INTENDED TO FACILITATE THE OPERATIONS OF
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.
JOHN WEALE, 59, HIGH HOLBORN.
BRADBURY AND EVANS, PRINTERS, WHITEFRIARS
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.
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
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.
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
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.
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
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.
.'. the distance of the lighthouse is 33 miles.
Xll EXPLANATION OF THE TABLES.
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.
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
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
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.
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.
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
^Natural Sine?, Cosines, &c., to every Quarter Poiiifc of the
Compass 1 .
Logarithmic Sines, Cosines, &c., to every Quarter Point of the
Xatural Sines, Cosines, &c., to every Degree and Minute of the
Traverse Table for Points and Quarter Points
Traverse Table for Degrees 41
Natural Cosines to Degrees, Minutes, and Seconds
Proportional Logarithms 101
For determining the Distance of an Object seen in the Horizon .117
For finding the Mean Time nearly when a specified Fixed Star
passes the Meridian 118
For finding the most suitable Time for taking the Altitude of a
Celestial Object, for the purpose of determining the TIME
at Sea . . . 130
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
NATURAL SINES AND TANGENTS TO EVERY QUARTER
POINT OF THE COMPASS.
*. CO CO CO CO
LOGARITHMIC SINES AND TANGENTS TO EVERY QUARTER
POINT OF THE COMPASS.
2 NATURAL 3I'NT5. [TABLE 3.