' DESCRIPTION AND USE

OF IIIE
* *

LONGITUDE SCALE

OR,

LUNAR CORRECTOR.

MK rgt; 7

a

m

mi

:

C- ilTHO*

the

description and use

OF THE

longitude scale,

or

lunar corrector,

FOR READILY

clearing the apparent lunar distances

FROM THE

effects of parallax and refraction;

AND

FOR FINDING THE APPARENT TIME

from the altitude of the sun, or a star.

david thomson.

THE SECOND EDITION, GREATLY ENLARGED AND IMPROVED,

CONTAINING ALL

THE TABLES

required with the nautical almanac, and scale,

IN FINDING THE

LONGITUDE

either by lunar observations, or chronometers.

london:
printed for the author,

andsolp, with the scale, by mr. bate, mathematical and philosophical
instrument maker, 17, poultry.

1823.

(Êntei-tïf at ^tatîoneré' î^aïL

Plummer and Brewis, Printers, Loye Lane, East Cheap.

preface.

The method of finding the Longitude at Sea by means of observations
between the Sun and Moon, or between the Moon and a Star, commonly
called Lunar Observations, is daily getting into more general practice ; and
probably, in a very few years, every person will be qualified to determine
the Longitude by this method, who undertakes to conduct a Ship from
one place to another, where he may be several days without seeing land;
for the error in the Longitude by account will frequently be greater in
24 hours, than the error in the Longitude when found by a good Lunar
Observation.

The operation of clearing the Apparent Lunar Distances from the effects
of Parallax and Refraction, in order to find the True Distance, has generally
been considered the only difficult part of the calculations which are necessary,
in finding the Longitude by the Lunar Observations, and a great variety
of methods have been invented with the view of rendering this part of the
process as simple as possible. In the year 1816, the Author of this pubhshed
a pamphlet, containing the description and use of a Sliding Scale invented
by him for the purpose of finding the True Lunar Distances, and the
Apparent Time from the Sun's Altitude, with the other necessary data.
This Scale met with the approbation of the Astronomer Royal, and also with
that of several other eminent Mathematicians and Astronomers, some of
whose names are given in the pamphlet before-mentioned; and the inventor
trusts that the Scale in its present improved state, will be found still more
worthy of attention.

The principal object in view in the alterations introduced on the Scale has
been to avoid as much as possible the distinction of cases in the correction
of the Lunar Distances. This has been so far effected, that it is presumed
the Rule now given can hardly be misunderstood by any person who pays

PREFACE.

the least attention to it; and the process is so short, that the Apparent
Distance may be Reduced in about two minutes, so correctly, as \ery
seldom to dilFer more than one or two seconds from the True Distance as
obtained by the most rigid calculation. The operation of finding the Apparent
Time by the Scale is also rendered more simple, and Examples are given of
the application of the Scale, in determining the Longitude both by Lunar
Observations and Chronometers.

The Tables are also more correct and extensive than in the former Edition,
and no other Tables are required in deducing the Longitude either from
Lunar Observations or Time-keepers. Table VL, which contains the third
correction to be applied to the Apparent Lunar Distances according to the
Rule now given, is new. The construction of this Table has been a work
of very considerable labour; this will be readily credited, when it is
considered that the third correction combines six small corrections when
applied to a distance between the Sun and Moon, and_five when the distance
is between the Moon and a fixed Star.

For the correctness with which the operations may be performed by the
Scales, the Inventor must acknowledge himself much indebted to Mr. Bate,
for the great pains taken by him in laying down new hnes to divide from.
These lines are perhaps more accurate than any of a similar nature hitherto
constructed, at least the Author has not met with any that could be
compared with them in this respect.

Scales of either Two or Three Feet in length may be had; the operations
are performed in the same manner by both. The result will, in general,
be more accurate when the Three Feet Scale is used; but the short Scale
will not cause any error of material consequence in the practice of Navigation,
for this error will very seldom amount to three miles of Longitude.

description op the lines

on the

three feet scale.

On the side marked Lunar Side are the following lines :

L On the upper fixed part is a line marked Apr. Dist. .This is a line of
LoKarithmic Tangents, commencing at the right hand at 28°, and mcreasing to-
wards the left up to 88° 40'; each degree from 28° to 70° is divided into 6
parts, so that each division is equal to 10 minutes of a degree. From 70 to
88° 40'each degree is divided into 12 parts; each division is therefore equal
to 5 minutes.

II On the fixed part, at the right hand extremity, is the Moon's Ho a. Par.
Each minute is divided into 6 parts, so that each division is equal to 10 seconds.

III.nbsp;On the upper edge of the slide is a line of Logarithmic Sines, marked
App. Dist. This line commences towards the left hand at 28°, and increases
towards the right up to 90°; each degree from 28° to 40° is divided into 6 parts
or to 10 minutes of a degree ; from 40° to 60° each division is equal to 15 mi-
nutes ; from 60° to 70° each degree is divided into 2 parts or to 30 minutes ;
from 70° to 85° each division is a degree; and there is no division between
85° and 90°.

IV.nbsp;On the slide to the left of the last mentioned line is the Moon's Hor. Par.
Each minute is divided into 6 parts; hence each part or division is equal to 10
seconds. The hor. par. on this line increases in a contrary direction to that on
the fixed part.

V.nbsp;On the lower edge of the slide is a line of Logarithmic Sines, marked
App. Alt. (for apparent altitude.) This line begins at 5°, and increases towards
the right hand up to 90° ; each degree from 5° to 20° is divided into 12^parts,
so that each division is equal to 5 minutes of a degree; from 20° to 40 each
division is 10 minutes; from 40° to 60° each division is 15 minutes; from 60°
to 70° eadi degree is divided into 2 parts or to 30 minutes ; from 70o to 85°
each division is a degree ; and there is no division between 85° and 90°.

VI.nbsp;On the lower fixed part is a line of numbers, marked Corr. (for Correc-
tion.) This line commences at the left hand ; the first division is 1 minute, and
the last or right hand division is 1° 40'. The minutes are subdivided as fol-

®nbsp;B

lows : from 1' to 20' each minute is divided into 12 parts, so that each division
is equal to 5 seconds of correction; from 20' to 50' each minute is divided into
6 parts or to 10 seconds; from 50' to 1° 20' each minute is divided into 4 parts
or to 15 seconds; and from 1° 20' to 1° 40' each minute is divided into 3 parts,
hence each part is equal to 20 seconds.

VII. On the fixed part below the line of correction is the complement of the
correction to 2°; this line is marked Cojip. Corr. The right hand division is
20 minutes, and the left hand one 1° 59'; and all the numbers in this line are
respectively equal to what the numbers on the line of correction which are at the
same vertical lines want of 2 degrees ; thus, the number on the line of correction
being 40 minutes, the number on the complement of correction is 1° 20'; or, if
the number on the line of correction be 1° 10', the number on this line is 50 ini-
nutes, and so on; also, if the number on the line of correction be 42' 30quot;, the
number on this line will be 1° 17' 30quot;, the two numbers together beino- always
equal to 2 degrees.nbsp;°

On the side marked Time Side are the following lines :

I. On the upper fixed part is a line of Logarithmic Co. Sines, marked Halp
Sum, beginning at 0° at the right hand, and divided as follows : from 0° to
there is no division ; from 5° to 20° each division is a degree ; from 20° to 30°
each degj-ee is divided into 2 parts or to 30 minutes ; from 30° to 50° each di-
vision is 15 minutes ; from 50° to 70° each degree is divided into 6 parts or to
10 minutes; and from 70° to the left hand division (viz. 89° 25') each degree is
divided into 12 parts, so that each division is equal to 5 minutes.

IL On the upper edge of the slide is another line of Co. Sines, marked Lat.
(for Latitude.) This line is divided and numbered exactly in the same manner
as that on the fixed part.

IIL On the lower edge of the slide is a line of Logarithmic Sines, marked
Di?f. (for difference.) This line begins at the left hand at 0° 35', and increases
towards the right up to 90°; from the beginning of this line up to 20° each de-
gree is divided into 12 parts, so that each part is equal to 5 minutes; from 20°
to 40° each degree is divided into 6 parts or to 10 minutes ; from 40° to 60° each
division is equal to 15 minutes ; from 60° to 70° each division is 30 minutes;
from 70° to 85° each division is a degree; and there is no division between
85° and 90°.

IV; On the lower fixed part, adjacent to the slide, is a line marked Time P. M.
This is a line of versed sines divided into time; from the left hand up to 4h. each
minute of time is divided into 2 parts or to 30 seconds ; from 4h. to 6h. each
division is 1 minute of time ; from 6h. to 8h. each division is 2 minutes ; from
8h. to Ilh. each division is 4 minutes ; and from llh. to 12h. each division is
20 minutes of time.

V. Below the last mentioned line, is a line marked Time A. M. This line is
of the same nature as the last, and the hours, amp;c. on this are respectively the
complements to 24h. of the hours, amp;c. immediately above them; that is, when
the hour on the P. M. line is 4, the hour on this is 20 ; or, if the time on the
P. M. line is 2h. 35m. the time on this will be 21h. 25m. and so on. The divi-
sions for the minutes, amp;c. on the P. M. line answer for this line, observing to
reckon from right to left; thus, if the time pointed out on the line above be

3h 17ra. 30s., the time on this will be 20h. 42m. 30s. The hours amp;c. on tiM
line are for astronomical time; if civil or nautical time be wanted, reject Uh.
from the time given by this line; the remainder will be tVe civil or nautical
time A. M.

The sliding brass clasp contains the Declination. This h a portion of a
line of Co. Sines. The left hand edge of the clasp is 0° ; the first division from
this towards the right is 5°; from 5° to 15° each division is a degree; from 15
to 32° each division is half a degree or 30 minutes.

description

of the

two feet scale.

The lines on this scale are of the same nature as those already described in the three feet
scale, and are marled in the same manner. The description begins as before on
the LcNAii Side.

T. Each degree of the A pp. Dist. on the fixed part, from 28° to 80°, is di-
vided into 6 parts, so that each part is 10 minutes of a degree ; above 80 each
division is 5 minutes.

II.nbsp;Each minute of Moon's Hon. Par. on the fixed part, and also on the slide,
is divided into 4 parts or to 15 seconds.

III.nbsp;The App. Dist. on the slide. From 28° to 40° each degree is divided
into 4 parts, or to 15 minutes of a degree ; from 40° to 60° each division is 30
minutes : from 60° to 80 each division is a degree ; the division between SO and
85° is at 82° 30', and there is no division between 85° and 90 .

IV The App Alts. From 5° to 10° each degree is divided into 12 parts, or
to 5 minutes of a degree; from 10° to 20° each division is Ip minutes ; from
20o to 40° each is 15 minutes ; from 40° to 90° the divisions are the same as
those in the line last described.

V.nbsp;The line of Cour. From 1 to 10 minutes each division is equal to 5 se-
conds of Correction; from 10' to 30' each division is 10 seconds ; from 30 to 50
each minute is divided into 4 parts, or to 15quot; ; above 50' each division is 30 se-
conds of Correction.

VI.nbsp;The CoMP. CoRR. This line will be easily understood by referring to the
description of the corresponding line on the 3 feet scale.

The Time Sibe.

I. The H iLP Sum. The first division to the left of 0° is at 5°; the division
between 5° and 10° is at 7° 30'; from 10° to 30° each division is a degree ; from

30° to 50° each dej?ree is divided into 2 parte; from 50° to 70° into 4 narts •
and from 70° to 80° into 6 parts, or to 10 minutes r from 80quot; to the end of the'
seale each division is 5 minutes of a degree,

TTnbsp;divided and numbered in the same manner as the

Halt Sirms'.

III.nbsp;The Diff. From the left up to 10° each division ia equal to 5 minutes •
from 10° to 20° each is equal to 10 minutes; from 20° fo 40° each degree is'
divided into 4 parts, or to 15 minutes; from 40° to 60° each division is 30 mi-
nutes; from 60° to 80° each division is a degree; the division between 80° and
85° is at 82° 30'; and the division next to 90° is 85°.

IV.nbsp;The Time P.M. From the left up to 3h. each division is 30 seconds of
time; from 3h. to 5h. each division is a minute; from 5h. to 8h. each division is
2 minutes; from 8h. to lOh. each is 4 minutes; from lOh. to 12h, each division
is 30 minutes.

V.nbsp;The Time A. M. This line has the same relation to the Time P. M. as
the corresponding line on the 3 feet scale, that is, the hours on the A. M. line
are merely the complements to 24h. of those on the P. M. line, and the minutes,
amp;c. increase in a contrary direction.

VI.nbsp;The Declination on the brass clasp. The left hand edge of the clasp
represents 0°; the firstlt;division from this towards the right is 4°; from 4° to 10°
each division is 2 degrees; from 10° to 32° each division is a degree.

REMARKS.

I.nbsp;In the first rule for correcting the Lunar distances by the scale, the line of
Cork, is only to be used; the use of the Comp. Coru. is explained afterwards.

II.nbsp;The apparent time whether before or after noon may always be found by
the P. M. line, for the time on that line is the horary distance of the observed ob-
ject from the meridian; therefore when the Sun is observed in the morning, the
time given on the P. M. line being subtracted from 24h. gives the astronomical
time, or from 12h. gives the civil or nautical time. The line of time A.M. is
only to save the trouble of these subtractions.

III.nbsp;If any difficulty be found in setting by hand a particular part of the slide
exactly to any given place on the fixed part, it may be very easily and accurately
done, by striking with any small thing (such as a pen knife) on the end of the
slide, or the fixed part, according as you wish the ^lide to move to the left or
right; and in reading off the corrections, amp;c. it will be found useful to trace the
divisions with the point of a pen-knife, or any thing else with a sharp point.

IV.nbsp;On the fines of Sinas and Co-sines, the space that is between 90° or 0°,
and the first division towards the left is always equal to 5°; but a degree next to
90° or 0° takes up a much less part of that space than a degree next to 85° or 5°
that is, 86° is farther to the right of 85° than 89° is to the left of 90° ; and sup-
posing the whole space to be divided into 100 parts, then on the lines of Sines
the number of those parts reckoned to the right from 85°, which the higher de-
grees will occupy respectively, is nearly as follows, 86° — 36, 87° = 63,
88° = 84, and 89° ir 96. And on the lines of Co-sines 1° will be 4 of the
same parts to the left of 0°, 2° = 16, 3° = 37, and 4° = 64. It will be pro-
per to attend to these remarks on the space between 85° and 90°, when correct-
ing a distance between 85° and 95°; the line of App, Dist. on the slide being a

f c- And in findine- the apparent time when the Latitude of the place
'robse ;Sn itt tLn the 'sjace between 0° and 5° on the line of Lax
sWuld be estoated in the proportion here given, tins bemg a hne of Co-smes.
itoYhe declination, when under 5°, must be thus estimated.

PROBLEM I.

Given the Apparent Distance of the Moon from the Sun or af.ed Star, together
^M the Apparent Altitudes of the ohjeets, and the Moon's horizontal parallax, to

find the true Distance.

RULE.

1 Set the ADoarent Distance on the sUde, to the Moon's Hor. Par. on the fixed
1 bet thenbsp;rent Altitude of the Sun or Star is the correc-

rr'vtiXS^ySnbsp;from the Apparent Distance.

9 Set the Moon's Hor. Par. on the slide to the Apparent Distance on the fixed
Id opposite to the Moon's Apparent Altitude is the second correction which
Fs lo bfaS to the Apparent Distance when less than 90°, but to be subtracted
when the Distance is greater than 90°.

3. Take the third correction from Table VL corresponding to the gwen D|stance
and AltUudes, this correction is always to be added to the Apparerit Distance.
These 3 corrections being applied to the Apparent Distance will give the True

Distance.nbsp;„ , . .

N. B. When the Apparent Distance is greater than 90°, subtract it from 180 ,
and work with the supplement on the Scale.

EXAMPLE L

Snnnose the Apparent Distance between the Moon and a Star is 60°, the Ap-
parenTCitudenbsp;Star 24°, and that of the Moon 16°, when the Moon's Hor.

Par. is 58': required the True Distance.

Moon's Hor. Par.

Apparent Distance

Star's Apparent Altitude 24° 0' First Correction -
Moon's Apparent Altitude 16° 0' Second Correction
Time Table VI. -nbsp;- Third Correction

Sum of Columnnbsp;-nbsp;-nbsp;-nbsp;_

Sum of — Column
True Distance

..n,nbsp;^ J- .nbsp;nn the slide being set to 58' of hor. par. on the

o'f W.' if: on the slide being set to the W^t distan^^
the fixed part, opposite to the apparent alftude of the Moon 16 , is the second

apparent distance 60°, under
Star and opposite to 16°, the apparent altitude of the Moon, is 1 30 , wlneh

^^wXatanged as above, it is plain from the Rule ^hatt^^^^^^^ c^.
rection is always to be placed in the Minus column, and that the secona cor

rection must be placed in the Plus column when the apparent distance

EXAMPLE II.

Let the apparent distance between the Sun and Moon be ind°. »i, c .

Moon's Hor. Par.

Apparent Distance

Sun's Apparent Altitudenbsp;20° 0'First Correction -
vZ^T tF^^^f * Altitude 42° 0' Second Correction-

From Table VI. .nbsp;. Third Correction

Sum of Column .nbsp;.

Sum of—Column -nbsp;. J,
True Distance

quot;^'ffquot; unbsp;^80° leaves the supplement

Moon's apparent altitude, =z 9' 11quot;nbsp;' to 42 , the

««». tl.. effect „?thV

SSe P rlWi ''5 quot;i'l Siquot;nbsp;co^cion Vi-quot; ftquot;

example in.

the apparent distance between the Moon and a Star is 52o av ^oquot;-

Moon's Hor. Par.
Apparent Distance

Mar's Apparent Altitudenbsp;First Correction -

FrZ Tab'irvT'

laoie VI. . . Third Correction

56'40»

Sum of Column

•Sum of — Column
True Distance

52o45'30quot; on the slide beins set to the hor ..

Again, the hor. par. 56'40quot; on the slide being set to the apparent distance

52° 45' 30quot; on the fixed part, the second correction 35' 49quot; is found opposite to
56° 15', the Moon's apparent altitude.

In Table VI. at the nearest apparent distance 52°, and under Star's altitude
18 , and opposite to the Moon's 56°, is 2' 17quot; of third correction; but under
Stars altitude 20°, and opposite to the Moon's 56°, the third correction is only
2 2quot;; therefore 3quot; is taken from 2' 17quot; for the 20' which the Star's altitude ex-
ceeds 18°, which makes the third correction 2' 14quot;. No allowance is required
for the odd minutes of Moon's altitude, nor for what the given apparent distance
exceeds 52°, as both these will not cause an error of more than 1quot; in the true
distance.

EXAMPLE IV.

Suppose the apparent distance is 77° 44' 4quot; between the Sun and Moon • the
Sun's apparent altitude 48° 18', that of the Moon 22° 43', and the Moon's hor.
par. 55' 22quot;: required the true distance.

Moon's Hor. Far.

Apparent Distance

Sun's Apparent Altitude 48° 18' First Correction —
Moon s Apparent Altitude 22° 43' Second Correction
From Table VI. - . Third Correction
Sum of 4- Column

Sum of — Columnnbsp;-nbsp;.nbsp;.

True Distancenbsp;-nbsp;.nbsp;.

rnbsp;'»PPai'ent distance on the slide being set to the hor. par.

otgt; 22 , the first correction 42' 18quot; is found opposite to the Sun's apparent alti-
tude. Again, the hor. par. 55' 22quot; on the slide being set to the apparent distance
on the fixed part, the second correction 4' 39quot; is found opposite to the Moon's
apparent altitude.

At apparent distance 76° (in Table VI.) the third correction for the given al-
titudes IS ]' 58quot;; but at apparent distance 80°, the third correction for the same
a titude IS 2' 5'; therefore 2quot; is to be added to 1' 58quot; for 1° 44' which the given

wprl wnbsp;2' 0quot; would be the third correction if the disLce

were tJetween the Moon and a Star, but the distance being between the Sun and
Moon the effect of the Sun's parallax will be found in Table P. = 2quot;, which is to
be subtracted from 2' 0quot;, the remainder 1' 58quot; is the third correction to be ap-
plied to the apparent distance.nbsp;'

EXAMPLE V.

Let the apparent distance between the Sun and Moon be 114° 50'42quot; the
^un's apparent altitude 38° 28', and the Moon's 23° 30', when the Moon's hor.
par. IS 58' 49quot;: required the true distance.

Moon's Hor. Par.

Apparent Distance ...
Sun's Apparent Altitude 38° 28' First Correction —
Moon's Apparent Altitude 23° 30' Second Correction 4-
From Table VI. - . Third Correction
Sum of Column

Sum of—Column -nbsp;-nbsp;. _

True Distance

Moon's Hor. Par.

Apparent Distance
Star's Apparent Altitude 43° 13' First Correction —
Moon's Apparent Altitude 66° 34' Second C:orrection
From Table VI. -nbsp;- Third Correction

Sum of Column
Sum of — Column
True distance

In the foregoing Examples, the corrections are all applied to the apparent
distance agreeable to the Rule; in these the line of correction has only been
used. The following method will be found rather shorter where the line of the
Com'p. Coru. is used, and by that means all the corrections may be added to
the apparent distance.

RULE.

1.nbsp;When the apparent distance is less than 90°, set the apparent distance on
the slide to the Moon's hor. par. as before ; then, instead of taking off the cor-
rection opposite to the apparent altitude of the Sun or Star, take off the com-
plement of the correction ; add this, together with the second and third correc-
tions (found as before) to the apparent distance, the sum, rejecting 2 degrees,
will be the true distance.

2.nbsp;When the apparent distance is greater than 90°, take the complements of
both the Jimt and second corrections from the scale ; add these, together with
the third correction, to the apparent distance, the sum, rejecting 4 degrees, will
be the true distance.

When the true distance is found in this manner, the number of figures re-
quired for reducing the apparent to the true distance is about the same as is
necessary for reducing the observed distance between the Sun and Moon to the
apparent distance, and the time required to perform the one operation is nearly
the same as that required for the other. That this method may be well under-
stood, the foregoing six Examples are worked by it, that the reader may com-
pare them with each other, and see the reason of applying the complements of
the corrections from the scale, instead of the corrections, when it is wished to
avoid subtractions.

EXAMPLE I.

Moon's Hor. Par.

Apparent Distance of Moon and Starnbsp;-nbsp;60

Star's Apparent Altitude 24° 0' Comp. 1st Corr.nbsp;4- 1

Moon's Apparent Altitude 16° 0' Second Corr.nbsp;

From Table VI. - - Third Corr.nbsp;

Sum rejecting 2° = True Distance

By referring to this example, as worked by the former method, it will be seen
that the first correction is 27' 15quot; to be subtracted. Now it is plain that the re-
sult will be the same if the complement of 27' 15quot; to 2° (that is 1° 32' 45quot;) be
added to the apparent distance, and 2° rejected on that account. The apparent
distance being less than 90°, the second correction is the same as before, and the
third correction is always the same in both methods.

example II.

Moon's Hor. Par.nbsp;55'0quot;

0 ' II

Apparent Distance of Sun and Starnbsp;-nbsp;104 0 0

Sun's Apparent Altitude 20° 0' Comp. 1st Corr. 1 40 37
Moon's Apparent Altitude 42 0 Comp. 2d Corr. 1 50 49
From Table VI. - - Third Corr. 2 45

Sum rejecting 4° = True Distancenbsp;- - 108 34 11

Here the apparent distance being greater than 90°, the complement of the
second correction, as well as that of the first, is taken from the scale. Now the
first correction being 19' 23quot;, its complement to 2° is 1°40' 37quot;; and the.second
correction being 9' IIquot;, the Complement of it to 2° is 1° 50' 49quot;. These com-
plements, together with the third correction, being added to the apparent dis-
tance, give the sum 107° 34' 11quot;, and 4° being rejected from that, the remainder
103° 34' 11quot; is the true distance.

example III.

Moon'8 Hor. Par.nbsp;56'40quot;

o / It

Apparent Distance of Moon and Starnbsp;-nbsp;52 45 30

Sun's Apparent Altitude 18° 20' Comp. 1st Corr. 1 37 37
Moon's Apparent Altitude 56 15 Second Corr. 35 49
From Table VI. - - Third Corr. 2 15

Sum — 2° = True Distancenbsp;-nbsp;- 53 1 11

example iv.
Moon's Hor. Par.nbsp;55'22quot;

O t if

Apparent Distance ofSun and Moon - 77 44 4
Sun's Apparent Altitude 48° 18' Comp. 1st Corr. 117 42
Moon's Apparent Altitude 22 43 Second Corr. 4'39
From Table VI. - - Third Corr. 1 58

Sum — 2° i: True Distance -nbsp;^nbsp;77 8 23

example v.
Moon's Hor. Par.nbsp;58' 49quot;

o t it

Apparent Distance of Sun and Moonnbsp;- 114 50 42

Sun's Apparent Altitude 38° 28' Comp. 1st Corr. 119 41
Moon's Apparent Altitude 23 20 Comp. 2d Corr. 149 13

From Table VI.nbsp;.nbsp;-nbsp;-4-34

Sum — 4° — True distancenbsp;- -nbsp;114 2 40

Moon's Hor. Par. CI'13quot;nbsp;^ ^ ^^

A nnarent Distance of Moon and Star -nbsp;- 5 3 r,7

Star's Apparent Altitude 44° 15' Comp. 1st CoiT. 1 0 $8

Moon's Apparent Altitude OG 34 Second Corr.nbsp; 50 2

Time Table VI. - - TMrd Corr. _^

Sum — 2° = True Distance - -nbsp;45 1 46

REMARKS.

L The scale is not adapted for distances less than 28°, nor for altitudes less
than 5°; indeed neither altitude should be less than 6 , on account that the
third corrections in Table VI. are not given for lower altitudes a thoug al-
lowance might be easily made in the third correction for an altitude between 5
anTe de^^rees ; but low altitudes should be avoided, if possible, m he practice
of the Lunar Observations, as they may cause a considerable error in the distance,
from the uncertainty of the refraction near the horizon.

IL If, when the Moon's hor. par. on the slide is set to the apparent distance
on the fixed part, the Moon's apparent altitude be found to the left of 1 minute
of correction, it shows that the second correction is less than 1 minute. The
number of seconds in this correction is to be found as follows :

Make a mark at the point of the slide that is opposite to 1 minute of correc-
tion; shift the slide to the right, until that point be opposite to 10nbsp;f ®
correction • then take off the correction opposite to the Moon s apparent altitude
Sfre'correction by 10, the quotient ^11 be the number of seconds contained
in the second correction. For example:

Suppose the apparent distance is 88° 50', the Moon's apparent altitude 15 , and
the hor. par. 56'; what would be the second correction ?

. Here when 56' of hor. par. on the shde is set to 88° 30' on the fixed part,

15° of altitude will be found to the left ofnbsp;o correction, and the pomt

of the slide that is opposite to 1 minute is 43° of altitude nearly. Now, 43° be-
oi tne biiue unbsp;oppos te to the Moon's apparent altitude

if ,quot;wil5Ve founT?nbsp;by 10, gives 22quot;.7 [Jr 23quot; nearly) of

second correction.

III. When the apparent distance is between 88° 40' and 91° 20', the second
correction cannot be found by the scale, but must be found by Table V.-See
the explanation of that table m the page opposite to it.

PROBLEM IL

Given the Latitude of a Place, together with the Suns true altitude and declination,
to find the Apparent Time of observation.

1 Add together, the Sun's altitude. Polar distance, and the Latitude of the
place of observation; find the Half Sum, and the Difference between the Half
Sum and the Sun's altitude.

11

2.nbsp;Set the left hand edge of the sliding brass plate to the Half Sum on the
fixed part.

3.nbsp;Set the Latitude on the Slide opposite to the Declination on the brass
plate ; then opposite to the Difference on the Slide will be found the Apparent
Tiaie on the fixed part.

EXAMPLE L

Jn Latitude 42° 0' N. when the Sun's declination is 10° N. and his true alti-
tude 34° 0': required the apparent time P. M.

o /

Sun's true altitude - - 34 0
Sun's Polar distance - - 80 0

Latitude - - - - 42 0

Sum - - - - 156 0
Half Sumnbsp;- . . 78 0

Differencs - - . 44 o Apparent time, St. 31m. Os. P. M.

Here the Half Sum of the Latitude, the Sun's polar distance, and altitude, is
78° 0', and the difference between this and the Sun's altitude is 44° 0'. Now,
the left hand edge of the brass clasp being set to the Half Sum 78° on the
fixed part, and then the Latitude 42° on the slide being set opposite to the De-
cimation 10° on tiie brass clasp, opposite to the Difference 44° 0' will be found
3h. 31m. Os. the apparent time P. M.

EXAMPLE n.

Suppose that in Latitude 29° 30' S. when the Sun's declination is 19° 30' N.,
ms true altitude is found to be 22° 0' east of the meridian: required the apparent

time of observation.

o /

Sun's true altitude - - 22 0
Sun s Polar distance . 109 30
Latitude - . . . 29 30

c ■ ■ - - 161 0

Half Sum - - . go 30

Difference - - - 58 30 Apparent time, 20h. 14m. 16s.

The left hand edge of the brass plate being put to the Half Sum 80° 30' on
the fixed part, and then the Latitude 29° 30' on the slide being set to the Decli-
nation 19° 30' on the brass slide, opposite to the Difference 58° 30' on the slide
the apparent time 20h. 44m. 16s. on the A. M. line; this is astronomical
time If civil or nautical time be required, reject 12h., and the remainder,
44m. 16s. will be the time required.

P M*^^® opposite to the difference 58° 30' were taken from the upper or
^•M. line, it will be 3h, 15m. 44s., and this being subtracted from 24h. the re-
mainder will be 20h. 44m. 16s. as before. The time given by the P.M. iiae
always the horary distance of the Sun from the nearest noon.

In the afternoon, in Latitude 34° 53' N. the Sun's true altitude was 14° 56',
and at the same time his declination was 16° 41' S.: required the apparent time
of observation.

EXAMPLE IV.
To find the apparent time A. M. or P. M. let there be give»,
0 •

The Sun's true altitude - 28 47
The Sun's Polar distance - 78 44
And Latitude - - - 56 34

Sum - - - - 164 5
Half Sum - - - 82 2i

Difference quot; - - 53 15 J App. time A.M. 20h. 24m. 27s. or P.M. 3h. 35m. B3s.

PROBLEM in.

Given the true altitude, together with the right ascension and declination, of a Star,
the Latitude of the place of observation, and the right ascension of the Sun, at the
time the Star's altitude is observed, to find the Apparent Time of observation.

RULE.

With the Latitude of the place of observation, the Star's true altitude, and de-
clination, find the Star's meridian distance in the same manner that the Sun's
meridian distance is found by the last Problem, always taking the meridian dis-
tance from the P. M. line when the Star is observed West of the meridian, but
from the A. M. line when the Star is observed East of the meridian.

Then to the complement to 24h. of the Sun's right ascension, add the Star's
right ascension, and its meridian distance; the sum (rejecting 24h. or 48h. if
necessary) will be the apparent time of observation.

EXAMPLE 1.

In Latitude 33° 51' N. the true altitude of Regulus, observed west of the me-
ridian, was 39° 21', and at the same time the Star's declination was 12° 50' N.,

its right ascension 9h. 58m. 51s., and the Sun's right ascension 3h. 22m. 8s.
required the apparent time of observation.

Star's true altitude 39 21nbsp;h- m. s.

Star's polar distance 77 10nbsp;„ , „ .nbsp;a oo o

T.»titade 33 51nbsp;Sun's R. A. -nbsp;- 3 22 8

150 22nbsp;Comp. of Sun's R. A.nbsp;20 37 52

Half Sum - - 75 11 Star's R. A - 9 58 51
Difference - - 35 50 Star's mend. dist. 3 23 44

Sum — 24h. r: Apparent time -nbsp;- -nbsp;10 0 2T

The operation for finding the meridian distance of the Star by the Scale, is the
same as it would be to find the apparent time P. M. if the Sun's true altitude
were found to be 39° 21' west of the meridian, and his declination, and the lati-
tude of the place, the same as here given. Now, the complement to 24h. of the
Sun's right ascension is 20h. 37m. 52s., to which the Star's right ascension
9h. 58m. 51s., and its meridian distance 3h. 23m. 44s. being added, the sum is
34h. Om.quot; 27s.^ from which reject 24h., the remainder lOh. Om. 27s. is the ap-
parent time of observation.

EXAMPLE 11.

In latitude 23° 31' S. the true altitude of Rigel observed east of the meridian
was 28° 42'; at the same time the declination of the star was 8° 25' S. ; its right
ascension 5h. 5m. 55s. ; and the Sun's right ascension 12h. 49m. 31s. : required
the apparent time of observation.

o /

Star's true altitude 28 42nbsp;h. m. s.

Star's polar distance 81 35nbsp;24 0 0

Latitude 23 31nbsp;Sun's R. A. - — 12 49 31

Sum . - 133 48 Comp. of Sun's R. A. 11 10 29
Half Sum - - 66 54 Star's R. A. -nbsp; 5 5 55

Difference - . 38 12 Star's Merid. Dist. 19 50 51

Sum - 24h. = Apparent time - - - - 12 7 15

Here because the star is observed to the eastward of the meridian, its meridian
distance 19h. 50m. 51s. is taken from the A.M. line; the complement of the
Sun's right ascension is llh. 10m. 29s.; and the sum of this, the star's right
ascension and meridian distance is 36h. 7m. 15s.; from which 24h. being sub-
tracted, the remainder 12h. 7m. 15s. is the apparent time of observation.

REMARKS.

What is called the star's meridian distance in the foregoing Examples, is
always the horary distance of the star reckoned to the westward of the meridian.
Hence, when the star is observed in the western hemisphere, its meridian dis-
tance will be less than 12h.; but when the observation is made in the eastern
hemisphere, the star's horary distance from the meridian, reckoning to the west-

ward, must be greater than 12h.; for it is the complement to 24h. of the horary
distance of the star to the eastward of the meridian.

11. It might be of advantage if the Nautical day were made to agree with the
Astronomical day, as mistakes sometimes take place, from the former being 24h.
in advance, with respect to the latter. If the hours of the Nautical day be con-
tinued to 24h. from noon to noon, that is, calling 8 P. M. 8h., and 8 A. M. 20h.,
and so on; then Nautical time is changed into Astronomical time, by merely al-
tering the date to the preceding day. Thus, Jan. 10, at 9 A. M. Nautical time,
will be Jan. 9, at 21h. Astronomical time. When apparent time is mentioned in
!*nbsp;any part of these directions, the apparent Astronomical time is always to be un-

(Jnbsp;derstood, unless where otherwise expressed.

:rnbsp;In the following examples of finding the Longitude by Lunars, a set of dis-

^nbsp;tances, with the altitudes of the objects observed at the same time, are supposed

j;. _nbsp;to be taken down on a slate, or a piece of paper; the observed distance and alti-

inbsp;tudes are the means of the observations as given by the instruments, corrected by
the index errors, if any.

jnbsp;It is supposed that the reader is acquainted with the usual methods of finding

'nbsp;the proportional parts of the variation in the Sun's declination, amp;c. in 24h. for the

Inbsp;given Greenwich time, or the parts of the variation of the Moon's semidiameter or

fnbsp;hor. par. in 12h. for any given time at Greenwich, past noon or midnight. These

'fnbsp;proportional parts maybe easily and correctly found on the Lunar Side of the

;nbsp;Scale, by attending to the following directions.

Vnbsp;The minutes on the line of Correction may be esteemed as hours, and then the
inbsp;seconds of correction become minutes of time; or the minutes of correction may

Vnbsp;be esteemed as seconds of the circle, and then the seconds of correction are to be
reckoned thirds of the circle.

1. To find the Correction of the Sun's declination for a given time past noon.

i

Inbsp;RULE.

■I

fnbsp;Set 30° of alt. on the slide to 24' of correction, and esteeming the minutes of

Inbsp;correction as hours, make a mark on the slide opposite to the given time past

fnbsp;noon. Then set 30° of alt. on the slide to the change of the Sun's declination in

Jnbsp;24h., and opposite to the mark that was made on the slide, will be found the pro-

rnbsp;portional part of that change for the given time past noon, which being added to,

Inbsp;or subtracted from the Sun's declination for the preceding noon, according as it is

!nbsp;increasing or decreasing; the sum or difference will be the declination at the given

time.

EXAMPLE.

Suppose the Sun's declination be found on any day in the Nautical Almanac
= 14° 10' 15quot;, and on the following day it is 1.4° 28' 55quot;, what would be the
Sun's decHnation at 9h. 35m. Greenwich time.

Here the change of the Sun's declination in 24h. is 18' 40quot;. And 30o of alti-
tude being set to 24' of correction, make a mark on the slide opposite to 9' 35quot;

(= 9h. 35ra.) then set 30° of alt. on the slide to the change of declination in 24h.
(=18' 40quot;) and opposite to the mark on the slide will be found 7' 27quot;, the pro-
portional part required, which is to be added to 14° 10' 15quot;, hence the declina-
tion at 9h. 35m. will be 14° 20' 42'.

In this way the proportional part of the variation of the Sun's declination may
be found to the nearest second. And in the same manner the proportional part
of the increase of the Sun's right ascension in 24h. may be found to the nearest
half second, for any given rime past noon.

EXAMPLE.

Suppose the increase of the Sun's right ascension in 24h. is 3m. 55s. what
would be the proportional part for 17h. 15m.?

Thirty degrees of aUitude on the slide being set to 24' of correction, and a
mark made on the slide opposite to 17' 15quot; (= 17h. 15m. the time past noon),
and then 30° of ahitude set to 3m. 55s., (the increase of right ascension in 24h.)
the mark on the slide will be opposite to 2' 49quot;, or 2m. 49s,, which is the pro-
porrional part required for 17h. 15m.

II. To find the proportional part of the variation of the Moon's semidiameter
or horizontal parallax in 12h. for any given time past noon or midnight.

RULE.

Set 10° of altitude on the slide to 12' of correction, and make a mark on the
slide opposite to the given time past noon or midnight, (esteeming the minutes of
correcrionas hours, and the seconds as minutes of time as before), then set 10°
of alt. on the slide to the variation of the Moon's semidiameter or hor. par. in
12h., and opposite to the mark on the slide will be found the proporrional part
required. Here the minutes of correction are to be used as seconds of variation.

EXAMPLE.

Let the variarion of the Moon's hor. par. in 12h. be 19quot;, what would it be in
7h. 40ra. ?nbsp;^

Here 10° of alt. being set to 12' of correction, and a mark made on the slide
opposite to 7'40quot; (or 7h. 40m.), and then 10° of alt. set to 19' (or 19quot; of varia-
tion) opposite to the mark on the slide is 12' nearly, which is to be called 12quot;, the
proportional part required.

It may be observed that any other particular division on the line of altitudes
Jiiay be used in place of 30° or 10°, these being merely used as appearing the
. ttost convenient in the foregoing problems for setting to 24' of correction, this
being esteemed 24h., or to 12' esteemed as 12h. These proporrional parts may
also be found on the line of correction with a pair of compasses, in a manner
^nich must be obvious to any person acquainted with the use of the line of
lumbers on Gunter's Scale.

h. m.

Estim. Astron. time at Ship, 4th July 2 20
Longitude in time, W.nbsp;- 1 21

Sun's Declin. for Greenwich time 22 54 0 N.
Moon's Semid. for do. do. 15 0
Moon's Hor. Par. for do. do. 55 3

Estim. time at Greenwich 'nbsp;3 41

Obs. dist. Sun and Moon's nearest limbs 59 5T 00
Sun's Semidiameter - - - 15 46
Moon's Semidiameter - - - 4- 15 0
Moon's Augmentation of semid. -4-14

Obsd. alt. Moon's up. limb 65 IS
Semid. dip . . — 19

Moon's Apparent Alt. 64 54

Apparent distance - - - - 60 28 00
Sun's App. alt. 52° 25' Comp. 1st corr. 1 9 52
Moon's App. alt. 64 54 Second corr. 4- 28 14
From Table VI.nbsp;Third corr.

1 1

Sum — 2° = True distance
Dist. in N. A. at III.
Dist. in N. A. at VI.

First difference
Second difference

App. time at Greenwich -
App. tinre at Ship

Longitude in time

Here the true distance is found in the manner explained at page 8, so as to
make all the corrections additive to the apparent distance; As the true distance
is greater than the distance in the Nautical Almanac at 3h. and less than the
distance at 6h., the apparent time at Greenwich must be greater than 3h. and
less than 6h., and the time after 3h. may be found by the Rule of Three : Thus,
As the difference of the two distances taken from the Nautical Almanac is to 3h.,
so is the difference between the true distance and the first distance taken from
the Nautical Almanac, to the Proportional part of time. The foregoing example
would stand as follows, As 1° 24' 17quot; : 3h. : : 0° 18' 56quot; : Oh. 40m. 26s. But
this proportional part is got with much more ease by the excellent table of Pro-
portional Logarithms which was constructed by Dr. Maskelyne, particularly for
that purpose. In this, as well as in the following examples, the time over the
first distance taken from the Nautical Almanac is added to the proportional part
of 3h. as found by Proportional Logarithms, which gives the apparent time at
Greenwich when the observation is made.

The apparent time at the ship is found from the Sun's altitude by Problem H.,
and this time being less than the time at Greenwich by Ih. 19m. 10s., the Longi-
tude at the place of observation is therefore 19° 47' 30quot; W.

The mode of finding the proportional part of 3h. by Proportional Logarithms
is perhaps the best that can be given for the purpose. But this proportional
part may be very easily found by the Scale, as follows:

rule.

s ä toun-inbsp;pquot;p«'»»«1 p*quot; »'

example.

Let the first difference be 0° 18' 5J -a the second difference P 24' Hquot;,
what would be the proportionalnbsp;„f ..section,

Set 90° of altitude to the «fis opposite to the first difference
and make a mark at the po'nt ^ tnenbsp;correction, and opposite to the

(18'56quot;); tben put 90 of alUtud^enbsp;^^^^^^ ^^nbsp;o^

mark on the slide wiu d« rpouired

40m. 26s. the proportional partnbsp;^^nbsp;to set

When the second difference ^^^^ !nbsp;he slide being maked op-
90° of altitude to half the ««c^ond dito

posite to the first difference setnbsp;part of 3h. as before,

to the mark on the slide will be halt the propnbsp;i-

example n.

Sun'sDeclin. for Greenwich time 22 8 0 N.
Moon's Semid. for do. do- 16 3
Moon's Hor. Par. for do.nbsp;do. o8 o3

Obs. dist. Sun and Moon's nearestlimbsnbsp;^ 00

Sun's Semidiameter - - -nbsp; 3

Moon's Semidiameter - quot; quot;nbsp;J ^^

Moon's Augmentation of semid.nbsp;-nbsp; __^

, . - -nbsp;103 36 31

Apparent distance - ^ ^nbsp;^ 27

.nbsp;103 3 21

.nbsp;102 6 32

,nbsp;103 43 30

Obsd. alt. Moon's up. limb 45 54

Semid. diP • ' ~
Moon's Apparent Alt. 45 35
o '

Obsd. alt. Sun'slowerlimb 24 43
Semid. — dip - -

Sun's Apparent Altitude 24 56
Sun's Corr. in Altitude — ^

h. m.

Estim. Astron. time at Ship, 5th June 12 30
Longitude in time, W.nbsp;- 2 27

Sun's R. A. at noon, 5th June
Add for 14h. 5Tm.

14 57 Sun's R. A. at time of observation 4 53 56

Moon's Hor. Par. at Greenw. time 54' 18quot;

o I 'I

Obs. dist. of Star from Moon's far. limb 67 39 10
Moon's Semid. at Greenw. time 14'47quot; ) n cc
Augmentationnbsp;- - 9 5quot;

Obs. alt. Moon's lower limb 37 25
Semid. — dip - n

Moon's Apparent altitude 37 36

Apparent distance - -nbsp;67 24 14

Star's App. alt. 21° 14'Comp. Istcorr. 1 38 42
Moon's App. alt. 37 36 Second corr. 13 47
From Table VI.nbsp;Third corr. 1 42

Obs.alt.ofSp.Vir.Wofmer,2I 18
Dip - -nbsp;—4

Star's Apparent altitude 21 14
Refraction - - — 2

Sum — 2° True distance
Dist. in N. A. at XII, h.
Dist. in N. A. at XV. h.

First difference
Second difference -

App. time at Greenwich
App. time at Ship

Longitude in time

67 18 25
65 52 24
67 22 13

1 26 1 P.L.
1 29 49 P.L.

h. m. s.

14 54 22 - - P.L.
12 27 24

Star's True altitude
Star's Polar distance
Latitude

3207

3019 Sum -

- Half Sum -

0188 Difference -

Sum — 24h. App. time 12 27 24

Note. Tables of the Distances of the Moon from the Planets Venus, Mars,
Jupiter, and Saturn, have been lately published at Copenhagen, under the direc-
tion of the celebrated Schumacher, Professor of Astronomy at the University of
that place. These Tables are adapted to the meridian of Greenwich, and the
directions for their use are in English.

An apparent distance of the Moon from a Planet is reduced to the true distance
in the same manner as a distance between the Moon and a fixed Star, execpt the
small correction for the effect of the Planet's parallax on the distance. This cor-
rection may be found as follows: With the apparent altitude of the Planet used,
as if the distance were between the Sun and Moon, take the eifcct of the Sun's
parallax from Table P, multiply this by the horizontal parallax of the Planet, and
divide the product by 9, tlie quotient will be the effect of the parallax of the Planet
on the distance, which is to be added to, or substracted from the Third Correction,
(or the apparent distance) according as the effect of the Sun's parallax should be
applied for the same distance and altitudes. The horizontal parallaxes of the
Planets are given in Schumacher's Tables, and also their Right Ascensions and
Declinations, so that the Apparent Time may be very correctly ascertained from
the altitude of a Planet.

on finbing the

longitude at se

Ry Chronometers.

In finding the Longitude by a Time-keeper, it is necessary that its error for
mean time at the meridian from which the Longitude is to be reckoned should
be known, at the time the Longitude is to be found, which error being applied
to the time shown by the Chronometer when an observation is made for the pur-
pose of finding the time at the Ship, will give the mean time at the first meridian
at the instant of observation. The diff'erence between this time, and the mean
time found at the ship, will be the Longitude in time, which will be East or West
accordmg as the time at the Ship is greater or less than the time at the first
meridian. Hence, if Time-keepers could be made to go exactly to mean time,
or to measure an equal portion of time during every mean day, no other method
of finding the Longitude would be required. But it is well known that even the
best Chronometers are liable to alter their rate of going; therefore entire de-
pendance should not be placed on any Time-keeper during a long voyage. A
Time-keeper is, however, of great utility to the practical Navigator both in long
and short voyages; for during a long voyage its error can be often ascertained
by Lunar Observations, and then the Longitude may be correctly found by the
watch when observations of the Lunar Distances cannot be had for that purpose.

In the following Examples the apparent time is found by Problem 11. or III,
according as the time is to be inferred from the altitude of the Sun or a Star,
The declination of the Sun, and the equation of time, may be taken from the.
,nbsp;A'l^anac for the time at Greenwich, as given by the Chronometer when

the observation is made. But when the Sun's right ascension is to be found,
the mean time at Greenwich should be reduced to the apparent time, by applying
to It the equation of time, with a contrary sign to that which is given in the
Nautical Almanac, and then the Sun's right ascension is to be found to the
Greenwich apparent time. This may be done by the Scale, as explained at
page 15. The necessary correction of the Sun's declination, and also that of the
equation of time, may generally be made at sight with sufficient exactness, or the
proportional parts of the variation of these in 24 hours may also be found bv
the Scale.nbsp;^

EXAMPLE I.

Suppose on the 2d September 1821, about 3h. 30m. P. M. nautical time, in
f-atitude 28° 42' N., and Longitude by account 38° 30' W. that the altitude of
tne Sun's lower limb is 37° 2', and the time by a Chronometer when the altitude
IS observed is 4h. 53m. 21s., the error of the Chronometer for mean time at

Greenwich Ih. 8in. 34s. slow; height of the observer's eye 16 feet: required
the longitude of the ship.

h. m. s.

Time of obs. by Chronometernbsp;4 53 21

Chron slow for M. T. at Greenwich 1 8 34

Obs. alt. of Sun's lower limb 37 2
Sun's Semid. —dip and corr. of alt. 11

M. T. of obs. at Greenw. 1st Sept. 6 1 55 Sun's True altitude
Sun's declin. at Greenwich time 8° 14' N. Sun's Polar distance

Latitude

Sum

Instead of applying the equation of time to the apparent time at Ship, it may
be applied with a contrary sign to the mean time at Greenwich, which will give
the apparent time at that place ; the difference between which, and the apparent
time at Ship, will be the Longitude in time.

EXAMPLE IL

On the 9th December 1821, at about 4 P. M. nautical time, in Latitude
36° 6' S., and Longitude per account 53° E., the observed altitude ot the Sun's
lower limb was 35° 13', at the same instant the time by a Chronometer, which
was fast for mean time at Greenwich Oh. Im. 24s., was OL 30m. 4s.; height of
of the eye 12 feet: required the longitude of the ship at the time of obser-
vation.

Sun's declin. at Greenwich time

Time of obs. by Chronometer
Chron. fast forM.T. at Greenwich

M. T. of obs. at Greenwich 8th Dec.
Equation of time (for sub.) add -

Appareat time at Greenwich
Apparent time at Ship

Longitude in time

22 45 S. Obs. alt. of Sun's lower limb

35 13

3 30 51 = 52° 42'45quot; E.

EXAMPLE in.

Julv 7 1822, about 9h. 40m. P. M. nautical time, in Latitude 35° 32 N. and
Longilude by account 10° W. the observed altitude oi Arctums west of the me-
ridi4 was 52° 40' when the time by a Chronometer was 12h. 11m 22s the
error of the Chronometer for Greenwich mean time Ih. 46m. 24s. last, and the
height of the observer's eye 17 feet: required the longitude of the ship.

Time of obs. by Chronometer -
Chron. fast for M. T. at Greenwich

Mean time of obs. at Greenwich -
Equation of time (for add in N. A.) sub.

Apparent time of obs. at Greenwich
h. m. s.

Comp. ofSun'sR.A.(ta24h.)16 58 24
Star's right ascensionnbsp;14 7 35

Star's meridian distance 2 35 45

Sum— 24h. = App. time
at Ship

Longitude in time

In this Example the apparent time of observation at the ship is found by
Problem III. to be 9h. 41m. 44s.; the difference between this and the apparent
time at Greenwich is Oh. 39m. Is.: hence the Longitude 9°45 15 W., the time
at Greenwich being farther in advance than the time at ship.

It is much to be desired that Navigators would attend more to observations of
the Stars, both for finding the time at ship and the latitude. But many, who
are in other respects good Navigators, scarcely know a star in the heavens, and
are thereby prevented from obtaining the latitude and longitude of the ship
during the night, when it may be of the utmost consequence that they should be
known.

' Captain Thomas Lynn has lately published very extensive and useful Tables
of the fixed Stars, which Navigators will find of great service to them in deter-
mining both the Latitude and Longitude.

explanation of the tables.

TABLE L

Refraction of the Heavenly Bodies in Altitude.

This Table contains the mean Refraction, which must always be subtracted
from the apparent altitude of an object when it is necessary to find the true al-
titude. For example, if the apparent altitude of a fixed Star be 12°, its true
altitude will be 11° 55'37quot;, the refraction for 12° of apparent altitude being

TABLE II.

Depression or dip cf the Horizon of the Sea.

The dip is to be subtracted from the observed altitude in order to find the ap-
parent altitude of an object, to which the refraction in altitude is to be applied.
In the examples in this boolc, where the apparent altitude is not required, the
^nbsp;sum of the dip and refi-action is subtracted from the observed altitude of a Star,

jnbsp;which gives its true altitude. When the altitude of the Sun's lower limb is ob-

;nbsp;served, the true altitude may be found by applying the differenee between the

Inbsp;Sun's semidiameter, and the sum of the dip and correction of the Sun's altitude,

to the observed altitude. The true altitudes need only be found to the nearest
minute when the operation of finding the Time is performed by the Scale.

TABLE III.

The S-un's Parallax in Altitude.

The Sun's Parallax in Altitude being taken from the Refraction in Altitude,
leaves the Correction of the Sun in Altitude. Thus, if the Sun's apparent alti-
tude be 12°, the Correction in Altitude will be 4' 23quot; — 9quot; z: 4' 14quot;.

TABLE [V.

Moon's Augmentation.

This Table is to be entered with the iVIoon's semidiameter, as found by the
Nautical Almanac, at the top, and the Moon's apparent altitude in the side
column, under the former, and opposite the latter is the Augmentation of the

Moon's semidiameter. For example, When the Moon's semidiameter in the
Nautical Almanac is 15' 40quot;, and her apparent altitude 57°, the Augmentation
is 13quot;.

TABLE V.

To find the Second Correction when the Apparent Distance is between
88° 40' and 91° 20'.

The explanation of this Table is given in the page opposite to it.

TABLE Vr.

Third Correction.

This correction is always additive to the Apparent Distance. The given Ap-
parent Distance is to be found in the Table ; then look for the apparent altitude
of the Sun or Star at the top, and the Moon's apparent altitude in a side column
under the former, and opposite to the latter is the third correction. For ex-
ample, When the Apparent Distance between the Moon and a Star is 84°, the
apparent altitude of the Star 46°, and that of the Moon 32°, the third Correc-
tion is 1' 50quot;.

The small Table, which is titled Table P., contains the effect of the Sun's
parallax on the distance; this never exceeds 9quot;, and it is to be added to, or
subtracted from, the third correction, according to the direction at the top of
the Table. Thus, at Apparent Distance 48°, the Sun's altitude 50°. and the
Moon's 10°, the effect of the Sun's parallax on the distance is 4quot; to be added to
the third correction, because it is found above the line in the column. But at
the same distance, if the Moon's apparent altitude be 50°, and that of the Sun
10°, the effect of the Sun's parallax on the distance is 8quot; to be subtracted from
the third correction, because it is found below the line in the column. When
the apparent distance is above 82° the effect of the Sun's parallax is always
subtractive from the third correction.

In taking out the third correction from this Table, when the given distance or
altitudes differ considerably from those in the Table, it is generally necessary to
make a proportion in the third correction. For example. Let the Apparent Dis-
tance be 45° 50', the Star's apparent altitude 50°, and the Moon's 30°. Here,
at Apparent Distance 44°, the third correction for the given altitudes is T 12quot;,
but at Apparent Distance 48°, for the same altitudes, the third correction is
T 16quot;. Hence the third correction for Apparent Distance 45° 50', is T 14quot;.

TABLE VII.

Proportional Logarithms.

quot;^his Table is given for the purpose of finding the Apparent Time at Greenwich
answering to a given True Distance between the Sun and Moon, or between the

Moon and a Star. For example, Let the given True Distance be 63° 18' 25quot;
when the distance of the same objects in the Nautical Almanac at 6h. is
62° 31' 40quot;, and at 9h. 64° 6' 15quot;, what would be the apparent time at Green-
wich ?

0 I II
True distance 63 18 25
Distance at 6li. 62 31 40
Distance at 9h. 64 6 19

First difference 0 46 45 P. L. 5855
Second difference 1 34 39 P. L. 2T92

h. m. s. ---

Time past 6h. 1 28 35 P. L. 3063
6

App. time at Greenwich 7 28 55

This Table is very useful in many other Problems in Nautical Astronomy,
which the reader may see exemplified in the different books on that subject, par-
ticularly in Mackay's Longitude.

TABLE VIIL
To turn Motion into Time, or Time into Motion.

The principal use of this Table is to turn Longitude into Time, or Time into
Longitude. The method of doing this will be obvious from the followmg ex-
amples :

I. What Time answers to Longitude 98° 45' 30quot; 1

o I IInbsp;h. m.nbsp;s.

98 0 0 =nbsp;6 32nbsp;0

45 0nbsp;=nbsp;3nbsp;0

30 =nbsp;2

Time requirednbsp;- - - 6 35 2

II. What Longitude answers to 7h. 42m. 28s. ?

h. m. s. °nbsp;'nbsp;quot;

7 40 0nbsp;= 115 0nbsp;0

2 0nbsp;=nbsp;30nbsp;0

28nbsp;=nbsp;7nbsp;0

Longitude required - - 115 37 0

TABLES r. IL in. IV.

TABLE I.

The Refractions of the Heavenly
Bodies on Altitude.

TABLE IV. Moon's Augmentation.

EXPLANATION AND USE OF TABLE V.

When the Apparent Distance between the Sun and Moon, or between the
Moon and a Star, falls between 88° 45' and 91° 15', the second correction
cannot be found by the scale, but must be found by Table V, which is very
easily done to the nearest half second as follows :

Rule. To the Log. of the Moon's hor. par. add the Log. of the Moon's ap-
parent altitude, and the Log. of the apparent distance ; the sum will be the Log.
of the second correction.

EXAMPLES.

1. Suppose the Moon's hor. par. is 56' 40quot;, and her apparent altitude 37° 30',
when the apparent distance is 89° 5': required the second correction.

Moon's hor. par. 56' 40quot; - Log. 0.042
Moon's app. alt. 37°30' - Log. 0.676
Apparent distance 89 5 - Log. 2.796

Second correction 0' 33quot; - Log. 3.614

IL Let it be required to find the second correction when the Moon's hor. par.
is 60' 25quot;, apparent altitude 57° 38', and the apparent distance 90° 55'.
Moon's hor. par. 60' 25quot; - Log. 0.014
Moon's app. alt. 57° 38' - Log. 0.534
Apparent distance 90 53 - Log. 2.812

Second correction 0' 47quot; - Log. 3.360
Or the complement of the second correction, will be 1° 59' 13quot;.

J^^ss^^tsstsaegsmsa^^^mmamti^

To find the Second Correction, when the Apparent Distance is between
88° 40' and 91° 20'.

Logs, of 5'» Apparent AUitude.

Logs, of App. Dist.

I Loffs. of 2nd Correction, '

Log.

0.055
054
05a
051
050
048

App. Dist.

Conjp. of 2nd Cor.