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TABLES

INTENDED TO FACILITATE THE OPERATIONS OF

NAVIGATION

AND

NAUTICAL ASTRONOMY,

AS AN ACCOMPANIMENT TO THE

rAVIGATION AND NAUTICAL ASTRONOMY,

VOLS. 99 AND 100 OF THE RUDIMENTARY SERIES.

BY J. R. YOUNG,

FORMERLY PROFESSOR OF MATHEMATICS JN BELFAST COLLEGE.

LONDOJST:

JOHN WEALE, 59, HIGH HOLBORN.

1859.

V

LONDON :

BRADBURY AND EVANS, PRINTERS, WHITEFRIARS

TABLES

FOB

NAVIGATION AND NAUTICAL ASTRONOMY.

EXPLANATION OP THE TABLES.

THE following Tables are intended to be used conjointly

with the " Mathematical Tables " published in the series

of Rudimentary Treatises in the various computations of

Navigation and Nautical Astronomy. They are eleven in

number, and the purposes they serve will be readily under-

stood from the following brief explanation ;

TABLES 1 AND 2.

Sines, Cosines, Sfc., to every Quarter Point of the Compass*

The first of these Tables exhibits the Natwral sines, cosines,

<fcc., of Courses to every quarter-point of the compass, and

the second furnishes the; Logarithmic sines, cosines, &c. of

the same angles. Here are two examples of their use :

1. A ship from latitude 37 3' N., sails S.W. by S. S. a

distance of 148 miles; required the latitude in and the

departure made ?

EXPLANATION OF THE TABLES.

EXPLANATION OF THE TABLES. Vll

These Tables are employed not only in plane sailing, but

also in parallel and mid-latitude sailings, as is sufficiently

exemplified in the treatise on Navigation and Nautical

Astronomy, to which the present collection is adapted.

And in all computations of the parts of a right-angled

triangle, provided the angles are expressed in degrees and

minutes seconds being disregarded, Table 5 may be used

to save the trouble of arithmetical calculation.

TABLE 6.

Natural Cosines to Degrees, Minutes, and Seconds.

This Table is employed in the Author's method of clearing

the lunar distance for the purpose of finding the longitude

at sea. The several columns of cosines are headed by the

degrees, the accompanying minutes being inserted in the

first column on the left of the page : this is equally a

column of the seconds, and is- accordingly headed by the

marks for minutes and seconds. As in the' ordinary trigo-

nometrical tables, the cosine of ag. arc or angle belonging

to any number of degrees and minutes is found in the

column of cosines, under the degrees, and in a horizontal

line with the minutes found in the first column.

Suppose this cosine to have been extracted from the

table ; then, if there are seconds also in the arc or angle,

we again refer to the same first column for these, and in the

same horizontal line with them, and in the column headed

" parts for " " which immediately follows the column from

which our cosine has been extracted, we shall find the

correction for that cosine : this correction is always to be

subtracted. The remainder will be the cosine of the given

degrees, minutes, and seconds. But in taking out a cosine

to degrees, minutes, and seconds, it will in general be

Vlll EXPLANATION OF THE TABLES.

better to enter the marginal column first with the seconds,

to write the " parts " for these on a slip of paper, and then,

entering the same column with the minutes, instead of

extracting the corresponding cosine, to place the slip under

it, and subtract the correction written thereon. The Table

extends from to 90 only, so that it does not give

immediately the cosines of obtuse angles r when therefore

the angle is obtuse, we must enter the Table with the

supplement of that angle, and regard the corresponding

cosine as negative. It was thought better that this trifling

amount of trouble should be incurred, than that the extent

of the Table should be doubled.

There is indeed a way of avoiding this enlargement of

the Table, and yet providing for the supplementary arcs ;

but as the Table would then have to be used in a peculiar

manner disturbing the general principle upon which the

extracts from it are made in the other cases it was thought

preferable, after due consideration, to reject it. The plan

is this. Suppose a marginal column added to the right of

each page, for the minut^ and seconds, proceeding upwards

from to 60, and that the degrees supplementary to those

at the top of the page (one degree in each case being

omitted in the supplement) were given at the bottom, as

in the ordinary tables of sines and cosines ; we should then

have to use the table as in the following instance. B/equired

the cosine of 115 41' 34" ? Eeferring to page 96, we

should find 115 at the bottom of the column headed 64,

from which column opposite the 41' on the right, we should

take the cosine 43397, then referring to the left for the 34",

we should extract the " parts ' ' 149, which we should have

to add to 43397: we should thus get 433546 for the

required cosine. As the Table at present stands, however,

we enter it with 64 18' 26", the supplement of 115 41' 34",

EXPLANATION OF THE TABLES. IX

and for 64 18' we find 433659, while the "parts'* for 26"

are found in a similar manner to be 113, which subtracted

from 433659 gives 433546 for the required cosine.

It has not been thought necessary to insert the decimal

point before each cosine ; indeed, in the operation for which

this Table is specially prepared, the numbers may always

be regarded as integers. (See p. 227, Naut. Ast.)

Ex. 1. Suppose the natural cosine of 37 21' 33" were

required :

Turning to the page containing 37 at the top (page 92),

we find the "parts " against 33" to be 98, and the cosine

against 21' to be 794944, subtracting the 98 from this,

we write down 794846 for the cosine required.

2. Again, suppose we wanted the cosine of 118 16' 43" :

Subtracting this from 180, the angle in the Table is

61 43' 17". Under 61, and against 17", the "parts" are

72, and against 43', the cosine is 473832 : subtracting the

72 from this, we find 473760 for the required cosine, which

is negative because the proposed angle is obtuse.

3. Eequired the angle whose cosine is 452801 ?

By the Table 452954 = cos 63 4'

Given cosine 452801

Parts for the sees. 153 35"

Hence the angle is 63 4' 35".

If given the cosine had been 452801, then the supple-

ment of the angle thus determined, namely 116 55' 25",

would have been the angle to which that cosine belongs.

X EXPLANATION OF THE TABLES.

TABLE 7.

Proportional Logarithms.

\

These are a peculiar kind of logarithms, first constructed

by Dr. Maskelyne to facilitate the operation of finding the

Greenwich time, at which a lunar distance taken at sea has

place. They are also useful in many other inquiries, in

which difference of time varies, as difference of angular

measurement. When difference of time is required the

Table is to be entered with difference of angular measure-

ment, and when difference of angular measurement is

required it is to be entered with the corresponding difference

of time. Sufiicient illustration of the mode of employing

the Table is given in the Nautical Astronomy, pages

236-7.

The last four figures in the Table are decimals, and the

greater part of the Table consists of these decimals alone ;

the decimal point however is suppressed, as well here as in

the Nautical Almanac, since in finding the Greenwich time

of a lunar distance the logarithms may be always regarded

as whole numbers.

1. Suppose the proportional logarithm of 2 h 8 m 16 s is

required, or the proportional logarithm of 2 8' 16": turning

to the proper page of the Table (p. 112) we find that for

each of these arguments the P. L. is -1472.

2. Suppose it be required to find the difference of time

corresponding to the P. L. '2954 : turning to page 109 we

see that this P. L. answers to the difference of time, l h 31 m

10 s . It also answers to the difference 1 31' 10" of angular

measurement. The Table extends from s to 3 h or 10800 s ,

or from 0" to 3 : the proportional logarithm of the extreme

number of seconds, namely 10800, being 0, the formal

insertion of it has iiot been thought necessary. For the

EXPLANATION OF THE TABLES. Xt

theory of proportional logarithms reference may be made to

the Nautical Astronomy, page 235.

TABLE 8.

For determining the distance of an object seen in the horizon.

This Table shows the utmost distance at which an object

on the surface of the sea can be seen by an eye elevated

above it ; the elevation of the eye being estimated in feet,

and the distance in nautical miles, allowance being made

for atmospherical refraction. If the object itself be elevated

above the surface, and its summit be just visible in the

remote distance, then, if the height of the distant object

thus lost to sight all but its top, be previously known, the

Table will enable us to find its distance, the height of the

spectator's eye being known.

Ex. 1. From the mast head, 130 feet high, a boat was

observed as a remote speck, just appearing in the horizon :

required its distance ?

In the Table opposite 130 feet is found 13'1 miles, the

distance required.

2. From the same height the top of a lighthouse known

to be 300 feet above the level of the sea, was discerned in

the horizon : required the distance from the lighthouse ?

For the height 130 the distance is 13 '1 miles.

300 19-9

.'. the distance of the lighthouse is 33 miles.

Xll EXPLANATION OF THE TABLES.

TABLE 9.

For finding the mean time (nearly) of the Meridian Transits

of the Principal Fixed Stars.

In this Table is recorded the mean time at which each of

the 100 stars there selected passes the meridian of the ship.

The times of transit are given only for every tenth day ;

but as the stars come to the meridian earlier every day, by

a uniform difference of time about four minutes it is

easy to find the time of transit on any intermediate day :

we have only to multiply the number of days after the day

of transit recorded in the Table by 4, and to subtract the

number of minutes in the product from the time of transit

on the day given in the Table. Or we may multiply the

number of days lefore the next tabular day by 4, and add

the resulting minutes to the time of that advanced day's

transit : for example,

Suppose it were required to find the mean time of transit

of o 2 Centauri on the fifth of November :

By the Table the time of transit Nov. 1, is 23 h 43 m

And for four days afterwards, we subtract 4x4= 16

.'. time of transit Nov. 5th is 23 27

Or, the time of transit Nov. 11, being by the Table 23 h 4 m ,

by adding 6 x 4 = 24 minutes, we have for the time on Nov.

5th, 23 h 28 m .

It will of course be understood that the times of transit

furnished by aid of this Table, are only the times nearly ;

but in no case will the time differ from the truth by more

*

than about two or three minutes, and the Table is therefore as

accurate as necessary for the purpose intended by it, which

EXPLANATION OF THE TABLES. Xlll

purpose is to apprise the mariner about what time he may

expect certain well known stars to appear on the meridian

whenever the weather permits his taking a star-altitude for

his latitude. Should the observer not be sufficiently ac-

quainted with the stars to avail himself readily of this

information, he is recommended to procure Mr. Jeans's

" Hand Book for the Stars."

But the right star may generally be detected when we

know, within about half a degree or so, what altitude it

ought to have when on the meridian, and this approximate

altitude may be found by help of the star's declination, and

the latitude by account ; thus :

1. FOB A MEBIDIAtf ALTITUDE ABOVE THE POLE.

In this case the star passes from the eastward towards

the westward, and ascends to the meridian.

When the latitude by account and the declination have the

same name. Add 90 to the declination, and subtract the

latitude by account ; or, which is the same thing, add the

colatitude to the declination, the result will be the ap-

proximate altitude, measured from the S. in IS", lat. and

from the N. in S. lat.

When the latitude by account and the declination have

different names. Add together the latitude by account, and

the declination, and subtract the sum from 90, or, which is

the same thing, subtract the declination from the colatitude,

the result will be the approximate altitude from the S. in

. lat. and from the N. in S. lat. If the sum of lat. and

dec. exceed 90 the star cannot appear above the horizon.

2. FOB A MEBIDIAN ALTITUDE BELOW THE PoLE.

In this case the star passes from the westward towards

the eastward and descends to the meridian. It can be visible

XIV EXPLANATION OF THE TABLES.

below the Pole only when the latitude and declination have

the same name.

From the sum of the latitude and declination subtract

90 : the remainder will be the approximate altitude

reckoned from the N. in N. lat. and from the S. in S. lat.

If the sum of lat. and dec. be less than 90 the star will

pass the meridian below the Pole under the horizon.

To assist in thus getting an approximate meridian alti-

tude, the stars' declinations each to the nearest degree

and minnte are given in the marginal column of the Table.

It is scarcely possible to mistake the star, because no other

will have nearly the same meridian altitude at the time.

The approximate altitude being found in this way, and

the index of the instrument set to it the sight being

directed to the proper point of the horizon the true

meridian altitude, and thence the latitude of the ship, may

be readily determined.

It is to be observed that if the mean time at ship be

A.M., we must add 12 h to that time for the corresponding

time in the Table, from the preceding noon, when the star

opposite that time will pass the meridian above the Pole.

If the mean time at ship be P.M. the star opposite that time

in the Table, will be on the meridian ahove the Pole ; and if

12 h be added to the time, the star opposite the result, will

be on the meridian below the Pole.

TABLE 10.

Best Time for talcing the Altitude of a Celestial Object, icith

the view of determining the TIME at Sea.

"When the time at the ship is to be deduced from an

jiltitude, it is desirable that the object observed should be

in such a position that a small error in the observation may

EXPLANATION OF THE TABLES. XVT

have the least possible influence on the magnitude of the

hour-angle. And this position is attained when the object

is on the prime vertical. If, however, the declination be of

a contrary name to the latitude of the place of observation,

the object will cross the prime vertical before it rises and

.after it sets, so that it cannot be observed on this circle at

all ; the observation should then be taken as soon after it

rises (or before it sets) as that the altitude of it is sufficient

to secure it from the fluctuating effects of the horizontal

refraction. The altitude should not be less than 6 or 7

ves.

The present Table points out, with accuracy enough for

the purpose, the time when the prime vertical is crossed

above the horizon, that is, when the declination is of the

same name as the latitude.

If the object observed be the sun, the table is to be

entered with that degree of declination which approaches

nearest to the sun's declination at the time, and which is

found at the top of the table; underneath this declination,

and opposite to the latitude found in the first column, is

the time before or after noon when it will be most advan-

tageous to take the altitude for TIME.*

If, however, the object be a star, we must first ascertain

the time when it passes over the meridian ; the preceding

Table will supply this information. Then, by aid of the

present Table, entering it with the star's declination and

the latitude of the place, we take out the corresponding

time, or hour-angle between the meridian and star, when

the latter is on the prime vertical. If the observation is to

* As the Table gives the hour-angle from the meridian in Time, if the

object be the sun, the time will be apparent ; the correction for Equation

of Time being applied, will convert it into mean time. For a star, no

such correction is requisite.

XVI EXPLANATION OP THE TABLES.

be made before the meridian transit, we must subtract the

latter time from the former ; if the observation is to be

made after the meridian transit, we must add the two

times together.

It is only when the latitude of the place is greater than

the declination of the object (both being of the same name),

that the object actually crosses the prime vertical; when

the two are equal, the object is on the prime vertical when

it is on the meridian ; when the latitude is less than the

declination, the Table shows the time of nearest approach to

the prime vertical, which will be the best for the altitude

to be observed.

"We may further observe that, when the latitude of the

ship is pretty nearly a mean between two consecutive

latitudes in the Table, the time will be obtained more

accurately by taking the mean of the times corresponding

to those two latitudes.

Ex. 1. At what time will the star a Leonis or Eegulus

bear due East on the 6th of February, in latitude 47 N. ?

Mean time of transit, Feb. 1 20 m (Table 9) 12 h 53 m

Time corresponding to dec. 12 N. and lat. 47 5 14

.

Mean time of star's bearing due East 7 39 P.M.

2. At what time will the star a Arietis bear nearest to

the West on Nov. 21, in latitude 17 32' N. ?

Time corresponding to dec. 22 and lat. 17 2 h 43 m

lat. 18 2 26

2) 5 9

lat. 17 30', + 2 35

Mean time of star's transit (Table 9) 9 56

,, ,, bearing nearest to W.

or

that is, at about 31 ra past midnight.

EXPLANATION OF THE TABLES. XVli

NOTE. The mean time of the meridian passage of each

of the planets, and also of the moon, is given in the

Nautical Almanac for every day in the year. (See Table at

p. 120 of the Nautical Astronomy.)

If a planet be favorable for observation, and there be any

doubt as to which of the planets it is, the doubt may be

removed by noticing what known star is nearest to it,

referring to Table 9 for the time of the meridian passage of

that star, and then finding, from the Nautical Almanac,

which of the planets it is that passes the meridian nearest

to that time.

TABLE 11.

For Finding the Altitude of a Celestial Object, most suitable

for ascertaining the Time at Sea.

This Table is intended to show, what altitude nearly an

object of given declination must have in a given latitude to

be most suitable for deducing the time from that altitude ;

that is to say, it points out approximately the altitude

which the object has when on the prime vertical, or when it

makes the nearest approach to it. "When the object is a

star, and the time most suitable for taking its altitude is

found from the last Table, the approximate altitude of it at

that time, as given by the present Table, will enable the

observer readily to discover it, even should he be but little

familiar with the constellations.

To these Tables is added a list of the Proper Names of

certain of the principal fixed stars, and to this is subjoined

the names and sounds of the letters of the Greek Alphabet.

*#* At the commencement of the foregoing Explanation,

reference is made to Law's " Mathematical Tables," as an

accompaniment to the present volume. The first eighteen

of these Tables, together with the eleven in the present

collection, will be found to contain all that is indispensably

necessary in the several operations of , Navigation and

Nautical Astronomy.

CONTENTS.

TABLE I.

PAGE

^Natural Sine?, Cosines, &c., to every Quarter Poiiifc of the

Compass 1 .

TABLE II.

Logarithmic Sines, Cosines, &c., to every Quarter Point of the

Compass 1

TABLE III.

Xatural Sines, Cosines, &c., to every Degree and Minute of the

Quadrant 2

TABLE IV.

Traverse Table for Points and Quarter Points

TABLE V.

Traverse Table for Degrees 41

TABLE VI.

Natural Cosines to Degrees, Minutes, and Seconds

TABLE VII.

Proportional Logarithms 101

XX CONTENTS.

TABLE VIII.

PAOE

For determining the Distance of an Object seen in the Horizon .117

TABLE IX.

For finding the Mean Time nearly when a specified Fixed Star

passes the Meridian 118

.

TABLE X.

For finding the most suitable Time for taking the Altitude of a

Celestial Object, for the purpose of determining the TIME

at Sea . . . 130

TABLE XL

For finding what Altitude of a Celestial Object is the most

suitable for deducing from that Altitude the TIME at Sea . 132

,

Proper Names of Certain of the Principal Fixed Stars . . .134

The Greek Alphabet . . . . . . . . .134

Note on using the Blank Forms for the different Computations of

Nautical Astronomy 135

TABLE 1.

NATURAL SINES AND TANGENTS TO EVERY QUARTER

POINT OF THE COMPASS.

POINTS.

SINE.

COSINE.

TANGENT.

COTANGENT.

!

o-ooooo

04907

09802

14730

1-00000

99880

99518

98918

o-ooooo

04913

09849

14834

Infinite

2035560

10-15319

674146

8

7|

7*

7J

1

1|

19509

24298

29028

33689

98079

97003

95694

94154

19891

25049

30335

35781

6-02734

399222

3-29483

279481

7

2

2J

U

s]

38268

42756

47140

51410

92388

90399

88192

85773

41421

47296

53451

59938

2-41421

2-11432

1-87087

1*66840

6

*. CO CO CO CO

55556

59570

63439

67156

70711

83147

80321

77301

74095

70711

66818

74165

82068

90635

1-00000

1-49661

134834

1-21850

1-10333

1-00000

5

3

4

COSINE.

SINE.

COTANGENT.

TANGENT.

POINTS.

TABLE 2.

LOGARITHMIC SINES AND TANGENTS TO EVERY QUARTER

POINT OF THE COMPASS.

POINTS.

SINE.

COSINE.

TANGENT.

COTANGENT.

o-ooooo

869079

8-99130

916652

10-00000

9-99948

9-99790

9-99527

o-ooooo

869132

8.99340

9-17125

Infinite

11-30868

11-00660

10-82875

8

7|

7|

7*

H

ll

9-29024

9-38557

9-46282

9-52749

9-99157

9-98679

9-98088

9-97384

9-29866

9-39878

9-48194

955365

1070134

10-60121

10-51806

10-44635

7

61

6*

9

2

2i

2

2*

959284

9-63099

9 67339

9-71105

9-96562

9-95616

9-94543

993335

9-61722

967483

972795

9-77770

10-38-278

10-32517

10-27204

10-22230

6

5*

3

4

9-74474

9-7750&

9-80236

9-82708

9-84949

9-91985

990483

9-88818

i 9-86979

9-84949

9-82489

9-87020

991417

995729

10-00000

1017511

10-X2980

10-08583

10-04270

1000000

5

41

4}

n

4

COSINE.

BINE.

COTANGENT. -

TANGENT.

POINTS.

3.

2 NATURAL 3I'NT5. [TABLE 3.

3

4

5

6

7

/

00 00

017 45

034 90

052 34

069 76

087 16

104 53

121 87

60

1

29

74

035 19

63

070 05

45

82

122 16

59

2

58

018 03

48

92

34

74

105 11

45

58

3

87

32

77

053 21

63

088 03

40

74

57

4

001 16

62

036 06

50

92

31

69

123 02

56

5

45

91

35

79

071 21

60

97

31

55

6

75

019 20

64

054 08

50

89

106 26

60

54

7

002 01

49

93

37

79

089 18

55

89

53

8

33

78

037 23

66

072 08

47

84

124 18

52

9

62

020 07

52

95

37

76

107 13

47

51

10

91

36

81

055 24

66

090 05

42

76

50

11

003 20

65

038 10

53

95

34

71

125 04

49

12

49

94

39

82

073 24

63

108 00

33

48

13

78

021 23

68

056 11

53

92

29

62

47

14

004 07

52

97

40

82

091 21

58

91

46

15

36

81

039 26

69

074 11

50

87

126 20

45

16

65

022 11

55

98

40

79

109 16

49

44

17

95

40

84

057 27

69

092 08

45

78

43

18

005 24

69

040 13

56

98

37

73

127 C6

42

19

53

98

42

85

075 27

66

110 02

35

41

20

82

023 27

71

058 14

' 56

95

31

64

40

21

006 11

56

041 00

44

85

093 24

60

93

39

22

40

85

29

73

076 14

53

89

128 22

38

23

69

024 14

59

059 02

43

82

111 18

51

37

24

98

43

88

31

72

094 11

47

80

36

25

007 27

72

042 17

60

077 01

40

76

129 08

35

26

56

025 01

46

89

30

69

112 05

37

34

27

85

30

75

060 18

59

98

34

66

33

28

008 14

60

043 04

47

88

095 27

63

95

32

29

44

89

33

76

078 17

INTENDED TO FACILITATE THE OPERATIONS OF

NAVIGATION

AND

NAUTICAL ASTRONOMY,

AS AN ACCOMPANIMENT TO THE

rAVIGATION AND NAUTICAL ASTRONOMY,

VOLS. 99 AND 100 OF THE RUDIMENTARY SERIES.

BY J. R. YOUNG,

FORMERLY PROFESSOR OF MATHEMATICS JN BELFAST COLLEGE.

LONDOJST:

JOHN WEALE, 59, HIGH HOLBORN.

1859.

V

LONDON :

BRADBURY AND EVANS, PRINTERS, WHITEFRIARS

TABLES

FOB

NAVIGATION AND NAUTICAL ASTRONOMY.

EXPLANATION OP THE TABLES.

THE following Tables are intended to be used conjointly

with the " Mathematical Tables " published in the series

of Rudimentary Treatises in the various computations of

Navigation and Nautical Astronomy. They are eleven in

number, and the purposes they serve will be readily under-

stood from the following brief explanation ;

TABLES 1 AND 2.

Sines, Cosines, Sfc., to every Quarter Point of the Compass*

The first of these Tables exhibits the Natwral sines, cosines,

<fcc., of Courses to every quarter-point of the compass, and

the second furnishes the; Logarithmic sines, cosines, &c. of

the same angles. Here are two examples of their use :

1. A ship from latitude 37 3' N., sails S.W. by S. S. a

distance of 148 miles; required the latitude in and the

departure made ?

EXPLANATION OF THE TABLES.

EXPLANATION OF THE TABLES. Vll

These Tables are employed not only in plane sailing, but

also in parallel and mid-latitude sailings, as is sufficiently

exemplified in the treatise on Navigation and Nautical

Astronomy, to which the present collection is adapted.

And in all computations of the parts of a right-angled

triangle, provided the angles are expressed in degrees and

minutes seconds being disregarded, Table 5 may be used

to save the trouble of arithmetical calculation.

TABLE 6.

Natural Cosines to Degrees, Minutes, and Seconds.

This Table is employed in the Author's method of clearing

the lunar distance for the purpose of finding the longitude

at sea. The several columns of cosines are headed by the

degrees, the accompanying minutes being inserted in the

first column on the left of the page : this is equally a

column of the seconds, and is- accordingly headed by the

marks for minutes and seconds. As in the' ordinary trigo-

nometrical tables, the cosine of ag. arc or angle belonging

to any number of degrees and minutes is found in the

column of cosines, under the degrees, and in a horizontal

line with the minutes found in the first column.

Suppose this cosine to have been extracted from the

table ; then, if there are seconds also in the arc or angle,

we again refer to the same first column for these, and in the

same horizontal line with them, and in the column headed

" parts for " " which immediately follows the column from

which our cosine has been extracted, we shall find the

correction for that cosine : this correction is always to be

subtracted. The remainder will be the cosine of the given

degrees, minutes, and seconds. But in taking out a cosine

to degrees, minutes, and seconds, it will in general be

Vlll EXPLANATION OF THE TABLES.

better to enter the marginal column first with the seconds,

to write the " parts " for these on a slip of paper, and then,

entering the same column with the minutes, instead of

extracting the corresponding cosine, to place the slip under

it, and subtract the correction written thereon. The Table

extends from to 90 only, so that it does not give

immediately the cosines of obtuse angles r when therefore

the angle is obtuse, we must enter the Table with the

supplement of that angle, and regard the corresponding

cosine as negative. It was thought better that this trifling

amount of trouble should be incurred, than that the extent

of the Table should be doubled.

There is indeed a way of avoiding this enlargement of

the Table, and yet providing for the supplementary arcs ;

but as the Table would then have to be used in a peculiar

manner disturbing the general principle upon which the

extracts from it are made in the other cases it was thought

preferable, after due consideration, to reject it. The plan

is this. Suppose a marginal column added to the right of

each page, for the minut^ and seconds, proceeding upwards

from to 60, and that the degrees supplementary to those

at the top of the page (one degree in each case being

omitted in the supplement) were given at the bottom, as

in the ordinary tables of sines and cosines ; we should then

have to use the table as in the following instance. B/equired

the cosine of 115 41' 34" ? Eeferring to page 96, we

should find 115 at the bottom of the column headed 64,

from which column opposite the 41' on the right, we should

take the cosine 43397, then referring to the left for the 34",

we should extract the " parts ' ' 149, which we should have

to add to 43397: we should thus get 433546 for the

required cosine. As the Table at present stands, however,

we enter it with 64 18' 26", the supplement of 115 41' 34",

EXPLANATION OF THE TABLES. IX

and for 64 18' we find 433659, while the "parts'* for 26"

are found in a similar manner to be 113, which subtracted

from 433659 gives 433546 for the required cosine.

It has not been thought necessary to insert the decimal

point before each cosine ; indeed, in the operation for which

this Table is specially prepared, the numbers may always

be regarded as integers. (See p. 227, Naut. Ast.)

Ex. 1. Suppose the natural cosine of 37 21' 33" were

required :

Turning to the page containing 37 at the top (page 92),

we find the "parts " against 33" to be 98, and the cosine

against 21' to be 794944, subtracting the 98 from this,

we write down 794846 for the cosine required.

2. Again, suppose we wanted the cosine of 118 16' 43" :

Subtracting this from 180, the angle in the Table is

61 43' 17". Under 61, and against 17", the "parts" are

72, and against 43', the cosine is 473832 : subtracting the

72 from this, we find 473760 for the required cosine, which

is negative because the proposed angle is obtuse.

3. Eequired the angle whose cosine is 452801 ?

By the Table 452954 = cos 63 4'

Given cosine 452801

Parts for the sees. 153 35"

Hence the angle is 63 4' 35".

If given the cosine had been 452801, then the supple-

ment of the angle thus determined, namely 116 55' 25",

would have been the angle to which that cosine belongs.

X EXPLANATION OF THE TABLES.

TABLE 7.

Proportional Logarithms.

\

These are a peculiar kind of logarithms, first constructed

by Dr. Maskelyne to facilitate the operation of finding the

Greenwich time, at which a lunar distance taken at sea has

place. They are also useful in many other inquiries, in

which difference of time varies, as difference of angular

measurement. When difference of time is required the

Table is to be entered with difference of angular measure-

ment, and when difference of angular measurement is

required it is to be entered with the corresponding difference

of time. Sufiicient illustration of the mode of employing

the Table is given in the Nautical Astronomy, pages

236-7.

The last four figures in the Table are decimals, and the

greater part of the Table consists of these decimals alone ;

the decimal point however is suppressed, as well here as in

the Nautical Almanac, since in finding the Greenwich time

of a lunar distance the logarithms may be always regarded

as whole numbers.

1. Suppose the proportional logarithm of 2 h 8 m 16 s is

required, or the proportional logarithm of 2 8' 16": turning

to the proper page of the Table (p. 112) we find that for

each of these arguments the P. L. is -1472.

2. Suppose it be required to find the difference of time

corresponding to the P. L. '2954 : turning to page 109 we

see that this P. L. answers to the difference of time, l h 31 m

10 s . It also answers to the difference 1 31' 10" of angular

measurement. The Table extends from s to 3 h or 10800 s ,

or from 0" to 3 : the proportional logarithm of the extreme

number of seconds, namely 10800, being 0, the formal

insertion of it has iiot been thought necessary. For the

EXPLANATION OF THE TABLES. Xt

theory of proportional logarithms reference may be made to

the Nautical Astronomy, page 235.

TABLE 8.

For determining the distance of an object seen in the horizon.

This Table shows the utmost distance at which an object

on the surface of the sea can be seen by an eye elevated

above it ; the elevation of the eye being estimated in feet,

and the distance in nautical miles, allowance being made

for atmospherical refraction. If the object itself be elevated

above the surface, and its summit be just visible in the

remote distance, then, if the height of the distant object

thus lost to sight all but its top, be previously known, the

Table will enable us to find its distance, the height of the

spectator's eye being known.

Ex. 1. From the mast head, 130 feet high, a boat was

observed as a remote speck, just appearing in the horizon :

required its distance ?

In the Table opposite 130 feet is found 13'1 miles, the

distance required.

2. From the same height the top of a lighthouse known

to be 300 feet above the level of the sea, was discerned in

the horizon : required the distance from the lighthouse ?

For the height 130 the distance is 13 '1 miles.

300 19-9

.'. the distance of the lighthouse is 33 miles.

Xll EXPLANATION OF THE TABLES.

TABLE 9.

For finding the mean time (nearly) of the Meridian Transits

of the Principal Fixed Stars.

In this Table is recorded the mean time at which each of

the 100 stars there selected passes the meridian of the ship.

The times of transit are given only for every tenth day ;

but as the stars come to the meridian earlier every day, by

a uniform difference of time about four minutes it is

easy to find the time of transit on any intermediate day :

we have only to multiply the number of days after the day

of transit recorded in the Table by 4, and to subtract the

number of minutes in the product from the time of transit

on the day given in the Table. Or we may multiply the

number of days lefore the next tabular day by 4, and add

the resulting minutes to the time of that advanced day's

transit : for example,

Suppose it were required to find the mean time of transit

of o 2 Centauri on the fifth of November :

By the Table the time of transit Nov. 1, is 23 h 43 m

And for four days afterwards, we subtract 4x4= 16

.'. time of transit Nov. 5th is 23 27

Or, the time of transit Nov. 11, being by the Table 23 h 4 m ,

by adding 6 x 4 = 24 minutes, we have for the time on Nov.

5th, 23 h 28 m .

It will of course be understood that the times of transit

furnished by aid of this Table, are only the times nearly ;

but in no case will the time differ from the truth by more

*

than about two or three minutes, and the Table is therefore as

accurate as necessary for the purpose intended by it, which

EXPLANATION OF THE TABLES. Xlll

purpose is to apprise the mariner about what time he may

expect certain well known stars to appear on the meridian

whenever the weather permits his taking a star-altitude for

his latitude. Should the observer not be sufficiently ac-

quainted with the stars to avail himself readily of this

information, he is recommended to procure Mr. Jeans's

" Hand Book for the Stars."

But the right star may generally be detected when we

know, within about half a degree or so, what altitude it

ought to have when on the meridian, and this approximate

altitude may be found by help of the star's declination, and

the latitude by account ; thus :

1. FOB A MEBIDIAtf ALTITUDE ABOVE THE POLE.

In this case the star passes from the eastward towards

the westward, and ascends to the meridian.

When the latitude by account and the declination have the

same name. Add 90 to the declination, and subtract the

latitude by account ; or, which is the same thing, add the

colatitude to the declination, the result will be the ap-

proximate altitude, measured from the S. in IS", lat. and

from the N. in S. lat.

When the latitude by account and the declination have

different names. Add together the latitude by account, and

the declination, and subtract the sum from 90, or, which is

the same thing, subtract the declination from the colatitude,

the result will be the approximate altitude from the S. in

. lat. and from the N. in S. lat. If the sum of lat. and

dec. exceed 90 the star cannot appear above the horizon.

2. FOB A MEBIDIAN ALTITUDE BELOW THE PoLE.

In this case the star passes from the westward towards

the eastward and descends to the meridian. It can be visible

XIV EXPLANATION OF THE TABLES.

below the Pole only when the latitude and declination have

the same name.

From the sum of the latitude and declination subtract

90 : the remainder will be the approximate altitude

reckoned from the N. in N. lat. and from the S. in S. lat.

If the sum of lat. and dec. be less than 90 the star will

pass the meridian below the Pole under the horizon.

To assist in thus getting an approximate meridian alti-

tude, the stars' declinations each to the nearest degree

and minnte are given in the marginal column of the Table.

It is scarcely possible to mistake the star, because no other

will have nearly the same meridian altitude at the time.

The approximate altitude being found in this way, and

the index of the instrument set to it the sight being

directed to the proper point of the horizon the true

meridian altitude, and thence the latitude of the ship, may

be readily determined.

It is to be observed that if the mean time at ship be

A.M., we must add 12 h to that time for the corresponding

time in the Table, from the preceding noon, when the star

opposite that time will pass the meridian above the Pole.

If the mean time at ship be P.M. the star opposite that time

in the Table, will be on the meridian ahove the Pole ; and if

12 h be added to the time, the star opposite the result, will

be on the meridian below the Pole.

TABLE 10.

Best Time for talcing the Altitude of a Celestial Object, icith

the view of determining the TIME at Sea.

"When the time at the ship is to be deduced from an

jiltitude, it is desirable that the object observed should be

in such a position that a small error in the observation may

EXPLANATION OF THE TABLES. XVT

have the least possible influence on the magnitude of the

hour-angle. And this position is attained when the object

is on the prime vertical. If, however, the declination be of

a contrary name to the latitude of the place of observation,

the object will cross the prime vertical before it rises and

.after it sets, so that it cannot be observed on this circle at

all ; the observation should then be taken as soon after it

rises (or before it sets) as that the altitude of it is sufficient

to secure it from the fluctuating effects of the horizontal

refraction. The altitude should not be less than 6 or 7

ves.

The present Table points out, with accuracy enough for

the purpose, the time when the prime vertical is crossed

above the horizon, that is, when the declination is of the

same name as the latitude.

If the object observed be the sun, the table is to be

entered with that degree of declination which approaches

nearest to the sun's declination at the time, and which is

found at the top of the table; underneath this declination,

and opposite to the latitude found in the first column, is

the time before or after noon when it will be most advan-

tageous to take the altitude for TIME.*

If, however, the object be a star, we must first ascertain

the time when it passes over the meridian ; the preceding

Table will supply this information. Then, by aid of the

present Table, entering it with the star's declination and

the latitude of the place, we take out the corresponding

time, or hour-angle between the meridian and star, when

the latter is on the prime vertical. If the observation is to

* As the Table gives the hour-angle from the meridian in Time, if the

object be the sun, the time will be apparent ; the correction for Equation

of Time being applied, will convert it into mean time. For a star, no

such correction is requisite.

XVI EXPLANATION OP THE TABLES.

be made before the meridian transit, we must subtract the

latter time from the former ; if the observation is to be

made after the meridian transit, we must add the two

times together.

It is only when the latitude of the place is greater than

the declination of the object (both being of the same name),

that the object actually crosses the prime vertical; when

the two are equal, the object is on the prime vertical when

it is on the meridian ; when the latitude is less than the

declination, the Table shows the time of nearest approach to

the prime vertical, which will be the best for the altitude

to be observed.

"We may further observe that, when the latitude of the

ship is pretty nearly a mean between two consecutive

latitudes in the Table, the time will be obtained more

accurately by taking the mean of the times corresponding

to those two latitudes.

Ex. 1. At what time will the star a Leonis or Eegulus

bear due East on the 6th of February, in latitude 47 N. ?

Mean time of transit, Feb. 1 20 m (Table 9) 12 h 53 m

Time corresponding to dec. 12 N. and lat. 47 5 14

.

Mean time of star's bearing due East 7 39 P.M.

2. At what time will the star a Arietis bear nearest to

the West on Nov. 21, in latitude 17 32' N. ?

Time corresponding to dec. 22 and lat. 17 2 h 43 m

lat. 18 2 26

2) 5 9

lat. 17 30', + 2 35

Mean time of star's transit (Table 9) 9 56

,, ,, bearing nearest to W.

or

that is, at about 31 ra past midnight.

EXPLANATION OF THE TABLES. XVli

NOTE. The mean time of the meridian passage of each

of the planets, and also of the moon, is given in the

Nautical Almanac for every day in the year. (See Table at

p. 120 of the Nautical Astronomy.)

If a planet be favorable for observation, and there be any

doubt as to which of the planets it is, the doubt may be

removed by noticing what known star is nearest to it,

referring to Table 9 for the time of the meridian passage of

that star, and then finding, from the Nautical Almanac,

which of the planets it is that passes the meridian nearest

to that time.

TABLE 11.

For Finding the Altitude of a Celestial Object, most suitable

for ascertaining the Time at Sea.

This Table is intended to show, what altitude nearly an

object of given declination must have in a given latitude to

be most suitable for deducing the time from that altitude ;

that is to say, it points out approximately the altitude

which the object has when on the prime vertical, or when it

makes the nearest approach to it. "When the object is a

star, and the time most suitable for taking its altitude is

found from the last Table, the approximate altitude of it at

that time, as given by the present Table, will enable the

observer readily to discover it, even should he be but little

familiar with the constellations.

To these Tables is added a list of the Proper Names of

certain of the principal fixed stars, and to this is subjoined

the names and sounds of the letters of the Greek Alphabet.

*#* At the commencement of the foregoing Explanation,

reference is made to Law's " Mathematical Tables," as an

accompaniment to the present volume. The first eighteen

of these Tables, together with the eleven in the present

collection, will be found to contain all that is indispensably

necessary in the several operations of , Navigation and

Nautical Astronomy.

CONTENTS.

TABLE I.

PAGE

^Natural Sine?, Cosines, &c., to every Quarter Poiiifc of the

Compass 1 .

TABLE II.

Logarithmic Sines, Cosines, &c., to every Quarter Point of the

Compass 1

TABLE III.

Xatural Sines, Cosines, &c., to every Degree and Minute of the

Quadrant 2

TABLE IV.

Traverse Table for Points and Quarter Points

TABLE V.

Traverse Table for Degrees 41

TABLE VI.

Natural Cosines to Degrees, Minutes, and Seconds

TABLE VII.

Proportional Logarithms 101

XX CONTENTS.

TABLE VIII.

PAOE

For determining the Distance of an Object seen in the Horizon .117

TABLE IX.

For finding the Mean Time nearly when a specified Fixed Star

passes the Meridian 118

.

TABLE X.

For finding the most suitable Time for taking the Altitude of a

Celestial Object, for the purpose of determining the TIME

at Sea . . . 130

TABLE XL

For finding what Altitude of a Celestial Object is the most

suitable for deducing from that Altitude the TIME at Sea . 132

,

Proper Names of Certain of the Principal Fixed Stars . . .134

The Greek Alphabet . . . . . . . . .134

Note on using the Blank Forms for the different Computations of

Nautical Astronomy 135

TABLE 1.

NATURAL SINES AND TANGENTS TO EVERY QUARTER

POINT OF THE COMPASS.

POINTS.

SINE.

COSINE.

TANGENT.

COTANGENT.

!

o-ooooo

04907

09802

14730

1-00000

99880

99518

98918

o-ooooo

04913

09849

14834

Infinite

2035560

10-15319

674146

8

7|

7*

7J

1

1|

19509

24298

29028

33689

98079

97003

95694

94154

19891

25049

30335

35781

6-02734

399222

3-29483

279481

7

2

2J

U

s]

38268

42756

47140

51410

92388

90399

88192

85773

41421

47296

53451

59938

2-41421

2-11432

1-87087

1*66840

6

*. CO CO CO CO

55556

59570

63439

67156

70711

83147

80321

77301

74095

70711

66818

74165

82068

90635

1-00000

1-49661

134834

1-21850

1-10333

1-00000

5

3

4

COSINE.

SINE.

COTANGENT.

TANGENT.

POINTS.

TABLE 2.

LOGARITHMIC SINES AND TANGENTS TO EVERY QUARTER

POINT OF THE COMPASS.

POINTS.

SINE.

COSINE.

TANGENT.

COTANGENT.

o-ooooo

869079

8-99130

916652

10-00000

9-99948

9-99790

9-99527

o-ooooo

869132

8.99340

9-17125

Infinite

11-30868

11-00660

10-82875

8

7|

7|

7*

H

ll

9-29024

9-38557

9-46282

9-52749

9-99157

9-98679

9-98088

9-97384

9-29866

9-39878

9-48194

955365

1070134

10-60121

10-51806

10-44635

7

61

6*

9

2

2i

2

2*

959284

9-63099

9 67339

9-71105

9-96562

9-95616

9-94543

993335

9-61722

967483

972795

9-77770

10-38-278

10-32517

10-27204

10-22230

6

5*

3

4

9-74474

9-7750&

9-80236

9-82708

9-84949

9-91985

990483

9-88818

i 9-86979

9-84949

9-82489

9-87020

991417

995729

10-00000

1017511

10-X2980

10-08583

10-04270

1000000

5

41

4}

n

4

COSINE.

BINE.

COTANGENT. -

TANGENT.

POINTS.

3.

2 NATURAL 3I'NT5. [TABLE 3.

3

4

5

6

7

/

00 00

017 45

034 90

052 34

069 76

087 16

104 53

121 87

60

1

29

74

035 19

63

070 05

45

82

122 16

59

2

58

018 03

48

92

34

74

105 11

45

58

3

87

32

77

053 21

63

088 03

40

74

57

4

001 16

62

036 06

50

92

31

69

123 02

56

5

45

91

35

79

071 21

60

97

31

55

6

75

019 20

64

054 08

50

89

106 26

60

54

7

002 01

49

93

37

79

089 18

55

89

53

8

33

78

037 23

66

072 08

47

84

124 18

52

9

62

020 07

52

95

37

76

107 13

47

51

10

91

36

81

055 24

66

090 05

42

76

50

11

003 20

65

038 10

53

95

34

71

125 04

49

12

49

94

39

82

073 24

63

108 00

33

48

13

78

021 23

68

056 11

53

92

29

62

47

14

004 07

52

97

40

82

091 21

58

91

46

15

36

81

039 26

69

074 11

50

87

126 20

45

16

65

022 11

55

98

40

79

109 16

49

44

17

95

40

84

057 27

69

092 08

45

78

43

18

005 24

69

040 13

56

98

37

73

127 C6

42

19

53

98

42

85

075 27

66

110 02

35

41

20

82

023 27

71

058 14

' 56

95

31

64

40

21

006 11

56

041 00

44

85

093 24

60

93

39

22

40

85

29

73

076 14

53

89

128 22

38

23

69

024 14

59

059 02

43

82

111 18

51

37

24

98

43

88

31

72

094 11

47

80

36

25

007 27

72

042 17

60

077 01

40

76

129 08

35

26

56

025 01

46

89

30

69

112 05

37

34

27

85

30

75

060 18

59

98

34

66

33

28

008 14

60

043 04

47

88

095 27

63

95

32

29

44

89

33

76

078 17