oneself transported 
to the bridge of a ship, perhaps on the other side of the 
world, and to visualize the given situation accurately. The 
table provides a means of checking work done on such 









g 1 





W (Watch = zone time in A.M. & 12 hrs. behind zone time in P.M.) 
+ C-W (Add 12 hrs. to C> if necessary, to get this) 
CP (- 12k. if 13 or over) 
db dC (& possibly =fc 12h. as follows): G.C.T. ** zone time 4- zone number in W. Long, 
or zone number in E. long. Add zone time before subtracting if zone 
time is < zone number in E. long. "Longitude West, Greenwich Time Best, Longi- 
tude East, Greenwich Time Least" (considering date as well). 
Estimate G.C.T. roughly. Then: 
1. If G.C.T. is 12h. > C.F. add 12h. with C.C. 
2. If G.C.T. is between 0* and 1* subtract 12h. with C.C. 
3. If G.C.T. is > 24^ in W. Long. \t means excess is time and date is one 
4. If G.C.T. is > Z.I. in E. Long, it means date is one less. 
1 G.C.T. & date 


Star Planet Moon 

Any Body by G.H.A. Method 

Eq. T (NA) 
+ Corr. 


+ RAM0 + 12 (GST at Oh 
GCT) (NA) 
+ Corr. (for amt. GCT> 0^) 



GHA (= GAT -12 if sun W. 
of G or 12 - GAT 
if sun E of G) 
Eor W 

GST (subt. 24 if > 24) 
RA (NA) 
GHA (= difference & if > 12 
subt. it from 24) 
Observation or diagram usual- 
ly tells E or W. Also, by 
rule, if: 
<12 then 
> 12 subt. from 
24 & GHA=E 
<12 then 
1> 12 subt. from 
( 24 & GHA=W 

GHA arc (always < 180) 
Eor W 
Long DR E or W 
If GHA & Long have same 
LHA = difference 
If GHA & Long have different 
names & 
Sum *> 180, LHA = 
360 -sum 
Observation or diagram usual- 
ly tells Eor W. 
Also, by rule, if: 
Same names <& 

GHA arc (always < 180) 
Long DR E or W 

GHA arc at . GCT (NA) W 
+ corr. (days, hours, minutes) 
+ corr. (seconds) 

LHA (same as for sun) E or W 

GHA e E or W 
(If < 180 = W 
If between 180 & 360, sub- 
tract from 360 & = E 
If> 360, subtract 360 & 
Long DR E or W 

LHA (same as for sun) E or W 

GHA> Long then LHA = 
GHA <Lbng then LHA - 
Different names & 

GHA-fLong = <180 f 
GHA'+Lang =*> 180, 
- LHA -like Long 


27. Finding G. C. T. 
and Date 

be used in checking his calculations of time. It ex- 
presses the first part of Table 19 in another way and like 
that table will be found more useful in doing book prob- 
lems than necessary in actual work. It gives at a glance the 
G. C. T. and date from a 12-hour face chronometer at any 
hour of a 12-hour face watch on zone time in any time zone. 
Until watches and chronometers are generally equipped 
with 24-hour dials, this matter will always be a potential 
source of confusion and error. Meanwhile the table may be 
used as follows: 

Suppose on July 4 you are in Long. 135 E. with no chronometer error. 
(Table 6 on page 41 shows this to be zone minus 9.) Suppose it is morning 
and navigator's watch on zone time reads 8 o'clock. Wanted G. C. T. and 
date. Look in vertical column of 9 zone for figure 8 in A. NT. section. 
Follow horizontally till central chronometer column is reached and figure 

11 found. This is, of course, chronometer face. Look at overprinting in the 
triangle where you started and find "C + 12 Day before." Apply this to 
chronometer, 11 + 12 23, and subtract one from date: G. C. T. is 23& 
00m 00s on 3 July. 

When the time is between 12 and 1, an additional step is necessary. Say 
zone time is 12:30 P. M. in zone + 7. This means half an hour after 12 
A. M. (upper half). Follow vertical column down to the figure 1 (lower 
half). Then follow horizontal to center, finding C = 8. Subtract 30 minutes. 
C then becomes 7:30. The overprinting in zone group is C -f 12. 7:30 4- 

12 = 19:30 = G. C. T. with no change of date. 


From a 12-hr, face chronometer at any hour of 
a 12-hr, face watch on zone time in any time zone. 

YA L N G. - G - * -* L O N G. 

* It'll +10 

*9 *6 *7 *6 -5*4 * 
V A T C H 


-1 -a -3 -4 -5 -6 -7 -8 -9 -10 -11 -fc 
C _ V A T C H 


4 5 6 7 8 9 10 IMS, 

6 T-N 8 9 10 II II 
6 7\8(y 10 11 I; 

6789 10 II 

7 8 

7 6 9 10 
3 4 5 *6 7 8 9 10 11 

2.34567 891011 


4 5 

3456769 10 II \t/[ 



5 6 

10 11 

i a 3 4 a 6 7 e> 9 10 n la 



Each figure in central vertical column gives corrected chronometer read- 
ing for zone time in same horizontal row. 

12 in zone groups is only an instant not a second over. You must follow 
vertical column down or up to other half to get succeeding hour of 1. 

1 in zone group includes the minutes of preceding and succeeding hour. 

2 to 11 inclusive in zone groups include the minutes of the succeeding 

Overprinting in the eight triangles tells what to do with the corrected 
chronometer time in order to get G. C. T. and Date. 


28. Abbreviations 

a. Altitude difference or intercept 

C. Course. 

C. C. ' Chronometer correction. 

C. F. Chronometer face. 

C. O. Chronometer at observation. 

C. S. Chronometer at stop time. 

C-W Chronometer minus watch. 

co-L Co-latitude. 

Corr. Correction. 

cos Cosine. 

cot. Cotangent. 

C. P. Computed point, 
esc. Cosecant. 

D. D. Daily difference, 
d. or Dec. Declination. 
Dep. Departure. 
Dev. Deviation. 
Diff. or D. Difference of. 
Dist. Distance. 

D. R. Dead Reckoning. 

E. East. 

E. P. Estimated position. 

Eq. T. Equation of Time. 

G. Greenwich meridian. 

G.A. T. Greenwich Apparent Time (or G. A. C. T.). 

G. C. T. Greenwich Civil Time. 

G. H. A. Greenwich Hour Angle. 


210 A b breviations 

G. S. T. Greenwich Sidereal Time. 

H. Meridian altitude. 

h. Altitude. 

h c Calculated altitude. 

h Observed altitude corrected. 

h s Sextant altitude. 

H. A. Hour angle. 

hav. Haversine. 

H . D . Hourly difference. 

H. E. Height of eye. 

H. P. Horizontal parallax. 

I. C. Index correction. 

L. or Lat. Latitude. 

L. A. N. Local apparent noon. 

L." A. T. Local apparent time (or L. A. C. T.). 

L. C. T. Local civil time. 

L. H. A. Local hour angle (Properly, only W.). 

L. S. T. Local sidereal time. 

Lo. or Long. Longitude. 

log. Logarithm. 

M. Observer's meridian. 

m. Meridional difference of latitude. 

Mag. Magnetic. 

M. D. Minutely difference. 

Mid. L. Middle latitude. 

N. North. 

N. A. Nautical Almanac. 

nat, Natural. 

Obs. Observed. 

O. S. M. The "opposite-the-sun" meridian. 

p. Polar distance. 

Abbreviations 211 

P. Parallax. 

p. s. c. Per Standard Compass. 

R. Refraction. 

R. A. Right ascension. 

R. A. M. O Right ascension of mean sun. 

R. W. Ran stop-watch. 

S. South. 

S. D. Semidiameter. 

sec. Secant. 

sin. Sine. 

t. Meridian angle. (E. or W. < 180) 

tan. Tangent. 

Var. Variation. 

W. West or Watch. 

z. Zenith distance. 

Z. Azimuth. 

Z n Azimuth on 360 scale. 

O Sun. 

( Moon. 


# Star. 

9 Venus. 

$ Mars. 

U Jupiter. 


T First Point of Aries or Spring Equinox. 

^ Difference from. 

> Greater than, 
Less than. 

29 Forms 

THE BEGINNER will do better work if he follows a 
certain form for each procedure. A form for line of po- 
sition by H. O. 211 has been prepared which I hope leaves 
nothing to chance. Roughly it is divided into three parts. 
The first covers the sextant and timepiece work and gen- 
eral data; the middle part is done by means of the Nautical 
Almanac; and the last section is worked entirely by the 
211 Table. Course and log reading should be noted in a 
moving ship to allow for run between, in case another sight 
is taken later for a fix and the first line has to be moved up. 
The stop-watch method is suggested as described at the 
end of Chapter 2. A space is provided for everything which 
may be needed, no matter what heavenly body is used. 
The small squares for D. D., H. D., M. D. and H. P. are 
for values which are found or calculated during the first 
consultation of the N. A. for the body being used and 
mean daily difference, hourly difference, minutely differ- 
ence and horizontal parallax. The proper entries should 
be made here before turning to the various correction tables 
in other parts of the almanac so that no looking back will 
be necessary. The abbreviations in the altitude correction 
section are: R, refraction; P, parallax; S D, semidiameter; 
H E, height of eye; I C, index correction, etc. After the 
student has used a form of this sort for some time, he will 
have become so familiar with the various things to remem- 
ber that he may then advantageously arrange matters in a 
more abbreviated way. 


Forms 213 

There are three other forms, designed for H. O. 211, 
which may be found occasionally convenient. A form for 
line of position by the cosine-haversine method with time 
and altitude azimuth should be useful when Bowditch 
alone is available. The remarkable, though at present rather 
impractical, formulas for Aquino's Fix may be easily ap- 
plied in a fprm here offered. 

A form for Dead Reckoning, two for Middle Latitude 
and two for Mercator sailing, should help the beginner. 

Forms for Day's Run and Day's Current show how each 
may be worked either by logs or by Table 3. 

When a word is followed by a different word in paren- 
theses, as Dep. (Lat.), it means: find the first in the column 
headed by the word in parentheses. 

H. O. 211 

(Form by J. F.) 











AT , h 


AT 1 






+FOR h. 

+FHR ff\ 

POD 1 


H s 





G.H.A. - W 
iF>160' - 










KD IF>>0 SUB- 





A "*" 








LAT. D.R. 




^L^UKES 7 - 5 ") 



FiND NO. wfta 


IF KNLAR 90't 

5 2* > LAT. TAKE- 1 FROM TOP 



a iTOWARDSl n \ LLS) : 

Forms 215 

H. O. 211 


* A 

d B A 

h A 

R >A B B 



Give <t>" same name as declination 

0' is N if body bears N & E or N & W 

0' is S if body bears S & E or S & W 

Combine by adding if different, subtracting if alike 

L = <P' (180* <P") for lower transit 

216 Forms 






2) l.v. 







H. O. 211 


90 = 89 60' .0 


l.CSC 2 1 A t = fc -r 2 = y% t = & found in A gives 

J /2 * = & X 2 = 

= & combined with 

G.H.A., gives 


When d & L have same name: 

p = 90 - d 
When d & L have opposite names: 

# = 90 + d 







, B 



Forms 217 

Altitude Azimuth 
H. 0. 211 


90 = 89" 60' .0 

l.sec 2 i/ 2 Z = &*2 = 

l.sec 14 Z = & found in B gives 

1/2 Z = & X 2 = 

When d & I have same name: 

p = 90 - d 
When d & L have opposite names: 

p = 90 + d 

218 Forms 

Line of Position 
H. O. 9 

(Bowditch: Cosine-Haversine Altitude, and Time and 
Altitude Azimuth) 










6 l.hav. 

6 n.hav. 





<-z n.hav. 

From 89 60' .0 




Z l.sin. 

Z from? pole 

Aquino's Fix 

(Form modified by J.F.) 

From Simultaneous Altitude and Gyro Azimuth 





l.sec. 89 60' .0 


B.l.tan. B = 

l.tan. C = 



l.sec. : 

b.l.tan. b = 

Long. Lat. 

Forms 2 1 9 

Middle Latitude Sailing 


Latitude Latitude Longitude 


1 II O / 

in o / 

// / // 
// / // 



' 2) ' 

" DLo ' /; 


' Lm 

// r 

' " Lcos. 













log (Dep.) 




T O 

, n 

LI ~\~ L% 

t tt _._ 2 = 



o / 

o / 


Forms 221 

Mercator Sailing 


Latitude Longitude 

From ' /' M.P. ' " 

To ' " M.P. ' " 

DL ' " m 




Ltan. l.sec. 







/ // 

' " M.P. 

Lt o / // M _p t 


m log 


222 Forms 

Day's Run 

Latitude Longitude 
By obs. or D.R. noon yesterday * 

By obs. or D.R. noon today * 


By Logs and Mid. Lat. Sailing 

DLo log 

Lm l.cos, 

(Dep.) log 

DL log log 

Course , l.tan. l.sec. 

In Quad 

Dist. log 

By Traverse Table and Mid. Lat. Sailing 

DLo (Dist,) 1 

Lm (CnJ.1 |Table 3: Dep. (Lat.) 



Day's Current 

(Given D.R. run in miles: DL * and Dep. ) 

Latitude Longitude 

Position by obs. noon yesterday N & 

Run by D.R. noon today DL * DLo 

Position by D.R. noon today 
Position by obs. noon today 

from N 
below / 


"(miles) Z>Z,0 

\ of Lo ) 

By Logs and Mid. Lat. Sailing 

DLo, Cur.. 


. log 

Dep. D.R.. 


nr, num. 

DLo, D.R. 


enter i __. 
above lo g 


fYmrse, Cur. 


Tn Quad. 


DIS* i^nr. 



= Drift (m.p.h.) 

By 1 raverse Table and Mid. Lat. Sailing 

= DL,, D. R . (DiM.) 

- above 

i 1 _ , , fCourse. 

Dep., Cur. (Dep.). 


_= Drift (m.p.h.) 

30. Problems 

IN CONTRAST to most texts, this Primer has not in- 
eluded problems with its explanations of the various 
procedures of celestial navigation. The reason for this is 
that it was hoped the student would more quickly acquire 
sound principles and a clear understanding of the subject 
as a whole if he were spared the detail of computations 
while first reading the book through as a sort of survey 
course. Plenty of problems can be found in Bowditch or 
Dutton or other texts and short-cut systems. Best of all 
are the problems one goes out and sets for himself. How- 
ever, a few illustrative examples will now be given of the 
more important procedures. I have taken data for the 
meridian latitude sights from Bowditch, 1933 edition, and 
for the time conversions from the 1939 Nautical Almanac. 
The rest are my own. The fact that most of my sights were 
taken from known positions may lessen interest but has 
the advantage of showing the degree of accuracy of the 

Problems 225 

Latitude by Meridian Altitude 
Case 1: L & d opposite names. L = z - d 

At sea, May 15, 1925, in Long. 0, the observed meridian 
altitude of the sun's lower limb was 30 13' 10"; sun bear- 
ing north; I. C., + 1' 30"; height of the eye, 15 feet. 


hs 30 13' 10" 

Corr. . . . . + 12' 02" 

ho 30 25' 12" Now subtracting from 90 or 

89 59' 60" gives 

z 59 34' 48" Now from G.C.T. and N.A. we get 

d 18 48' 30" N. andz - d gives 

L 40 46' 18" S. 

226 Problems 

Latitude by Meridian Altitude 
Case 2: L & d same name and L>d.L=z+d 

At sea, June 21, 1925, in Long. 60 W., the observed 
meridian altitude of the sun's lower limb was 40 04'; sun 
bearing south; I. C., + 3' 0"; height of the eye, 20 feet. 

hs 40 04' 00" 

Corr. . . . . + 13' 21" 

ho 40 17' 21" 

89 59' 60" 


z 49" 42' 39" 

d 23 26' 48" N. and z + d gives 

L 73 09' 27" N. 

Now subtracting from 90 or 

Now from G.C.T. and N.A. we get 

Problems 227 

Latitude by Meridian Altitude 

Case 3: L & d same name and d > L. L d z 

At sea, April 14, 1925, in Long. 140 E., the observed 
meridian altitude of the sun's lower limb was 81 15' 30"; 
sun bearing north; I. C., - 2' 30"; height of the eye, 20 feet. 

hs 81 15' 30" 

Corr. . . . . + 9' 00" 

ho 81 24' 30" Now subtracting from 90 or 

89 59' 60" gives 

z 8 35' 30" Now from G.C.T. and N.A. we get 

d 9 10' 48" N. and d - z gives 

L ~ 35' 18" N. 

228 Problems 

Latitude by Meridian Altitude 

Case 4: L & d same name, lower transit. 
L = 180 - (d + z) = h + p 

June 13, 1925, in Long. 65 W., and in a high northern 
latitude, the observed meridian altitude of the sun's lower 
limb was 8 16' 10", sun below the pole; L C., 0' 00"; height 
of the eye, 20 feet. 

hs ....... 8 16' 10" 

Corr. .... + 5' II" 

8 21' 21" Now subtracting from 90 or 
89 59' 60" gives 

z ........ 8 r 38' 39" Now from G.C.T. and N.A we get 

d ..... ... 23 10' 56" N. and d + z = 

104 49' 35" and subtracting from 180 or 
179 59' 60" gives 

L ..... ... 75 10' 25" N. 

90" ..... 89 59' 60" 
d ....... -23 10' 56" 

p ..... . . 66 49' 04" 

ho ...... + 8 21' 21" 

L ..... . . 75 10' 25" N. 


Latitude by Phi Prime, Phi Second 
(H. 0.211) 


At anchor, in evening of August 21, 1935, in Lat. 42 
12' N., Long. 87 48' W., observed star Deneb, bearing 
about ENE, as follows: sextant altitude 58 56'; I. C. 0' 0";- 
H. E. 12 feet; G. C. T. 22 August, P 39 m 20 s . 

G.H.A. at 1 Aug. O h 
-1- for 22 d 
+ for l h 39 m 
+ for 20 s 


358 51'. 4 
20 41'. 9* 

d = 45 3'. IN 

R 0'.6 
H.E. -3'. 4 


404 27'. 4 W. 
-360 00'. 

ho 58 52' 

44 27'. 4 W. 


8748'.0 W. 


43 20'. 6 E. 

t 43 20'. 5 E 



/ \1/ 
^r s\ 

+ n 

d '45 3'. IN 

B 15089 A 15014 

ho 58 52' 
<" 54 l'.5N 

> B 5822 

A 6754 

31435 A 

5 5822 

-<r A 9192 

<' 11 49'. 5 N 

<- B 932 

L 42 12' N 

* Beginning with the 1936 N. A., the G, H. A. of stars is given for each 
day of the month, thus shortening this calculation by 1 step. 



Longitude by Time-Sight 
(H. 0.211) 

On Lake Michigan shore, in morning of June 24, 1935, 
in about Lat. 42 12' N., Long. 87 48' W., observed sun's 
lower limb bearing about E., as follows: sextant altitude 
14 10'; I. C. 0' 0"; H. E. 87 feet; G. C. T. 24 June, 11* 
46 m s . 

G.H.A. at 10 h 
+ for i h 4< 


329 29'. 6 
3 m 2630'.0 

d =* 2326'N 
from 90 00' 

R.P.S.+ 12'. 1 
H.E. - 9'.1 

355 59'. 6 W. 
from 360 00'. 

p * 66 34' 

+ 3'.0 
hs 14 10'. 

ho 14 13' 

4 0'.4E. 











34 ; 








29'. 5 





16'. 5 







1. CSC % t ' 


t = 


28748 & -s- 2 = 
14374 & found in A gives 
45 54'. 5 &X2 = 
91 49'. E. & combined with 
4 0'. 4 E. gives 

87 48'. 6 W. 

Problems 231 

Altitude Azimuth 
(H. 0.211) 

At anchor, in morning of June 16, 1936, in Lat, 42 
12' N., Long. 87 48' W., observed sun's lower limb, bear- 
ing about E., as follows: sextant altitude 33 31' 10"; I. C. 
0' 0"; H. E. 12 feet; G. C. T. 16 June, 13* 32 m 6 s . 

d = 23 21'. 6 N R.P.S. + 14'. 5 

from 9000'.0 H.E. - 3'. 4 

66 38'. 4 + 11'. 1 

hs 33 31'. 2 

ho 33 42'. 3 

h 33 42'. 3 B 7993 

L 42 12' B 13030 


p 66 38'. 4 Lv. 21023 

2) 142 32'. 7 

s 71 16'. 3 B 49346 

-66 38'. 4 + 

s-p 4 37'. 9 B 142 

Liu 49488 
Lv. 21023 

L sec' 1 A Z = 28465 & -s- 2 

1. sec y% Z = 14232 & found in B gives 

y 2 Z = 43 54' & X 2 = 

Z = N. 87 48' E. 

(Merely a coincidence that this = Longitude) 

232 Problems 

Azimuth by H. O. 214 

Make 5 columns headed by: Decimals, t } d> L> and Base. 
Data taken from Sun problem, which follows: 

* 19 47.7 E., d 23 26'.8 N., L 42 12' N. 

Enter these as degrees and decimals in left column. 

Use H. 0. 214, Vol. V, entering with t 19, d 23, L 42 
(the next lower whole degrees) on page for L and d of same 
name and get Base Z 134 .3. 
Enter this at top of the four remaining columns. 
Use H. 0. 214 with base d fe L but 1 greater t 
" " " " " tkL " 1 " d 

^ d jo L 

and write resulting azimuths under Base in proper columns. 

Find differences between above and Base which give 
changes for 1 increase of each variable. 

Mark results ( + ) or ( ) according as azimuth is increas- 
ing or decreasing with the higher entry figures. 

Convert each to minutes of arc. 

Multiply each by the decimal fraction of a degree by 
which corresponding quantity of given data exceeds 
quantity used to get Base. 

Mark results ( + ) or ( ) as before. 

Take algebraic total for correction. 

Apply to base Z in 5th column for Z (on 180 system). 

Convert to Z on 360 system = Zn. 










(19) 134. 3 
(20) 132. 5 

(23) 134. 3 
(24) 132. 8 

(42) 134. 3 
(43) 135. 8 

134. 3 


(-) l-8 = 

(-) l-5 = 

(+) l-5 = 

N132.5E =Z 
132. 5 = Zn 


X .45 


(-) 86' A 

(-) 40' .50 
(-) 86' .4 (Add.) 

(+) 18' .0 

(-) 126' .9 
(+) 18'. (Sub.) 

(-) 108' .9- 
148'.9 = 
(-) l.8 = Corr. 

234 Problems 

Line of Position from Sun 
(H. O. 211) 

At anchor, in forenoon of June 21, 1936, in about 
Lat. 42 12' N., Long. 87 47' W., observed sun's lower 
limb, bearing about SSE, as follows: sextant altitude 64 
52' 10"; I. C. 0' /A ; H. E. 12 feet; G. C. T. 21 June, 16 h 
33 m 34 s . 

G.H.A. at 16 h 59 35'. 8 
-f for 33 m 8 15' 
+ for 34 s 8'. 5 

d = 23 26'. 8 N. 

R.P.S. - 3'. 6 
H.E. + 15'. 6 

+ 12' 
hs 64 52'. 2 

G.H.A. 67 59'. 3 W. 
Long.D.R. 8747'.OW. 

ho 65 4'. 2 

L.H.A. , 19 47'. 7 E. 

t 19 47'. 7 A 47031 
d 23 26'. 8 N B 3744 

A 40017 
- B 2204 B 2204 A 50775 

50775 A -) 

K 24 45' N < 
L 42 12' N 

-A 37813 
B 2046 

E>L 17 27' 

be 65 4'.0 
ho 65 4'. 2 

-f A 4250 ->- B 375 14 

A 13261 
Zn 132 32' 

a .2 mile toward 

(See diagram on opposite page) 



Line o Position from Sun 

(H. O. 9) 

Previous problem worked by Cosine-Haveisine method, 
with "time and altitude" azimuth formula. 

t 19 47'. 7 

L 42 12' N 
d 23 26'. 8 N 

18 45'. 2 

1. hav. 8.47052 


1. cos. 9,86970 


1. cos. 9.96258 

1. hav. 8.30280 

8 n. hav. .02008 
n. hav. .02654 

Lsin. 9.52975 

1. cos. 9.96258 

z 24 56' 15" -4- z n. hav. .04662 

from 89 59' 60" 

he 65 3'. 7 
ho 65 4'. 2 

1. sec. 10.37506 
Zl. sin. 9.86739 
Z = 47 28 7 (from S. toward E.) 

a . 5 mile toward 

(This method requires 9 consultations of the tables, whereas H.O. 211 requires 
but 7.) 



Line of Position from Moon 
(H. 0.211) 

At anchor, in evening of October 29, 1936, in Lat. 42 
12' N., Long. 87 48' W., observed moon's lower limb, 
bearing about ENE, as follows: sextant altitude 7 58' 40"; 
I. C. 0' 0"; H. E. 12 feet; G. C. T. 29 October, 23 h 4 m 1 s ! 
(See diagram on opposite pdge.) 


. 14 24'. 4 

H.D. 11'. 4 

H.P. 61'. 5 

G.H.A. at 23* 

+ for 4 m 
+ for 1 s 





354 32'. 2 
57'. 6 
.2 ' 

d at 23* 165'.2N 
+ for 4 m 1 s 1' 



+ 70'. 7 

- 3'. 4 

d 166'.2N 

+ 67'. 3 
7 58'. 6 

9 5'. 9 

355 31' W. 
360 GO' 

4 29' E. 
87 48' W. 

92 17' E. 

t 92 17' 

d 16 6'.2N 

A 34.5 
B 1738 

A 55694 

1772. 5A->B 55289 



97 49' N 
42 12' N 

9 5'. 8 
9 5'. 9 


B 55289 

B 24816 
-<-A 80105 

A 1772 


A 1222 

a . 1 mile toward Zn 76 28' 

(This calculation illustrates the exception to Rule 1: since t > 90, take 
K from bottom; and the exception to Rule 2: since K is same name as 
and > L, take Z from top. Whenever the first exception applies, the second 
will also be called for. See Fig. 40, page 136.) 



(This diagram is to be 
used with the Problem 
on Page 236.) 

Line of Position from Planet 
(H. 0.211) 

On board S.S. Western States, making passage Chicago 
to Mackinac, in evening of July 15, 1937, in D. R. posi- 
tion Lat. 43 30' N., Long. 86 51' W., observed planet 
Jupiter, bearing about SE, as follows: sextant altitude 
7 13'; I. C. 0' 0"; H. E. 35 feet; G. C. T. 16 July, 2* 
7 m 9 s . 






M.D. 15'. 0468 

D.D. 1'.4 

R 7'. 2 
H.E. 5'. 8 

at O h 359 14' . 1 
for2 11 30 4'. 8 
for7 m 145'.3 
for 9 s 2'. 3 

d at O b 21 56'. 9 S. 
+ for 2 h 7 m 9 8 . 1 

d = 21 57 7 S. 

- 13' 
hs 7 13' 

391 6'. 5 
- 360 O'.O 

ho 7 

31 6'.5W. 
86 51' W. 

55 44'. 5 E. 



t 55 44'. 5 
d 21 57' 

K 35 36' S 
L 43 30' N 

K>L79 6' 
he 6 58'. 5 
ho 7 O'.O 

A 8275 

B 3268 A 42736 

11543 A -> B 19238 
4- A 23498 

1.5 miles toward 

B 72332 
A 91570 

A 11543 



Line of Position from Star 
(H. O. 211) 

At anchor, about an hour before midnight o October 
29, 1936, with full moon, in Lat. 42 12' N., Long. 87 
48' W., observed star Capella, bearing about ENE, as fol- 
lows: sextant altitude 51 51' 40"; I. C. 0' 0"; H. E. 12 
feet; G. C. T. 30 October, 4 h 53 m 31 s . 

G.H.A. at O 11 

320 14'. 8 

d = 4556M N. 


- O'.S 

+ for 4* 53 m 

73 27' 


- 3'. 4 

-f- for 31 s 

7'. 8 

- 4'. 2 

393 49'. 6 


51 51'. 7 

360 00' . 


51 47'. 5 


33 49'. 6 W. 




53 58'. 4 E. 



53 58'. 4 

60 21' N 
42 12' N 

18 9' 

51 47'. 
51 47'. 5 

A 9218 
B 15771 

A 14355 

24989 A -> B 8259 

<- A 6096 

B 8259 A 24989 

B 2216 
-<- A 10475 ->- B 20856 

.5 miles toward 

A 4133 
Zn 65 24' 

(This calculation illustrates the exception to Rule 2: since K is same 
name as and > L, take Z from top. See Fig. 40, page 136.) 

(For Problem on Page 239) 

(For Aquino's Fix) 

Aquino's Fix 

At anchor, in evening of June 20, 1936, in Lat. 42 12' 
N., Long. 87 48' W., observed star Vega as follows: sextant 
altitude 41 16' 30"; simultaneous gyro azimuth 70 50'; 
I. C. 0' 0"; H. E. 12 feet; G. C. T. 21 June, 2* 7 m 9 s . 

(As I had no gyro, this azimuth was actually determined by calculation, 
and the above is merely to illustrate the method which follows.) 

G.H.A. at 0* 

350 18'. 2 

d - 38 43'. 4 N. 


- 1'.2 

+ for 2 h 7 m 

31 50'. 2 


- 3'. 4 

+ ff\i- QS 

f ) 1 i 

At f. 

382 10'. 7 


41 16'. 5 

3/rnO AA/ A 


4-1 11' O 


22 10'. 7 W. 

ho 41 12' 

1. sec 12354 1. tan 9.94222 

Zg 70 50' 

1. esc 2477 1. sec 48371 89 


38 43' 

1. sec 10777 

1. tan 9 . 90397 C = 20 33' . 5 
1. sec 38436 4- 

t 65 37' . 7 E. -<- 1. esc 4054 

G.H.A. 22 10'. 7 W. b. 1. tan 0.28833 b = 62 46' 

Long. 87 48'. 4 W. 


Lat. 42 12'. 5 N. 



Fix from Two Stars * 
(H. O. 214) 

In evening of Nov. 2, 1941, in D. R. position Lat. 42 15' 
N., Long. 87 42' W., observed stars as follows: 

Capella: bearing about N. E. by N., sextant altitude 11 
30' 00", G. C. T. Nov. 3, O h l m 30 s . 

Deneb Kaitos: bearing about S. E. by E., sextant altitude 
1 1 04' 50", G. C. T. Nov. 3, O h 3 m 22 s . 

In each case: I. C. 0' 0", H. E. 12 feet. 




Deneb Kaitos 

11 04' .8 

(-) 4' .9 

H 3' .4 

10 56' .5 

* A practice sight from known position, 42 12' N., 87 48' W., on Lake 
Michigan shore, using an imaginary D.R. position to illustrate the method. 




FIG. 45. Fix bv H. 0. 214, D. R. Position. (See opposite page.) 

Problems 243 

Solution using the D.R. position and 3 altitude corrections for each body. 
Capella Deneb Kaitos 

d = 45 56'. 2 N (from N.A.) d = 18 18'. 3 S 

G.H.A. at O h = 323 45'. 4 W G.H.A. at 0* = 31 41'. 4 W 

+ for 1* 15' + for 3 m 45'. 1 

+ for 30 s 7'. 5 + for 22 s 5'. 5 

324 07'. 9 W 
from 359 60'. 

G.H.A. at obs. 35 52' . 1 E, add G.H.A. at obs. 32 32' . W, from 

LoJXR. 8742'.OW Lo.D.R. 8742'.OW 

t 123 34'. IE t 5510'.OE 

(from H.O. 214, Vol. V.) 

(p. 73) (p. 63) 

Enter with Take out Enter with Take out 

L 42 N] (h 11 06'. 5 L42N) (hll27'.4 

d 46 N ) < A d 78, A 1 43 d 18 S > | A d 78, A t 59 

t 124 j (Z35.9 t 55 j (z 127. 4 

Alt. com for: Alt, corr. for: 

L(15'&Z36) = 12M(+) L(15'&Z127) = 9'.0 (-) 

d(3'.8&78) = 2'.9(-) d(18'.3&78) = 14'. 2 (-) 

t (25'. 9 & 43) = ll'.2(+) t (10'&59) = 5'. 9 (-) 

20'.4(+) 29'.1 (-) 

h from table 11 06 ; . 5 h from table 1 1 27' .4 

hcforD.R. 11 26'. 9 hcforDJR. 10 58', 3 

ho 11 21'. 9 ho 10 56'. 5 

= (away) 5.0 miles a ~ (away) 1.8 miles 

(Plotted from D.R. position: 42 15' N & 87 42' W gives fix at 42 12' N 
87 48' W.) 



FIG. 46. Fix by H. 0. 214. Assumed Position. (See opposite page.) 

Problems 245 

Solution using an assumed position and 1 altitude correction for each body. 
Capella J) e neb Kaitos 

d - 45 56'. 2 N (from N,A.) d - 18 18'.3 S 

G.H.A. at O h = 323 45'.4 W G.H.A. at O h 31 41'. 4 W 

+ for l m ' 15' +for 3 m 45'. 1 

+ for 30 s 7'. 5 + for 22 s 5'.5 

324 07'. 9 W 
from 359 60'. 

G.H.A. at obs. 35 52' . 1 E, add G.H.A. at obs. 32 32' . W, from 

Lo assumed 88 07' . 9 W Ao give t a\Lo assumed 87 32' . W 

\whole no.J 

t 124 E t 55 E 

(from H.O. 214, Vol. V.) 

(p. 73) (p. 63) 

Enter with Takeout Enter with Takeout 

L 42 N) fhll06'.5 L42N) (hll 27'.4 

d 46N Ad78 d 18 S Ad78 

t 124 J (Z35.9 t 55 J [Z 127. 4 

Alt. corr. for: Alt. corr. for 

d (3'. 8 & 78) = 2'. 9 (-) d (18'. 3 & 78) - 14'. 2 (-) 

h from table ll^'.S h from table 1127'.4 

he for A.P. 11 03'. 6 he for A.P. 11 13'. 2 

ho 11 21'. 9 ho 10 56'. 5 

= (toward) 18.3 miles a = (away) 16.7 miles 

Plotted from A.P. \ Fix at / Plotted from A.P. ' 

42 N&8807',9WJ 42 12 ; N & 87 48' W \42 N& 87 32'. W 

246 Problems 

(H. O. 127) 

At anchor, in evening o May 5, 1937, in Lat. 42 12' N., 
Long. 87 48' W., observed an unknown star as follows: 
sextant altitude 39 9' 10"; I. C. 0' 0"; H. E. 12 feet; 
azimuth about 99; G. C. T. 6 May, P 32 m 4 s . 

R - T .2 
H. E. - 3' .4 

- 4 7 T 
hs 39 9' .2 

ho 39 4' .6 

1. Entering H. O. 127 with L 42, Z 100 and h 40, we obtain: 

d = 19 .3 and t = 3 h 32 m 

2. t converted to arc = 53 

3. Long. 87 48' - 53 = 34 48' = G. H. A. (approximate) at 


4. Enter N. A. at "Correction to be added to tabulated 

G. H. A. of stars" with G. C. T. l h 32 m and obtain 
23 3' .8 = increase of G. H. A. since O h 

5. G.H. A. at observation 34 48' 
Increase since O b 23 3' .8 

G. H.A. atO h 11 44' .2 W. 

6. Look in N. A. "Stars" for May 1937, date 6, to find one 
with approximately d 19 18' N. and G. H. A. at O h 
11 44' .2 W. The nearest combination found is 
d 19 30' .3 N. with G. H. A. 10 13' .3. This is close 
enough and identifies this star as Arcturus. 

Problems 247 

Time of Sun on Prime Vertical 
(H. O. 211) 

When declination is less than latitude and the same 
name,* the time sun will be due E. or W. may be found as 

Estimate G. C. T. when sun will be on Prime Vertical, 
E. or W. as you may desire. 

Take d for this G. C. T. from N. A. 

Estimate L and Lo in which ship will be at this G. C. T. 

Find L. H. A. (f) of sun when on P. V. by the following: 

esc d 

sec h 

-= csc/z 


sec d 

Combine t with D. R. Lo to get G. H. A. of sun on P. V. 

Use method for transit to find G. C. T. of this G. H. A. 

This G. C. T. will be approximate time (as good as D. R. 
Lo) when sun will be on P. V, and longitude sight can best 
be made. 

The advantage of taking the observation for longitude 
at this time is that any ordinary error in the latitude used 
will not affect the accuracy of the result. 

* If d > L and same name, body never crosses P.V, 
If d & L are opposite names and 

d < L body crosses P.V. below horizon, 
d > L body never crosses P.V. 

Required time of sun on P. V., 31 July, 1942, in after* 
noon, at D. R. position Lat. 41 46' N., Long. 86 51' W. 
Estimated G. C. T. for above: 22 h on 31 July. 
Declination for this: 18 15' .7 N. 


By H. O. 211: 



18 15'. 7 N 
41 46'. ON 

t 6819'.OW 

+ Lo 8651'.OW 

G.H.A. 155 10'. OW 
G.C.T. 22 h 


A 50404 
A 17646 

A 32758 

B 2243 

B 5430 
A 3187 

To find G.C.T. for this 
when G.H.A. = 



26 m to increase G.H.A. by 

56 s to increase G.H.A. by 

G.C.T. 22 h 26 m 56 s Time of sun on P.V. 

For +6 zone and wartime this became 5 h 26 m 56 s P.M. 

155 10' 
148 26' 

6 44' 
6 30' 


Civil to Sidereal Time 249 

Civil to Sidereal Time Conversion 

On July 13, 1939, when local civil time is 9 h 3 m 30 s in 
longitude 85 15' W. (5 h 41 m ), what is the local sidereal 

G.S.T. of O h G.C.T., July 13 (NJL p. 3) 19* 19 m 54 s . 5 

"Reduction" (here an increase) for 5* 41 m (Table VI, p. 289) . + 56 s . 

L.S.T. of (4 L.C.T. July 13 19* 20 m 50*.5 

AddL.C.T 9* 3* 30* 

"Reduction" (here an increase) for 9 h 3 m 30 s (Table VI, p. 289) l m 29 s . 3 

28 11 25 m 49 s . 8 
Reject 24^ 

L.S.T 4 h 25m 498 ' 8 

In case the above is not obvious to the student, a detailed 
explanation of each step with roughly accurate diagrams 
will be found below. Dotted curves represent sidereal 

NOTE: In the N. A., Table VI and the table at bottom of pages 2 and 3 
have two purposes: 

1. Conversion of Civil to Sidereal (add) 

2. Correction to G.S.T. for L.S.T. 

a. W. Longitude: add. 

b. E. Longitude: subtract. 
Starting with G.S.T. of O h G.C.T., 

To find L.S.T. of O h L.C.T.: ,_ n., v T Q T 

For W. Long, add factor for Long, in Tune from either table because L.b. 1 . 

is later = a passage of time into future. 

For E. Long, subtract factor for Long, in Time from either table because 

L.S.T. is earlier = a passage of time into past. 


Civil to Sidereal Time 

G. S. T. of 0* G. C. T. 

19 m 54 s . 5. 

At O h L. C. T. we want the 
L. S. T. But then O will have 
moved through 5 h 41 m of longi- 
tude since first diagram. (Angle 
between dotted meridians is 
same as angle between G and 
M meridians.) So T will have 
also moved 5 h 41 m plus a cer- 
tain amount. (Table VI.) This 
extra amount is because T 
"gets around" faster than O 
and for a given duration there 
are always more units of 
sidereal time than of civil 
time. For L. S. T. we start our 
dotted curve at local meridian 
M, thus discarding that part of 
dotted curve in first diagram 
which lies between G and M, 
or 5 h 41 m . Hence L. S. T. at 
O h L. C. T. is same as G. S. T. 
at O h G. C. T. plus "Reduc- 
tion" of 56 s for the longitude. 
This totals 19 h 20 m 50 s .5. 

Sidereal to Civil Time 


But we must figure for the 
L. C. T., when O has moved 
9 h 3 m 30 s since second diagram. 
Then T will have also moved 
9 h 3 m 30 s plus a certain 
amount. (Table VI.) So we ex- 
tend the dotted curve 9 h 3 m 
30 s + l m 29 s .3. Adding this to 
L. S. T. of O h L. C. T. gives 
28* 25 m 49 s .8. 

As the last amount is over 24 h , 
and we do not count sidereal 
dates, we discard 24 h leaving 
L. S. T. = 4 h 25 m 49 s .8 at 
L. C. T. 9 h 3 m 30 s . 

Sidereal to Civil Time Conversion 
On July 13, 1939, when local sidereal time is 4 h 25 m 
49 s .8 in longitude 85 15' W. (5* 41-), what is the local 
civil time? 

G.S.T. of 0* G.C.T., July 13 (N.A. p. 3) 19* 19m 

"Reduction" (here an increase) for 5* 41m (Table VI, p. 289) . + 


L.S.T. of 0. L.C.T. July 13 * 20m 50s 5 

L.S.T. given (+ 24 for subtracting above) ^o n ** m w .& 

Sidereal time interval since 0* L.C.T ^ 4m 5*.3 

Reduction (actual) for 9^ 4m 59S.3 (Table V, p. 287) - lm 293.3 


3m 30s 


Sidereal to Civil Time 

In case the above is not obvious to the student, a de- 
tailed explanation of each step with roughly accurate 
diagrams will be found below. Dotted curves represent 
sidereal time. 

G. S. T. of Q h G. C. T. 19 h 
19 m 54 s .5. 

At O h L. C. T. we want the 
L. S. T. But then O will have 
moved through 5 h 41 m of longi- 
tude since first diagram. (Angle 
between dotted meridians is 
same as angle between G and 
M meridians.) So T will have 
also moved 5 h 41 m plus a cer- 
tain amount. (Table VI.) This 
extra amount is because T 
"gets around" faster than O 
and for a given duration there 
are always more units of 
sidereal time than of civil 
time. For L. S. T. we start our 
dotted curve at local meridian 
M, thus discarding that part of 
dotted curve in first diagram 
which lies between G. and M., 
or 5* 41 m . Hence L. S. T. at O b 
L. C. T. is same as G. S. T. at 
O h G. C. T. plus "Reduction" 
of 56 s for the longitude. This 
totals 19*20* 50 s .5. 

Sidereal to Civil Time 
We know the L. S, T. at the 
unknown L. C. T. is 4 h 25 m 
49 s .8. We want to know the 
interval in sidereal time be- 
tween this L. S. T. and the 
L. S. T. of 0* L. C. T. as 
shown in second diagram. In 
order to subtract 19 h 20 m 50 s .5 
from 4 h 25 m 49 s .8 we add 24* 
to the latter. (Other combina- 
tions may not require this.) 

This subtraction gives the de- 
sired interval in sidereal time 
between L. S. T. at the un- 
known L. C. T. and L. S. T. 
of O h L. C. T. It is 9 h 4* 59 s .3. 
This is a slightly larger quan- 
tity than the civil time for the 
same duration. To convert it 
to civil it must be reduced by 
an amount l m 29 s .3 found for 
it in Table V. Subtracting this 
gives the interval now in civil 
time.Itis9 b 3 m 30 8 . 

This shows L. C. T. 9 h 3 m 30 s 
at L. S. T. 4* 25 m 49 s .8. 


254 Transit 

A Uniform Method for More Exact Time 
of Local Transit of Any Body 

Find G,H.A. (W. or E. to 180) at 0* G.C.T. of date on which local transit 
is desired and compare with Lo. 

Rule I, for West Longitude: If G.H.A. (W.) at O h G.C.T. of date on which 
local transit is wanted is < Lo. W., use G.H.A. in calculation as of one date 

Rule II, for East Longitude: If G.H.A. (E.) at O h G.C.T. of date on which 
local transit is wanted is < Lo. E., use G.H.A. in calculation as of one date 

When neither rule applies, use G.H.A. in calculation as of date on which 
local transit is wanted. 

When L.H.A. (transit), G.H.A. = Lo. (always expressed as W. to 
360 ). 

Find G.H.A. of planet or star at O h G.C.T. of date ds found above and ex- 
press as W. or E. to 180. Find G.H.A. of sun or moon on date as found above 
at whatever hour the G.H.A. is closest under Lo. W. to 360. These are start- 
ing positions. 

Calculate the _ angular distance to be timed from body's starting position, 
west to its position at transit, adding an east to a west angle if necessary. 

The appropriate "Correction to G.H.A." table must now be used in each 
case (sun, moon, planet or star) to add up the total G.C.T. equal to this an- 
gular distance, using correct H. D. for moon or V. p. M. for planet. This gives 
G.C.T. of local transit^ 

Apply longitude in time, or zone description with reversed sign, to G.C.T. 
for L.C.T. or Z.T, of local transit. 

Transit 255 

Required: L. Tr. Denebola, June 3, 1939, Lo. 88 30' W. 
(G.H.A. (W.) 6/3 0* G.C.T. = 74 03'. 3 = <Lo. W.: Rule I) 

When L.H.A. = 0, G.H.A. = Lo. = 88 3(X W. 

At G.C/L 6/4 O h , G.HA = (-) 75 02 ; ,5 W. 

13 27'. 5 to go 
N.A. p. 214: O h 53 m represents (-) 13 17'. 2 

10'. 3 to go 
" 41 8 " -10'.3 

G.C.T. 6/4 O h 53 m 41 s = G. Time of L. Tr. 
Lo. W. (-) 5* 54 m 00 s 

L.C.T. 6/3 18 h 59 m 41 8 = L Time of L. Tr. 

Required: L, Tr. Betelgeux, Dec. 9, 1939, Lo. 90 E. 

(G.H.A. (W.) 12/9 O h G.C.T. = 348 51'. 
from 350 60'. 

G.H.A. (E.) = 11 09'. - <Lo. E.: Rule II) 

When L.H.A. = 0, G.H.A. = Lo. = (360 - 90) = 270 00'. W. 

At G.C.T. 12/8 0*, G.H.A. - 347 51 ; .9 W. 
from 359 60'. 

G.H.A. (E.) 1208M (+) 12 08'. IE. 

282 08M to go 
N.A. p. 216: 18 h 45 m represents (-) 282 01' .2 

6'. 9 to go 
27 fl .5 " {-)6'.9 

G C T 12/8 18 h 45 m 27 s . 5 = G. Time of L. Tr. 
Lo. E. (+) 6 h 00* 00 B ,0 

L.C.T. 12/9 Q h 45 m 27 8 .5 = L. Time of L. Tr. 

256 Transit 

Required: L. Tr. Canopus, Nov. 15, 1939, Lo. 24 14'. W. 

(G.H.A. (W.) 11/15 & G.C.T. - 317 31'. 2: No rule, use same date) 

When L.H.A. = 0, G.H.A. = Lo. 24 14' W 

At G.C.T. 11/15 0*, G.H.A. - 317 31' .2 W, 
from 359 60'. 

G.H.A. (E.) = 42 28' . 8 (+) 42 28' . 8 E. 

66 42' 8 to 20 
N.A. p. 214: 4 h 26* represents (-) 66 40'! 9 

G.C.T, 11/15 4 h 2&* 07 s . 5 = G. Time of L. Tr 

Lo. W. (-) l h 36^ 56 s . 4 

L.C.T. 11/15 2*49 m 11M L. Time of L. Tr. 

Required: L. Tr. Aldebaran, Dec. 20, 1939, Lo. 45 E. 

(G.H.A. (W.) 12/20 0* G.C.T. = 19 32'. 2: No rule, use same date) - 
When L.H.A. - 0, G.H.A. - Lo. - (360 - 45) - 315 00' W 

At G.C/T. 12/20 0*, G.H.A. = (_) 19 33'*. 2 W.' 

N.A. p. 216: 19* 38 m represents (-) 295 18'! 4 t0 g 

- 8'. 4 to go 

G.C.T. 12/20 19* 38 m 33 fl .5 - G. Time of L. Tr 
Lo. E. (-f) 3 h 00* 00 8 .0 

L.C.T. 12/20 22 h 38 m 33 s . 5 - L. Time of L. Tr. 

Transit 257 

Required: L. Tr. Saturn, June 1, 1939, Lo. 176 19'. 4 W. 

(G.H.A. (W.) 6/1 0* G.C.T. = 222 41'. 8: No rule, use same date) 

When L.H.A. = 0, G.HiA. Lo. = 176 19' 4 W 

At G.C.T. 6/1 O h , G.H.A. = 222 41'. 8 W 

from 359 60'. 

G.H.A. (E.) - 137 18'. 2 (+) 137 18' .2 E. 

(H.A.Va,p.M. = 15'.0369) ^IF^ to go 

N. A. p. 159 : 20 h represents (-) 300 44' . 4 

12 53'. 2 to go 
" 161 51 m (-) 12 46'. 8 

6'. 4 to go 
25*. 5 (-)6'.4 

G.C.T. 6/1 20 h 51 m 25 s . 5 = G. Time of L. Tr. 

Lo. W. (-) ll h 45 m 17 s . 6 

L.C.T. 6/1 Qh 06 m 07a 9 = 

Required: L. Tr. Moon, Jan. 28, 1939, Lo. 15 26'. 3 W. 

(G.H.A. (W.) 1/28 O h G.C.T. = 99 07'. 9: No rule, use same date) 

When L.H.A. = 0, G.H.A. = Lo. = 15 26'. 3 W. 

At G.C.T. 1/28 19 h G.H.A. = (-) 14 40' .2 W. 

46M to go 
(H.D. = 14 29'. 3) 
N.A. p. 133: 03 m represents (-)43'.4 

2' . 7 to go 

G.C.T. 1/28 19 h 03 m 11 B .0 = G. Time of L. Tr. 

Lo. W. (-) I h 01 m 45".2 

L.C.T. 1/28 18 h Ol m 25 fl .8 - L. Time of L. Tr. 

258 Transit 


Comparison of this method with that given in Button 
(7th. Ed., 1942, pp. 282-3) for Denebola shows the present 
procedure requires ten lines to Button's sixteen. My 
method for finding the date with which to enter the N. A. 
usually requires but one line (never more than three) 
whereas Button's requires seven. Bowditch (Revised Ed., 
1938) does not explain how to pick the entering date. 
Neither Button nor Bowditch presents a uniform method 
for more exact time of local transit of any body. 

Lack of space prevents adding rare sun problems where, 
near the 180th meridian, the longitude exceeds the sun's 
G.H.A. (W. or E.) at O h G.C.T. 

Moon problems, on days when the moon is known to 
miss any transit, come out with a time of transit for the 
following day. 

Selected Bibliography 

Published by the U. 5. Government Printing Office, Washington, D. C.: 
The American Nautical Almanac. Issued annually. 
The American Air Almanac. Each section covers four months. 
Published by the U. S. Hydrographic Office, Washington, D. C.: 

9. American Practical Navigator. Originally by Nathaniel Bowditch, 

LL.D. Revised Edition of 1938. 

66. Arctic Azimuth Tables. For declinations to 23 and latitudes 70-88. 
71. Azimuths of the Sun ("Red"). For declinations to 23 and latitudes 

to 70. 
120. Azimuths of Celestial Bodies ("Blue"). For declinations 24-70 and 

latitudes to 70. 
127. Star Identification Tables. For latitudes and altitudes to 88 and 

azimuths to 180. 
202. Noon-Interval Tables. 
205. Radio Navigational Aids. 

211. Dead Reckoning Altitude and Azimuth Table. Ageton. 
214. Tables of Computed Altitude and Azimuth. Vols. I-VIIL Each vol- 
ume covers 10 latitude and declinations to 74 30', with star identi- 
fication tables. 
2102-B. Star Identifier. 

3000. (OZ to 15Z.) Position Plotting Sheets. In 16 sheets covering latitudes 

Published by the U. 5. Naval Institute, Annapolis, Maryland: 

Navigation and Nautical Astronomy. Capt. Benjamin Button. 7th Edition. 


Mathematics for Navigators and Manual for Self-Study of Button's Naviga- 
tion and Nautical Astronomy. B. M. Dodson and D. Hyatt. 1940. 

Nautical Astronomy. An Introduction to the Study of Navigation. 1940. 

U. S. Naval Institute Proceedings. Published monthly. 

Other Publications: 

Wrinkles in Practical Navigation. S. T. S. Lecky. 22nd Edition. George 
Philip & Son, Ltd. London. 1937. 


260 Selected Bibliography 

Cugle's Two-Minute Azimuths. For decimations to 23 and latitudes to 65. 
Charles H. Cugle. In 2 volumes. E. P. Button & Co., Inc., New York, 1936. 

How to Navigate Today. M. R. Hart. Cornell Maritime Press. New York, 1943. 

Navigation for Mariners and Aviators. David Polowe. 2nd Edition. Cornell 
Maritime Press. New York. 1942- 

Piloting, Seamanship and Small Boat Handling. Charles F. Chapman. Motor 
Boating. New York. 1943. 

The Ensign. Official Publication of the U. S. Power Squadrons. Secretary- 
Robert A. Busch, 155 Worth St., New York, N. Y. 


Abbreviations, 209-211 
Altitude, 55 
angular, 61 
apparent, 56 
corrections for, 56-60 
of a celestial body, 63 
parallax in, 59 
sextant, 55 
true, 55 
Altitude azimuth, 231 

form for, 217 
Altitude difference, 129 

hour, 32, 36 
Angle of reflection, 63 

rules for, 206 
Aphelion, 6 

Apparent magnitude, 6 
Apparent noon, 106 
AQUINO'S fix, 144-145,240 

form for, 218 
Arc scale, 63-64 

first point of, 10, 13,34 
Artificial horizon, 187-188 
Assumed vs. D. R. position, 134-135 
Astrolabe, 67 

Astronomical triangle, 79-81 
finding parts of, 80 
in polar regions, 151, 153 
Augmentation, 59 
Azimuth, 117421, 192-193 
altitude, 119, 121 
compass, 122 

Azimuth (Cont.) 

formulas for, 119 

means for finding, 117-121 

of Polaris, 120-121 

plotting of, 193-194 

time, 119, 121, 190 
Azimuth circle, 77 


transit of, 25 
Book problems 

reference rules for, 205-206 


tabular summary, 157-158 
Canopus, 23 

Celestial co-ordinator, 81 
Celestial equator, 10 
Celestial globe, 16 
Celestial horizon, 55 
Celestial navigation 

equipment, 51 
Celestial sphere, 9, 33 

declination, 14 

ecliptic, 32 

equinoctial, 32 

hour circle, 32 
Celestial triangle, 80 

gnomonic, 170 

great circle, 170 

ocean, 183 

polar, 171 
Chronometer, 48-49, 181 

historical, 49 



Chronometer error, 114, 116 

great, 32 

hour, 32 

small, 32 

of equal altitude, 123-125 
Compass, 69-78 

conversion, 73-74 

correction, 73 

deviation, 72 

dumb, 78 

error, 70-76 

gyro, 77 

heeling error, 72 

historical, 69 

local magnetic disturbance, 72 

magnetic, 69 

marking of card, 70 

methods of steering, 76 

polar regions, 151 

table of points, 78 

variation, 70 
Compass card, 70-74 
Compass error, 74, 117, 122, 180 

deviation, 72 

methods of determining, 74-76 
Computed point, 133 
Correction tables, 53-54 
Cosine-haversine formula, 130 
Course, 161 
Current, 161, 174 

drift of, 174 

problems, 175-178 

real ocean, 174 

set of, 174 


calculation of, 207-208 
change of, 43-46 
determination of, 42 

Greenwich, 42 

international line, 44 

civil, 29 

complete rotation, 28 

fractional, 27 

measurement by sun noons, 24 

sidereal, 28 

star, 12 

sun, 12 
Day's current 

form for, 223 
Day's run 

form for, 222 
Day's work, 180-181 

compass error, 180 

dead reckoning, 180 

fixes, 180 

radio time signals, 181 

reports, 181 

sun observations, 180 
Dead reckoning, 102, 161, 173, 180 

form for, 220 
Declination, 8-9 

northern, 14 

southern, 14 

star, 52 

sun, 52 
Departure, 162 

taking a, 162 
Dip, 55 
Distance, 161 


aphelion of, 22 
movement of, 11-12 
orbit of, 19, 22-23 
perihelion of, 22 
speed of, 24 



Ecliptic, 10 

obliquity of, 10 

definition of, 5 

eccentricity of, 22 

foci of, 5 
Equinoctial, 10 
Equinox, 9, 26 

apparent motion of, 26 

autumnal, 9 

fall, 14 

precession diagram, 20 

precession of, 19-22, 26 

vernal, 9, 14, 21 

westward shift, 14 

navigational, 182 

Fix, 180 

AQUINO'S, 144-145 

noon, 142-143 

on equator at equinox, 143-144 

on equator not at equinox, 144 

running, 132 

special, 141-147 

star, 180 

star altitude curves, 145-147 

within equal altitudes, 142 

zenith with Sumner arcs, 141 
Fixed stars, 6 
Forms, 212 

altitude azimuth, 217 

day's current, 223 

day's run, 222 

dead reckoning, 220 

latitude, 215 

line of position, 214, 218 

longitude, 216 

mercator sailing, 221 

middle latitude sailings, 219 

Great circle, 31 

Great circle sailing, 169-170 

Greenwich civil time 

calculations of, 207-208 
Globe map, 168 
Gnomonic charts, 170 
Gnomonic projection, 171 

H. O. 211, 136 
H. 0.2/4,137-140 
Heavenly bodies, 7 

artificial, 66, 187-188 

celestial, 55 

dip of, 56 
Hour angle, 36 

calculations for position, 36 

Greenwich, 32 

local, 33 
Hour circles, 33 

Identification, 155-156, 246 
sidereal time method, 156 

Index correction, 55 

Intercept, 129 

International date line, 44 

difference from 180th meridian, 

Interval to noon, 106 

KEPLER'S Laws, 23 

Latitude, 7, 103-113 

by meridian altitude, 225-228 

by noon sun, 105 

by phi prime, phi second, 110- 

112, 229 

by Polaris, 109-110 
calculations of, 103, 191 
difference of, 161 
form for, 215 



Latitude (Cont.) 

interval to noon, 106 

measuring transits, 109 

meridian altitude, 104-105 

meridian sight, 108 

parallel, 9 

reduction to meridian, 108 

TODD'S method, 105 

WEEMS' method, 107 
Line of position, 99, 126-127 

calculating, 193 

chord method, 126 

forms for H. O. 211, 214, 218 

from moon, 236 

from planet, 237-238 

from star, 239-240 

from sun, 234-235 

SAINT-HILAIRE method, 127 

tangent method, 127 
Logarithms, 92-96 
characteristic, 94 
examples of, 96 
mantissa, 94 
natural, 92-93 
Longitude, 7, 114-115 

by time sight, 114, 230 
chronometer error, 116 
difference of, 161 
form for, 216 
time relationships, 38 
time sight, 115 
Lubber's line, 78 
Lunar distances, 48 


apparent, 6 
Mercator chart, 100, 151, 168, 171-172 

in polar regions, 151 
Mercator sailing, 167 
Meridian, 7 

crossing the 180th, 47 

Meridian (Con.) 

opposite-the-sun, 45 

reduction to, 108 

time frame, 45 
Meridian angle, 32 
Meridian sight, 108 
Meridional parts, 168 

motion of, 16-19 

observation of, 189 

phases of, 16-19 

plane of orbit, 18 

Napier diagram, 75-76 
NAPIER'S rules, 87 
Nautical Almanac f 6, 51-54 

correction table, 53-54 

tables in, 52 

uses, 51 

Nautical mile, 99-100 

abbreviations, 209-211 

form of procedure, 212 

practical points, 183-195 

problems, 224-248 

stars and planets, 196-204 
Navigational stars 

declinations of, 17 

apparent, 106 

Greenwich civil, 43 

interval, 106 
North Pole 

conditions at, 148-150 
Notices to mariners, 183 


sun, 180 

diagram and discussion of, 13-16 



Parallax, 55, 57-59 

geocentric, 57 

horizontal, 59 

in altitude, 59 
Pelorus, 78 
Perihelion, 6 
Planets, 196 

locating, 202-204 

observation of, 189 

orbits of, 22 
Plotting, 193-195 

table of, 78 
Polaris, 14 

estimated, 174 

from moon, 236 

from planet, 237-238 

from star, 239-240 

from sun, 234-235 

line of, 214, 218, 234-239 
Polar chart, 171 

polar great circle chart, 151 
Polar position finding, 148-154 

correction position lines, 152 
Polar regions 

astronomical triangle, 151 

compass behavior in, 151 
Position finding, 99 

in polar regions, 151-154 

north pole, 148-150 

polar, 148 

standard of accuracy, 183 
WEEMS' polar system, 153 
Position plotting sheets, 101 
Problems, 224-248 

altitude azimuth, 231 
AQUINO'S fix, 240 
azimuth, 232-233 
civil to sidereal time, 249-251 
current, 175-178 

Problems (Con t.) 

fix from two stars, 241-245 
identification, 246 
latitude, 225-228 
latitude by phi prime, phi sec- 
ond, 229 

line of position, 234-239 
longitude, 230 

sidereal to civil time, 251-253 
time conversion, 249-253 
time of sun, 247-248 
transit, 254-258 

Quadrant, 67 


angle of, 63 
Refraction, 55, 56-58 
Right ascension, 10-11, 13-14 
Rhumb line, 101, 161 

Sailings, 161, 162-166 
composite, 172 
great circle, 169-170 
mercator, 167-169 
form for, 221 
middle latitude, 166-167 

form for, 219 
parallel, 165 
plane, 162-163 
spherical, 164-166 
traverse problem, 164 
SAINT-HILAIRE method, 128-131 
computed point, 133 
fix, 132, 133 
procedure, 129-130 

arc, 63-64 
vernier, 64 
Semidiameter, 55, 59-60 



Sextant, 61-68, 184-187 

adjustments, 65 

artificial horizon, 66 

bubble, 67 

care of, 185-186 

description of, 61-64 

designation, 65 

historical, 67 

index correction, 65 

index error, 65, 186-187 

index mirror, 63 

latitude, 191-192 

moon observation, 189 

Navy type, 66 

planet observation, 189 

reading of, 63-66 

star observation, 190 

sun observation, 188 
Sextant angles, 62 
Sextant vernier, 63 
Short-cut systems, 134-140 

assumed position, 134-135 

D. R. position, 135 

great circle of, 31 

small circle, 32 
Solar system, 4 

historical, 3, 5 

speed of light, 23 
Solstice, 9 

summer, 14 

winter, 14 
Star day, 12 
Stars, 196 

coordinates, 26 

declination, 52 

declination of navigational, 17 

fixed, 6 

identification, 155, 246 

locating, 197-202 

navigational, 14 

Stars (Cont.) 

north, 21 

observation of, 190 

right ascensions of, 21 

timing sights of, 190 

taking sights with, 42 
Sumner lines of position, 123-127 

declination, 52 

observations of, 188-189 
Sun day 

true, 12 
Sun noons, 24 

Tabular summary, 157-158 
Terrestrial triangle, 80 

apparent, 24-25, 29, 37 

calculations of t, 36 

civil, 30-31, 37 

civil units, 29 

conversion, 249-253 

civil to apparent, 52 
civil to sidereal, 249-251 

equation of, 25 

finding apparent, 37-38 

Greenwich civil, 25 

Greenwich mean, 25 

kinds of, table, 30 

local apparent, 37 

local civil, 25 

longitude relationships, 38 

mean, 25 

minus zone, 39 

observer's watch, 41 

plus zone, 39 

radio signals, 181 

relations between zone and civil,. 

relationships, 38 



Time (Cont.) 
rules for, 206 
sidereal, 30-31, 35, 52 

and practical navigation, 38 
or star, 26 

conversion to civil, 251-253 
units, 29 
solar, 25 
standard, 39 
zero zone, 39 
zone, 39-41 

of mean sun noons, 40 
Time difference, 36 
Time frame, 44 
Time zones, 39-40 
TODD'S method, 105 
Transit, 25 

problems, 254-258 
Traverse, 164 

astronomical, 79-81 
celestial, 80 
oblique spherical, 90 
right spherical, 87 
spherical, 85 
terrestrial, 80 
Trigonometry, 82-91 
NAPIER'S rules, 87 
plane, 82, 85 
spherical, 85 

Trigonometric functions, 83-84, 86- 

Vernal equinox, 9 
Vernier, 63 

rule for reading, 64 

scale, 64 
Vertex, 170 
Visible stars 

northern declination, 15 

southern declination, 15 


second-setting, 49 


method, 107 
plotting charts, 173 
polar position finding, 153 
Universal Plotting Charts, 194 

Work reports, 181 


leap, 28 
sidereal, 27 
solar, 27 

Zenith distance, 125 
Zero zone, 39 

signs of, 14 
Zone time, 39-40 





By Capt. David Polowe 

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