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A
NEW AND ACCURATE METHOD
OF
FINDING A SHIP'S POSITION AT SEA,
BY PROJECTION ON MERCATOR'S CHART,
WHEN THE LATITUDE, LONGITUDE, AND APPARENT TIME AT THE SHIP ARE UNCERTAIN; ONE
ALTITUDE OF THE SUN, WITH THE TRUE GREENWICH TIME, DETERMINES,
FIRST,
THE TRUE BEARING OF THE LAND;
S E CONDLY,
THE ERRORS OF LONGITUDE BY CHRONOMETER,
CONSEQUENT TO ANY ERROR IN THE LATITUDE;
THIRDLY,
THE SUN'S TRUE AZIMUTH.
WHEN TWO ALTITUDES ARE OBSERVED, AND THE ELAPSED TIBIE NOTED, THE TRUE LATITUDE IS
PROJECTED; AND IF THE TIMIES BE NOTED BY CHRONOMETER, THE TRUE.
LONGITUDE IS ALSO PROJECTED AT THE SAME OPERATION.
THE PRINCIPLES OF THE METHOD BEING FULLY EXPLAINED AND ILLUSTRATED BY
PROBLEMS, EXAMPLES, AND PLATES,
Bitt |N~uks fr   radifat, teg  a 0 5nirpbl fnromt |Aitl 0m bsulbiltl
BY CAPT. THOMAS H. SUMNER.
THIRD EDITION, REVISED.
BOSTON.
PU]BLISHED BY:1  TfIOMAS GROOM & CO., 82, STATE STREE1IT,
1851.




LETTER FROM CAPTAIN-W. HOMANS.
SHIP THOMAS H. PERKINS, at Sea, Nov. 9, 1844.
Messrs. Thomas Groom & Co.
GENTLEMEN: Agreeably to promise, I now write my brother, G. B. Wellman,
to pay you one dollar for the work of Captain Sumner, and never have I spent a
dollar with more satisfaction, or ever got more value received for the money. I
have now given the work a fair trial; and am perfectly satisfied that it is the
greatest discovery in the art of Navigation since the discovery of the longitude by
chronometer, by Harrison, almost a century ago, and places Capt. Sumner, if not
equal, certainly only second, to Nathl. Bowditch, and I look upon it as what has
been for a long time among the desiderata of Navigation. It has been a source of
anxiety to the Navigator, to ascertain his latitude when meridian altitudes could
not be got. The ordinary modes of double altitudes for that purpose, published in
most works on Navigation, have not always answered the purpose sufficiently to
entitle them to implicit confidence. The work of Capt. Sumner has supplied a
remedy; for any one with a reasonable knowledge of the rules and art of Navigation, can ascertain his latitude by following Capt. Sumnerl' rules, whether he has
a chronometer or not; and if he has one, he can ascertain his latitude and longitude
at any time when the sun, moon, or stars are visible, and there is a clear horizon.
At the present time but few vessels are navigated without a chronometer, and Capt.
Sumner has contributed to increase the value of that instrmiaent four fOld.
I have not the honor of a personal acquaintance with Capt. Sumner, but he is
one of those web-foots whose acquaintance I should feel an edifying pleasure in
cultivating; at the same time, I am proud to see and acknowledge that this discovery has been made by an American ship-master. This work should be in the
hand of every master of a vessel, aye, and mate too, and should bathe never-failing
accompaniment of Bowditch's great work on navigation.
Wishing you and Capt. Sumner every success in the sale of this work which its
merits entitle it to,
I am your very humble and obedient servant,
WILLIAM HOMANS.
Entered according to Act of Congress, in the year 1843,
By CAPT. THOMAS H. SUMNER,
In the Clerk's Office of the District Court of the District of Massachusetts.
BOSTON:
WILLIAM CHADWICK, PRINTER,
18, EXCHANGE STREET.




RECOMMENDATIONS.
The following letter from Professor PEIRCE, of Harvard College, Cambridge, to
J. INGERSLL BOWDITCH, Esq., President of the American Insurance Company,
is published by permission.
CAMBRIDGE) 16th March, 1843.
My dear Bowditch, —I have examined Capt. Sumner's processes, and they are
founded on perfectly correct principles. I think his methods are especially valuable, because they require but one formula; and the Geometry is so simple and
obvious, that it can easily be made intelligible to any man of good sense.
Yours, truly,
(Signed)                                  BENJAMIN PEIRCE.
In obedience to a resolution adopted at the last stated meeting of the "Naval
Library and Institute," in the following words, to wit: "Resolved, That a Committee of three members be appointed to investigate the'New Method of Finding
a Ship's Position at Sea,' by Capt. Sumner, and report at the next meeting."
The Committee respectfully submit the following REPORT:
We have carefully examined the subject referred to, and find that Capt. Sumner's
method of ascertaining " The Bearing of the Land; of finding a Ship's Position by
pr9jecting two of those Bearings, and of projecting the Sun's true Asimuth, &c.,"
are all founded-on spherical principles, as applied to Nautical Astronomy.
And your Committee is of opinion, that in practice, Capt. Sumner's discovery
(for we can call t nothing else) will prove a useful auxiliary to the present knowledge of Navigators, and, as such, would recommend it to the attention of all
persons interested in the promulgation and improvement of Nautical Science.
(Signed)        JAMES ALDEN, Lt. U. S. Navy,             )
SAMUEL R. KNOX, Lt. U. S. Navy,.Committee.
GEO. H. PREBLE, Passed Mid. U. S. N. $
"Naval Library and Institute,"
Navy Yard, Boston, April 30th, 1843.
I certify the above to be a true Copy of the Report,
(Copy)         (Signed)           W. WHELAN, Recording Secretary.
Navy Yard, Boston, 9th May, 1843.
Extract from the Nautical Magazine, published monthly at London, of an extended
notice of this work, by H. Raper, Lieut. R. N., attached to the Royal Hydrographical Department, and author of a celebrated practical treatise on Navigation:
"This Method consists in a new use or application of a single altitude, observed
for the longitude by chronometer; and as the suggestion is highly ingenious and
very useful when the ship is near the land, it will be rendering a service to our seamen, few of whom can yet have heard of it, to make it known to them in the pages
of the Nautical Magazine."
"As this Method arises out of the employment of the chronometer, it may be
said greatly to enhance the utility of that instrument."




4
Letter from Captain STURGIS.
BOSTON, July 8th, 1843.
Captain Thomas H. Sumner, Sir: I have examined with much care and attention your "New Method of Finding a Ship's Position at Sea," which I consider a
valuable discovery. I have no knowledge of any nautical work that directs otherwise than that observations be taken when the sun bears East or West; therefore
I think your Method, when published, will be universally adopted by navigators.
I therefore recommend it to all who feel an interest in the promotion of nautical
science, as well as practical navigators. Wishing you every success in your
undertaking,     I am, very respectfully, your obedient servant,
(Signed)                           JOSIAH STURGIS,
Captain U. S. Revenue Cutter Hamilton.
Letter from Capt. ALEXANDER V. FRASER, R. S.
Bureau of Revenue Marine, Treasury Department,
Dear Sir:                                   WASHINGTON, Oct. 31, 1843.
I have tested the accuracy of your methods, by calculations made from my own
observations, and cheerfully bear testimony that the system is based on correct
principles, and the value enhanced by the extreme simplicity.
I am Sir, very respectfully, your obedient servant,
(Signed)         ALEXANDER V. FRASER, Capt. Rev. Service.
Extract from a letter of W. W. Hunter, Lieutenant U. S. Navy.
Dear Sir:                                  WASHINGTON, Nov. 10, 1843.
Your book contains self-evident proof of the propriety of its title; exhibiting
indeed with beautiful simplicity a new and accurate method of finding a ship's
position at sea.      *         *        *         *         *        *
3*     *      *     Estimating your labors to have been highly serviceable
and honorable to our country,
I am, very respectfully, your obedient servant,
(Signed)                W. W. HUNTER, Lieut. U. S. Navy.
Extract of a letter from M. F. MAURY, Lieut. U. S. Navy.
Hydrographical Office, WASHINGTON CITY, October 9, 1843.
Sir: I have examined your book, entitled "A New Method of Finding a
Ship's Position at Sea," and think highly of it. It may be considered as the
commencement of a new era in practical navigation.
An order has been given to supply every ship in the Navy with it.
Respectfully, your obedient servant,
(Signed)                         M. F. MAURY, Lt. U. S. Navy.
Extracts from letters of Lieut. MCLAUGHLIN, U. S. N.
WASHINGTON CITY, September 14, 1843.
Dear Sir: I have examined your book very carefully, and look upon it as one
of the most valuable acquisitions which have been added recently to the science of
Navigation. It has supplied a want which hap long been felt by the navigator,
and should be in the hands of every man intrusted with the care and safety of a
ship.
I cannot too strongly recommend it as a work which should be in the library of
every ship in the Navy. I am, very respectfully,
Sir, your obliged and obedient servant,
(Signed)                   JOHN J. McLAUGHLIN, Lt. U. S. N.




CONTENTS.
Page.
TO THE READER,........................................   7
NOTE AND EXAMPLE BY T. H. SAFFORD, JR.,..................   10
I.  THE PRACTICE OF THIS METHOD,....................   12
To find the bearing of the land, &c., by one altitude, at any hour,  14
To find the latitude and longitude by chronometer, by two altitudes, at one operation,.................................  21
To find the latitude by two altitudes; times noted by " watch,"..  27
To find the latitude and longitude by chronometer without pro-  29
jecting,..............................................  31
How to allow for possible errors of latitude,..................  32
Table of the preference which obtains among any sets of double
altitudes,...........................................  39
Table and Rules, showing errors of longitude, &c.,............  43
II.  THE ADVANTAGES,........................................     46
III.  THE PRINCIPLES,...........................................  52
Explanation of them,.....................................  52
To project a parallel of equal altitude in latitude,..............  56
To project two parallels, in latitude, showing their intersections,.  60
To project a parallel, in latitude and longitude by chronometer,
showing the bearing of the land,..........................  62
To project two parallels, in latitude and longitude by chronometer,
showing the true latitude and longitude,....................  64
To project two parallels on a terrestrial globe, showing the latitude,..............................................  65
To project the change of station between two observations,......  66
To project the sun's azimuth,............................  67
To project errors of longitude, &c.,..........................  68
To project any arc of a parallel of equal altitude,..............  69
To project two arcs, change of station, the latitude, &c.,.......  70
IV.   APPLICATION TO MOON, PLANETS, AND FIXED STARS,.............  79
The most simple method to do this, in most cases,............  80
Another process,.......................................  80
V.   CURRENTS IN THE GULF AND STREAM OF FLORIDA,..............   89




NOTICE TO THE SECOND EDITION.
THE present edition has been carefully revised; a few pages
have been introduced upon the preference which obtains among any
sets of double altitudes which can be observed; the attention of the
reader is therefore directed to the tables A, B, and C, with the
remarks annexed.
NOTICE TO T         TH THIRD EDITION.
THtI edition has also been carefully revised, and several errors
of the printer have beet corrected. Two additional pages have
been furnished, written by T. H. Safford, Jr., of Cambridge, which
the publishers trust will add to the value of this already celebrated
work on navigation.




TO THE READER.
IT is not so much the object of this work to present the navigator
with a new method of "double altitudes," as to afford him an accurate method of finding, by one altitude of the sun, taken at any hour
of the day, with the chronometer time, the true bearing of the land,
when the latitude, longitude, and time at the ship, are from any
cause uncertain; and to place him on his guard, when near a
dangerous coast, (and all coasts are dangerous when the latitude is
not accurately known,) against those errors of longitude by chronometer,
which arise from an'erroneous latitude used in finding the apparent
time at the ship; directing, particularly, his attention to the fact, as
shown in these pages, that when the latitude is uncertain, a single
altitude of the sun, at any time of day, when not less than say 7~
high, is, with a good chronometer, as useful as a MERIDIAN OBSERVATION for the latitude; and the errors above alluded to are
rendered apparent.
When a single altitude is thus calculated, one half of the calculation of a double altitude is finished; and a second altitude furnishes
all that can be desired, both latitude and longitude.
Hence, it seemed proper to explain "double altitudes" by this
method of projection; and it is believed, that it has the advantage
over any other method, in the simplicity of the calculation, and in
the fact, that ship-masters universally understand, and daily practice,




8
the numerical calculation, namely, that of finding the apparent time
at the ship, which is the only one used.
Many navigators, having taken morning sights for the chronometer, and supposing the observation useless without "the
latitude," wait for the meridian observation, in order to deduce the
longitude by chronometer; or, if the sun be obscured till afternoon,
think a single altitude under such circumstances is of small value;
and, by the common methods, with good reason; for then the
latitude by dead reckoning from the preceding noon must, in
general, be used to find the apparent time at the ship; and here is
the source of error; because, 26 to 30 hours having probably
elapsed, in such time the ship may have sailed from 250 to 300
miles; if two days have passed without a meridian observation or
double altitudes, then from 400 to 500 miles, in frequent cases;
and while sailing so far, a small current, with a small error in the
log, (if one is used, but they are much out of fashion in these days,)
will easily make an error in the distance of half a mile an hour,
and of half a point in the course, and cause an error in the latitude
by dead reckoning, and consequently in the longitude by chronometer.
Suppose the course and distance sailed by log to be east 300
miles, from a place in lat. 50~ N., when the sun's declination is
23'28 S.; but, from accidental causes, the true course and distance
sailed is 315 miles E. i N.; the difference of latitude is, in this
case, 30'9 miles; but the dead reckoning gives 50~ N. for the latitude to be used in finding the apparent time at the ship, if an
observation should be now taken; but, with the most favorable
altitude, (which is the one that is nearest to the east or west points,
when observed,) the least error of longitude by chronometer is, in
this instance, 1~ 2'; and with other, greater altitudes, it may be
many times as much, even when the chronometer is strictly accurate.
None of the works on navigation within the writer's knowledge,
exemplify or even hint at this important source of error, but merely




9
direct the observations to be taken when the sun bears as nearly
"east or west as possible;" but it is impossible, for nearly seven
months in the year, to observe the sun in east or west points.
It is hoped, that the method by projection, which explains these
errors, and renders a single altitude, taken at any bearing of the
sun, available, in a similar manner as a meridian observation, will
supply a want which every practical navigator must have frequently
experienced.
The latter pages of this treatise are intended to show the principles upon which this method depends; and the problems there
solved, and the examples given, merely as illustrations of those
principles, in the order in which they are explained.
It has been endeavored, to explain them in such a manner, that
practical persons may be led, step by step, to the entire understanding of the theory.
Notwithstanding these might prove amply sufficient to enable
many readers to put the principles in practice, yet, since there may
be persons who might not be inclined to give them the attention
necessary for this purpose, the more Practical Rules are first introduced, so that all such persons may proceed at once to the adoption
of this method.
The present seems a fit opportunity to offer the chart relating to
the currents in the Gulf of Mexico, at the end of the book; if it
serve no better purpose, it will put the mariner on his guard, in
those parts so fatal to the commercial interests.
[2]




NOTE AND EXAMPLE.
BY T. H. SAFFORD, Ja.
CAMBRIDGE, July 28, 1851.
Messrs. Groom  Co.:
Gentlemen,-I have examined Capt. Sumner's method,
and made the annexed remarks upon it. I find it to be
founded on correct principles, and it seems to me likely to
become generally useful.
Yours respectfully,
T. H. SAFFORD, JR.
ON SUMNER'S METHOD OF FINDING A SHIP'S PLACE AT SEA.
It is always best to take the observation, if but one is
taken, when the bearing of the sun is near eight points from
the course you are sailing. Also, an observation should not
be taken too near noon, as the bearing of the land may
be much affected by small errors in the altitude. The
following rule may be useful for finding the bearing of the
land.
RU LE.
Calculate the longitude from the altitude, using the latitude by
account, instead of the two nearest degrees, as given by Capt.
Sumner; also, when you take out the natural cosine of the sum or
difference of the latitude and declination, take out its natural sine,
and when you take out the log rising from Table xxiii. take the




11
log. half elapsed time. To the log. rising add the log. sine of the
latitude and the log. cosine of the declination, which will give a
logarithm, which call A.
Find the number corresponding to A, which subtract from the
nat. sine of the sum or difference of the lat. and dec. before taken
out, unless the latitude is of the same name as, and less than, the
declination, when add it to this nat. sin. Take the log. of this last
sum or difference, and add it to the log. half elapsed time, together
with the log. secant of the lat. and the log. secant of the dec. The
sum, adding 5 00000, will be the log. cotangent of the sun's
bearing from the north.
It will be known in which quarter the sun was when the observation was taken, by the common rules, or by the compass. Project
the place found by the latitude by account, and from this the bearing
of the sun.   Then draw a line at right angles to this, which will
be the bearing of the land.
E x A M P L E.-Same as Capt. Sumner's First.Example.
Lat. 510 37' N.                    sec. 0-20696            sec. 0-20696
Dec. 23 23 S.                     sec. 0 03722             sec. 0'-0722
Sum. 75 00        nat. cos. 25882             nat. sin. 96593
OAlt. 12 10       nat. sin. 21076
H. M. 8.
12 00 00        diff.  4806      log. 3-68178
1 34 48                    log. rising 3-92596   log. H. El. T. 0-39565
10  25  12                  sin. lat.  9-89425
-    3 37                    cos. dec. 9-96278
10 21 35                            3-78299  num.  6068
10 47 13 chro. time.                              90525  log. 4-95677
T-5B660
Lon. 25' 38" = 60 24/        0( bear. S. by E. I S. = S. 140 12' E. cot. 10'59660
The log. cotangent of the sun's bearing is the log. of the error
of long. for one mile of lat.: in this case =  3 miles 9 tenths.




I.
THE   PRACTICE   OF THE METHOD BY PROJECTION,
WITH EXAMPLES FROM ACTUAL OBSERVATION.
IT is a direct inference from the principles herein stated,
that there is, by the common rules, but one proper instant
in one day, namely, when the sun bears north or south, at
which a single observation to find the latitude can be taken
with a correct result, unless the apparent time at the ship is
accurately known.
And when the latitude is uncertain, that there are only two
proper instants in one day, namely, when the sun bears east,
and when he bears west, at which his altitude can be taken
to find the longitude by chronometer with accuracy.
All deviations from these bearings of the sun, at the'time
of observation, (in such circumstances,) are attended with
errors of latitude or longitude, proportional to the angular
distance of the sun from these points; and these errors are
frequently very great.
To remedy this difficulty, and render a single altitude of
the sun, taken at any angle from the meridian, or from
the east and west points, available, when the latitude and
apparent time at the ship are, from accidental causes, uncertain, (the time of observation by chronometer being
given,) the method of projection affords a substitute for a
parallel of latitude, or a 9neridian of longitude; namely, a line
diagonal to either of titse, and which is called a PARALLEL
OF EQUAL ALTITUDE; which, when projected on Mercator's
chart, according to the rules given, shows a ship to be on
such projected line, corresponding to the observed altitude;
in a similar manner as a ship is found to be on a certain




13
parallel of latitude by a meridian observation; consequently,
the projected line shows the BEARING OF THE LAND, in a similar
way as a parallel of latitude.
And likewise if two altitudes be observed, the times being
noted by chronometer, and the two lines corresponding to
the two altitudes, be projected as before, then both the
true latitude and true longitude is found at the intersection
of the two projected lines.
The following remarks are offered upon the corrections
to be made at sea to any observed altitude of the sun, or
other body.
These corrections are commonly four; namely, for parallax, semi-diameter, dip, and refraction.
The first two are found for the given time in the Nautical Almanac, and we have only to take them upon trust,
as they are there laid down.
But with regard to the last two, accuracy depends, in a
great degree, upon the observer himself.
With regard to dip, it should be remembered, that in
the large ships built at the present day, the eye is so
elevated (especially from the poop-deck) that the dip is
fully 1' more than has been usual.
With some persons it has become a habit always to add
12' to an observed meridian altitude of the sun's lower limb,
and from inattention they neglect to subtract the proper
refraction, when the altitude is observed for the purpose of
finding the apparent time for the longitude by chronometer:
important errors will often occur, if the proper corrections
be not strictly applied for different altitudes observed.
In all observations for the latitude by two altitudes, by
whatever method, all these four corrections should be
applied with particular care.
Both dip and refraction being subtractive, if the proper
corrections be not attended to, the Oror will be so much
the greater.




14
The following THREE PRACTICAL PROBLEMS are deducted from
the principles of the method.
PROBLEM I.
The latitude, longitude, and apparent time being uncertain, and one altitude of the sun being observed, at any
hour, when sufficiently high above the horizon, the chronometer time being noted, and declination given; it is required to project, on Mercator's chart, a line, diagonal to
the parallels of latitude, and meridians of longitude, called
a parallel of equal altitude, which shall pass through the position of the ship, and show by inspection,
1st. The bearing of the land.
2nd. The errors of longitude by chronometer, to which
the ship is subject for any in the latitude by dead
reckoning.
3d. The sun's true azimuth.
RULE     I.
1st. Select two latitudes, one of which is the next degree
(without any odd minutes) less, and the other, the next
degree greater than the latitude by dead reckoning.
2d. Find, in the usual way, the ship's longitude by chronometer, upon the supposition that she is in the least latitude
assumed; and project this position on your chart in a point,
which call A.
3d. Find, in the same way, the ship's position, supposing
she is in the greatest assumed latitude; and project this
position also on your chart in a point, which call A'.
4th. Join these two points by a straight line, which
produce as far as necessary; this line is an arc of a "parallel
of equal altitude;" and it passes through the true position of
the ship; and whatever land it passes through, bears from
the ship in the same direction as the line lies on the chart.
5th. The error of longitude by chronometer, at the time
of observation, to which the ship is subject for an error of
latitude by dead reckoning, (when such latitude is used to




15
find the apparent time,) amounting to one degree, is the
difference of longitude between the points A and A'; for
half a degree, half that difference of longitude, &c.
6th. Erect a perpendicular upon the projected straight
line AA', on that side next towards the sun, and it will be
in the direction of the sun's true bearing; and the angle it
makes With the meridian is the true azimuth.
NOT E.
It will thus appear, that the ship is always situated on a line,
which is perpendicular to the sun's true bearing or azimuth.
It follows, that the nearer an observation is taken to noon,
the more accurately a bearing of the land is ascertained,
by this method, if the CHRONOMETER ITSELF BE ERRONEOUS.
If the observation be taken near to noon, (at other times
of day a mistake would not be likely to happen,) and
the declination and latitude in are both of the same name;
the sun bearing south in north latitude, or north in south
latitude, neither of the two assumed latitudes must be
greater than the sum of the declination and the complement
of the sun's true central altitude; but if the sun bears
north in north latitude, or southin south latitude, then
neither of the assumed latitudes must be less than the
diference between the declination and the complement of the
sun's altitude; but when the declination and latitude in
are of different names, neither assumed latitude must be
greater than the difference between the declination and the
complement of the sun's altitude.
It may happen, when the altitude is near noon, that the
difference of longitude of the two points, A and A', to be
projected, may be greater than the extent in longitude of
the "particular" chart in use; in such case, the points may
be projected on a "general" chart; or, what would be better,
assume two latitudes, which are less than one degree
distant; namely, one on each side of the latitude by dead
reckoning; the one being only ten miles greater, and the
other only ten miles less than the latitude by account; or,
if it be very near noon, five miles greater and less: taking
care not to assume a latitude too great, or otherwise, as
mentioned above.




16
With this restriction, it is immaterial what two neighboring latitudes are chosen, they may be either both greater
or both less, or one greater and the other less than the latitude
by dead reckoning; but, in general, when the sun is not
very near the meridian, it will be more convenient to
assume latitudes, one less, the other greater, and which
have no odd minutes; because their logarithms are more
readily taken from the tables, as they are always at the
top or bottom of the page.
It is recommeded to use Bowditch's third method,
for finding the apparent time; because it is something
shorter, and there is a convenience in placing the logarithms, which neither of his first two methods admit.
The result, however, will be nearly the same, by either
method. It is proper to mention, however, that method 1,
(Bowditch,) has its own advantages, as explained in the
Navigator.
Several of the logarithms used in finding the apparent
time are the same for all the operations; which contributes
to shorten the method, in particular when two altitudes
are observed.
EXAMPLE I.
On 17th December, 1837, sea account, a ship having run
between 600 and 700 miles without any observation, and
being near the land, the latitude by dead reckoning was
51~ 37' N., but supposed liable to error of 10 miles on either
side, N. or S.; the altitude of the sun's lower limb, was
12~ 02' at about 10i A.M., the eye of the observer being
17 feet above the sea; the mean time at Greenwich, by
chronometer, was 10h 47m 138 A.M.
Required, the true bearing of the land: what error of
longitude the ship was subject to, by chronometer, for the
uncertainty of the latitude: the sun's true azimuth.
dip. - 4' 3"     semi-dia. + 16' 8"
refra. - 4' 23"  px      +      8"
-8' 26"             + 16' 16"
- 8' 26"
correction of alt. obs'd. +  8' 00"




IT
Obs'd alt. 0 L. L.         12 02!
Correction              +       8'
True alt. s)-  centre,     12~ 10'
lst, The latitude by. dead reckoning was 51~ 37" N.
he latitude te text degree less, without odd minute     ia
1~' N.; and that, the next degree greater  52s N.
2d and 3d, Find the, longitude of these two point~ as
O'S ALTITUDE 12~ 10'.
Po tbe poit A    w  if   e 61~ 
Lat.   51~     N.     -    -   -       sec. 020113
Dec.   23  23  S.    -   -    -    -   sec. 003722
Sum. 74    23       nat, cos. 26920
0 Alt. 12  10       nat. sin. 21076diff. - - 5844     log, 3-76671
I.  M.  8.
1 43   59  from noon       -log rising    4 00606
12  hours.
10  16  01   app. time at ship.
3  37  equa. time.
10  12  24   mean time at ship.
10  47  13      do.    by chro.
34  49 =    8~ 42i' west of Greenwich.
For the poW   A' in laitude 62  N.
Lat.   52~.         -        - see. 0-21066
Dec.   23  23  S.    -    -   -    -   see. 003722
Sum. 75    23       nat. cos. 25235
Alt, 12   10       nat. sin, 21076
diff. -    4159    log, 3-61899.  11..
1  28  28j from noon  =    log rising  =  3 86687
12 hours.
10  31  31j app. time at ship.
3  37  equa. time.
10  27  54A  mean time at ship.
10  47  13     do.     at Greenwich, by chro.
19  18  =-   40 49' west long.
[3]




18
4th. On Mercator's chart, Plate II, project the point
A in latitude 51~ N., longitude, 8~ 42~' W.; and project
the point A' in latitude 52~ N., longitude 4~ 49i' W.
5th. Join the points A and A' by a straight line; and
the: ship's position is upon this line: which, referred to the
compass, is found to tend E.N.E. true; and shows, that
Smalls light bears E.N.E. true from the ship.
6th. The error of longitude to which the ship was
subject for 10 miles error of latitude, is 39 minutes, as
projected.
7th. The sun's true azimuth is 2 points from south, or
S.S.E., as projected.
NOT E.
The ship's true position, in latitude and longitude, at
the time of this observation, is shown on the plate, as was
actually proved by making Small's light. (See page 49.);/Had it been required to make Tuskar light, a northerly
course might have been shaped, and such a "departure"
made from the straight line A A', (30 miles in this case,)
as would have brought Tuskar to bear E.N.E., and then
the course again altered to E.N.E., and Tuskar would have
been made as shown on the plate. But, as the wind was
S.E., when this observation was taken, it was preferable to
make Small's light.
Thus, if the projected line does not pass through your
port, a proper course may be shaped, by which you will
ultimately make the land as desired, in the same manner
as by a meridian observation; which may place you on
a parallel of latitude, which passes some miles to the north
or south of your port, or the head land you wish to make.
The accuracy of your work may be very readily verified,
by assuming a third or intermediate latitude, (for instance,
the latitude by dead reckoning,) and finding a third point,
which project as before; then, if the three points be not
very nearly in the same straiglt line, you have made a mistake
in your work. It should be observed, however, that the
middle point should in truth be a little out of the straight
line, and in a direction from the sun, agreeably to the
curves laid down in Plate I; but with altitudes which are




19
not great, it would be a very trifling difference from a
straight line, indeed, scarcely perceptible.
When the line is projected, your latitude by dead reckoning gives your approximate position in the line; that is,
your approximate distance from  the head-land through,which the line passes.
If the line runs parallel to a coast, it gives your true
distance from such coast. In the above example, the true
distance from the Irish coast is found to be 30 miles; and
the true bearing of Smalls, E.N.E.
ExAMPLE II.
On 4th April, (sea account,) 1840, at about 1, P.M., the
correct central altitude of the sun was 60~ 32', the mean
time at Greenwich, by chronometer, 6h 13m 56s P.M., the
latitude, by dead reckoning was 32~ 20' N., but supposed
liable to an error of 10 miles on either side.
Required, the true bearing of the land; the error of
longitude by chronometer consequent to 10' error of latitude; and the sun's true azimuth.
1st. The two latitudes to be used in the calculation, are
32~ N., and 33~ N.; one on each side of the latitude by
dead reckoning.
2d. and 3d. Find the longitudes of the points A and A',
having those two latitudes, as follows:
For the point A in latitude 32~ N.
Lat.  32~    N.    -        --    sec. 0'07158
Dec.   5 35 N. -      -   -   -   sec. 0-00207
Diff.  26 25     nat. cos, 89558
O Alt. 60 32     nat. sin. 87064
diff. - - 2494  log, 3-39690
H.  M.  B.
0 55 51 p. m.     =     log rising =  347055
-+     3 21 equa. time.
0 59 12 mean time P.M. at ship.
6 13 56      do.   by chronometer.
5 14 44     - 78~ 41' west longitude.




20
For the poi  A' in latitude 33~ -.
Lat.  33~    N.         -          see. 0 -07641
Dec.   5 85 N. -      -       -     se. 0-00207
Diff. 27 25      nat. cos. 88768
d Alt. 60 82      nat. sin. 8r064
diff.? - 1704   log, 3-23147
A'  ~.   g.  _._
0 46 28 P.M.          log riia,  tt  3 30995
+      3 21 equa. timoe
9 49 44 mean time P. M. at ship.
6 13 56      do.   chronometer.
5 24 12 =     81 03' west loagitude.
4th. On Merrtor chart, plate IV, project the point A,
in latitae 32~ N., longitude 786 41' W., and project the
point AI, in latitude 33~ N., longitude 81~ 03' W.
th. Join the points A aid A' by a straight line; this
line passes through the postion of the ship; it is found to
ted N.W    by W. i W., showing that the land, situated
abot 10 miles southwesterly, from Charleston light, bears
N.W. by W. j W.
6th. The error of longitude by chronometer, to which
the ship was subject from 10' error of the latitude, is seen
to be 24' as set off on the plate.
th. The sun's true azimuth is also projected; his bearing
being nearly S.S.W. i W.
NOT E.
If it be required to make Charleston light, then sail
northerly from the line AA', until a departure from AA'
is made equal to about 10 miles, then haul up N.W. by
W. | W. until the light is seen.




21
PROBLEM         II.
Two correct altitudes of the sun's centre being observed;
and the times of observation noted by chronometer; the
declinations at both times, and the latitude by dead reckoning being given, and A.M. and P.M. times of observation
noticed;
Required to project the two corresponding "straight lines"
on Mercator's chart, showing their mutual intersection;
the true latitude and the longitude by chronometer, and
likewise the results of problem I.
R   L E.
1st. Proceed as in problem  I, to project the first
straight line AAM, which will correspond to the first observed altitude.
2d. Proceed in exactly the same manner, to project the
second straight line, (using the very same assumed latitudes,
as before,) and it will correspond to the second observation;
name the two points in it, which have the assumed latitudes, B and B'; taking care to correct the declination,
and likewise the equation of time, if any change has taken
place in them between the observations.
3d. These two straight lines will be found to intersect
each other, for the most part between the two assumed
latitudes; if they do not intersect, then produce them to
an intersection; and the intersected point will be in the
latitude and longitude of the ship, if she has not changed
her station between the observations.
4th. If the ship has changed her station; then, set off the
distance sailed between the observations, in the direction
of the true course made good, from any point in the straight
line AA', through the point set off, draw a straight line
parallel to AA, and produce it until it intersects the
straight line BB'; this new intersection with BB' is the
position of the ship in latitude and longitude, at the time
of the second observation, corrected for change of station.
5th. The other requisitions of the problem are explained
in problem I.




22
NOT E.
It follows, from the principles hereafter explained, that
all observations by double altitudes are, in general, to be
preferred, when the observations are taken at right angles
to each other, by compass
If it be preferred, you may correct the first altitude for
change of station by the usual methods, before finding the
apparent times; and then the intersection of the two straight
lines, AA' and BB' will be the ship's position in latitude
and longitude at the time of the second observation.
EXAMPLE       I.
January 1st, 1839. The sun's correct central altitude,
A.M., was 14~ 23'; the mean time at Greenwich, by chronometer, being 11h 8m 188 A.M.; and when the time by
chronometer was Oh 6m 448 P.M., his correct central altitude
was 19~ 33' A.M.; the latitude, by account, being 43~ 45' N.;
between the observations the ship sailed only one mile
N.E. by E.
Required the true latitude, and the longitude by chronometer, at the time of the second observation, &c.
1st. The two latitudes to be used are 43~ N. and 44~ N.
FOR THE FIRST ALTITUDE 14~ 23' A.M.
To find the point A, in latitude 43~ N.
Lat.  43~    N.        -    -   - sec. 013587
Dec. 23 03   S.   -   -   -   -   sec. 0-03613
Sum. 66 03       nat. cos. 40594
0 Alt. 14 23     nat. sin. 24841
diff.- - 15763  log, 4-19736
H.  M.  8,2 40 63             log rising  =    4-36936
12 hours.
9 19 57 app. time A.M. ship.
+     3 42 equa. time.
9 23 39 mean time at ship.
11   8 18 mean time chronometer,
1 44 39   =   26~ 9' 45" west longitude of A.




23
To find the point A, in ltitude 440 N.
Lat.   44~     N.     -                sec. 0 14307
Dec.   23  03  S.   -      -  -        -  sec. 0-03613
Sum.   67  03       nat. cos. 38993
0 Alt. 14  23       nat. sin. 24841
diff.- - 14152     log, 4-15082
H.  M.  S.
2  32  40     -       log rising   =      4-33002
12 hours.
9  27   20  app. time A.M. at ship.
+-     3  42   equa. time.
9  31  02   mean time at ship.
11  08   18     do.    chronometer.
1  37   16 =    24~ 19' west longitude of A'.
FOR   THE   SECOND    ALTITUDE. 19~ 33' A.M.
To find the point B, in latitude 43~ N.
Lat.   43~ 00  N.     -    -           sec. 0-13587
Dec.   23  03  S.      -  -            sec. 0-03613
Sum. 66    03       nat. cos. 40594
0 Alt. 19  33       nat. sin. 33463
diff.- -  7131     log, 3-85315
H.  M.  S.
1   46  28     =       log rising   =      4-02515
12 hours.
10  13  32   app. time at ship, A.M.
+      3  43   equa. time.
10, 17  15   mean time at ship, A.M.
12  06  44      do.    chronometer.
1 49   29  =    27~ 22' 15" west longitude of B.




24
To find the oint B' in laWtude 44~. N.
Lat.  44~ 00 N.     -   -   -   - sec. 014307
Dec.  23 03 S.        - - -sec. 0*03613
Sum. 67 03        nat. cos. 38993
O Alt. 19 33      nat. sin. 33463
diff. -  5530   log, 3-74273
H.  M.  S.
1 34 21             log rising  =    3-92193
12 hours.
10 25 39 app. time at ship, A.M.
+      3 43 equa. time.
10 29 22 mean time at ship.
12 06 44      do.   chronometer.
1 37 22       24~ 20' 30" west longitude of B.
the point A, in latitude 43~- N.
2d. On Mercator's chart,  longitude 26~ 9' 45" west.
plate V, project         the point A' in latitude 44~ N.
I  longitude 24~ 19' west.
Join A and A', and this line will pass through the position of the ship at the time of the first observation.
[also the point B, in latitude 43~ N. and in
Proect  longitude 27~ 22' 15" west.
roject and the point B' in latitude 44~ N., and in
longitude 24~ 20' 30" west.
Join B and B', and this line will pass through the position of the ship at the time of the second observation.
3d. It is seen that these two lines intersect each other
in latitude 44~ 1' N., which is the true latitude, and the
true longitude is 24~ 18' west.
4th. No correction is required for change of station, since
the course sailed was on thefirst projected straight line AA'
tending N.E. by E., the same as the course sailed.
5th. The other requisitions of the example are projected
as in problem I.
EXAMPLE       II.
On December 21st, 1838, the sun's correct central altitude
was 20~ 23' A.M., when the chronometer time was 1h 34m"




25
P.M.; and his correct central altitude was 25~ 10' P.M.,
when the mean Greenwich time was 5h 55m 34s P.M.; the
latitude by account being 36~ 08' N., and between the
observations the ship sailed E.N.E. { E., 25 miles. Required
to project, on Mercator's chart, the true latitude, and the
longitude by chronometer, &c.
1st. The two latitudes to be assumed, less and greater,
than 36~ 8' N. are 360 N. and 37~ N. The declination is
23~ 27' S. and equation of time as under.
FOR   THE PIRST ALTITUDE, 206 23' A.M.
For the point A, in latitude 36~ N.
Lat.   36 00   N.    -    -   -      sec. 0-09204
Dec.   23  27  S.  -    -   -    -   sec. 0 03744
Sum. 59   27       nat. cos. 50829
0 Alt. 20  23      nat. sin. 34830
diff.    15999   log, 4 20409
H.  M.  S.
2 33   20    =     log rising,          4-33357
12 hours.
9  26 40   app. time at ship, A.M.
-   1 40   equa. time.
9 25 00 mean time at ship.
13  34 00       do.   chronometer.
4  09 00   =   62 15" west longitude of A.
For the point A', in latitude 37~ N.
Lat.  37  00   N.    -    -   -    - sec. 0-09765
Dec.  23   27  S.  -    -   -    -   sec. 0-03744
Sum. 60   27       nat. cos. 49318
0 Alt. 20  23      nat. sin. 34830
diff.    14188   log, 4 16101
H.  M.  S.
2 26   37    =     log rising,   =      429610
12 hours.
9  33  23  app. time at ship, A.M.
-   1 40   equation.
9  31  43 mean time at ship.
13  34  00      do.   by chronometer.
4  02  17 =    60~ 341' west longitude of A'.
[4]




26
FOR THE SECOND ALTITUDE, 25~ 10' P.M.
For the point B, in latitude 36~ N.
Lat.  36  00  N.    -    -   -    -sec. 0-09204
Dec.  23  27  S.   -   -   -    -   sec. 0 03744
Sum. 59   27       nat. cos. 50829
0 Alt. 25  10     nat. sin. 42525
diff.     8304  log, 3 91929
H.  M.  S.
1 49   27 P.M. =    log rising,  =     4-04877
1 30 equation time.
1 47   57 mean time at ship, P.M.
5  55 34       do.   chronometer.
4  07  37 =    61~ 544' west longitude of B.
For the point B', in latitude 37~ N.
Lat.  37 00 N.       -     -     -  - sec. 0-09765
Dec.  23  27 S.   -         -  -  -  sec. 0-03744
Sum. 60   27      nat. cos. 49318
) Alt. 25 10      nat. sin. 42525
diff.     6793   log, 3 83206
H.  M.  S.
1 39   28    =      log rising,  =     3-96715
-      1 30   equation time.
1 37   58 mean time at ship, P.M.
5 55 34        do.   chronometer.
4  17 36 =     64~ 24' west longitude of B'.
Project on Mercator's chart, plate VI, the four points A,
A', B, B'; in their respective latitudes and longitudes; and
join A and A'; B and B'. If the ship had not changed
her station, her true place would be at their intersection.
3d. But since this is not the case, from any point, as P,
in the line AA', set off the point D, 25 miles E.N.E. | E.,
the course and distance made good between the observations; through D, parallel to AA', draw    an indefinite
straight line DB; the intersection of DB with BB', is the
ship's place at the second observation.




27
4th. The correction in miles of the first altitude, for change
of station, is the perpendicular upon AA  from B to C,
equal to 10' additive.
The true latitude is 36~ N.; and the longitude, by chronometer, is 61~ 54k' west of Greenwich.
NOTE.
This observation was found to be accurate by meridian
observation of the sun, as follows:
O's observed altitude was 30~ 24' bearing S.
11 correction.
300 35!
59 25 zenith distance N.
23 27 declination S.
The ship's intermediate )
position between the obser- 35 58 latitude N.
vations was at noon.
PROBLEM III.
When two altitudes of the sun are observed, and the
times are noted by a "well-regulated watch," for the elapsed
time, the course and distance being given; and it is required to find the latitude only; we may consider, in such
case, that the "watch" shows mean time at any meridian,
which we may pleasp to assume; in the same way as the
"chronometer" shows mean time at the known meridian of
Greenwich.
We may, then, assume, as a first meridian, that of Greenwich; and set the hands of the watch, previously to any
observation, to the Greenwich mean time, as nearly as we
may be able to estimate it, according to the supposed longitude
of the ship by account. And then we may regard the
"watch" as a "chronometer."
Having done this, proceed exactly as in the two last examples, to find the longitudes of A, A', B, B'; and project
them  by that rule, and the true latitude will be found as




28
before; but the longitude from Greenwich will not be
found; since the watch shows only the approximate time at
Greenwich.
But it is not absolutely necessary to set the hands of the
watch to the Greenwich time, but merely see how much
the watch is fast or slow, of such' estimated time, and allow
for the error, accordingly. Thus it is quite immaterial
what hour and minute the watch shows; it is only required,
that the going of the watch between the observations is
known to be accurate.
By assuming that the watch shows Greenwich time, the
resulting longitudes will be such as correspond in degrees
to the chart in use; otherwise, any other meridian might
be assumed with equal success.
The above method will be found the most simple, and is
to be always used when the sun is the object observed;
but to familiarize the learner with the principles of this
method, so that he can apply the method in all cases to the
fixed stars and planets, the following example is given, in
which the four points are projected by a different process;
for the rule to do this, the reader will refer to problem VI.
and Example (fo     )
EXAMPLE.
On the 4th of January, 1839, sea account, between 2 and
3 o'clock, P.M., the sun's true central altitudes were 15~ 10'
and 106 30'; and. the elapsed time was 46m 5s; the course
and distance made good between the observations, being
E.N.E., 6 miles; the latitude, by account, 46~ 25' N. Required the true latitude, at the time of the second observation; the correction of the first altitude for change of
station; the sun's azimuth at each observation.
1st. The elapsed time is 46m 5 -  11~ 31' 15", or the
difference of longitude of Z' West of Z.
2d. The two latitudes to be assumed, will be 46~ N. and
47~ N.
3d. Find the apparent times "from noon," as follows:




29
FOR   THE FIRST ALTITUDE, 15~ 10' P.M,
For the point A, in latitude 46~ N.
Lat.   46  00  N.     -    -    -      sec. 0-15823
Dec.   22  51 S.   -    -    -   -    sec. 0'03552
Sum.   68  51       t nat. cos. 36068
0 Alt. 15  10       nat. sin. 26168
diff.     990      log, 3 99585
H.  M.  a.
2   "9  12            log rising,         4-18960
2   9   12 =  32~ 18' diff. longitude east of Z, be-?
cause P.M., of a point A, in latitude 46~ N.
For the point A' in latitude 47~ N.
Lat.   47  00  N.     -      -    -    sec. 0-16622
Dec.   22  511 S.   -    -    -   -    sec. 0-03552
Sum.   69  514      nat. cos. 34435
O Alt. 15  10       nat. sin. 26163
diff.      8272    log, 3 91761
H.  M.  A.
1  58  55            log rising,   =      4-11935
1  58  55 == 29~ 43t' diff. longitude east of Z, be- 
cause P.M., of a point A' in latitude 47~ N.
FOR   THE   SECOND    ALTITUDE, 10~ 30' P.M.
For the point B, in latitude 46~ N.
Lat.   46  00  N.     -          -  -  - sec. 0 15823
Dec.   22  511 S.   -        -    -    sec. 0-03552
Sum. 68    514      nat. cos. 36068
0 Alt. 10  30       nat. sin. 18224
diff     17844     log, 4-25149
H.  M.  S.
2  55   22     =      log rising,   =     4-44524
2  55   22 =  43~ 50i' diff. longitude east of Z', be-1
cause P.M., of a point B in latitude 46~ N. 




30
For the point B', in latitude 47~ N.
Lat.  47 00 N.      -    -   -    - sec. 0-16622
Dec.  22 51i S.   -    -   -    -   sec. 0-03552
Sum. 69 51a       nat. cos. 34435
0 Alt. 10  30     nat. sin. 18224
diff.    16211   log, 4-20981
H.  M.  S.
2 48 23       =     log rising,  =     4-41155
2 48   23 = 42~ 5' diff. longitude east of Z', be- 
cause P.M., of a point B', in latitude 47~ N.
A in lat. 46 N., is east of Z,  - 32~ 18'
Z is east of Z', -              11 311
A is, also, east of Z',  -  -   43~ 491'
Project A, on the chart, plate VII., in latitude 46~ N.,
and in any longitude; and mark it, east of Z, 32~ 18'; and
east of Z', 43 49'.
A', in lat. 47~ N., is east of Z  - 29~ 433'
A         46~ N., is east of Z -  32  18
A' is west of A  -   -   -       2 34~
Project A', in lat. 47~ N., and 2~ 34i' west of A; mark
it 29~ 43}' east of Z'.
A is east of Z'  -   -   - =    43~ 491'
B in lat. 46~ N. is east of Z' -  43  501
B is east ofA  -   -    -            1
Project B', in lat. 46~ N., 1i' east of A, and mark it
43~ 50j' east of Z'.
A is east of Z'  -   -   -   - 43   49'
B', in lat. 47~ N., is east of Z' -  42  5b
B' is west of A      -    -      1 431
Project B', in lat. 47~ N., 1~ 43i' west of A, and mark
it 42~ 53' east of Z'.
5th. Join by straight lines A, A', and B, B', from any
point, as P, in AA', set off the point D, with the course and
distance, E.N.E., 6 miles; through D, draw a line parallel
to AA', and its intersection with BB', is the true latitude




31
at the time of the second observation, namely, 46~ 15k' N.
6th. The correction of the first altitude is 5 miles subtractive, as projected by the rule; the bearings of the sun
are also projected.
If the sun's true azimuth could be always exactly observed,
it would be necessary to find the longitude of only one
joint in each straight line; and that might be found by
using the latitude by dead reckoning to get the apparent
time; for then, if a line be drawn on the chart, through
this point in the exact direction of the sun; and a second
like be drawn through the same point, and perpendicular
to the first line; it would show the bearing of the land
&c. as before; and if two altitudes were taken, and the
lines thus projected, the latitude and longitude would be
very easily found-but unless under very favorable circumstances this cannot be done with sufficient accuracy;
and the only safe way is to find the position of two points
in each line as directed.
It is evident, that, having found the position in latitude
and longitude of the two points in either straight line,
the bearing of one from the other is easily found by case I.
Mercator's sailing; and thence the bearing of the land; but
still it is preferable to project them.
The latitude may be found by this method of calculation
from  two altitudes, without projecting the straight lines, as
follows. Note the elapsed time by watch or chronometer,
(in example, problem III, this was 46m 5s;) with both altitudes, find the apparent times from noon, using either one
of the two assumed latitudes; take the difference of these
times; (in example, problem III, the times for o's altitude,
15~ 10' and 10~ 30', in assumed latitude 46~, were 2h 9m 12S
and 2h 55m 228; their difference is 46m 16;) which difference
of times (46m 10s) would have been the elapsed time, if the true
latitude had been 46~; but 46m 10s is 5s greater than the
elapsed time noted by watch; do likewise with both altitudes, using the other assumed latitude; (we have, example
problem  III, for latitude 47~, the Ames Ih 58m 55", and
2h 48m 23s; their difference is 49m 288;) which difference




32
would have been the elapsed time if the true latitude had
been 47~; but 49m 28' is 3m 23' greater than the elapsed
time noted, (46m 5';) therefore, in this example, the further
north we assume a latitude the greater we make the error
in the elapsed time; and we may already see that the true
latitude is not so much as 46~ N. Now take again the
difference between 46m 10', (the elapsed time if the latitude
be 56~,) and 49m 28', (the elapsed time if the latitude be
47~;) this difference is 3m 18'; then make this proportion;
if 3m 18' is the error of elapsed time, caused by an error of
latitude of 1~, what is the error of latitude (to be subtracted
from 46~ N.) caused by an error of 5' of elapsed time?
By proportional logarithms (Table XXII, Bowditch,) we
have
As -     -    3   18    ar. co.  log 8-2632
is to -   -   1~                    *4771
so is  -   -   m   5'              3-3345
To the ans.   1' 31"           log 2 0748
46~ 0' 00"
450 58' 29" latitude of the intersection
of AA' and BB'.
By reference to plate VII, we see that the straight lines,
AA' BB', intersect each other a little to the south of latitude 46~; no change of station being allowed; to find the
true latitude, according to the example, we should have:jrst corrected the first altitude for change of station; and
then proceeded as above. The longitude also may be
found when the times are noted by chronometer, by a
similar process.
What are the errors to which observations calculated by
the method of projection are subject?
The only calculation used, is that for finding the apparent time. For this purpose, three things are necessary;
the latitude, the declination, and the altitude; if all these
be accurate, we have the true apparent time; and if the
chronometer be right, we have no error; now, by assuming
the latitudes used, no error exists in this element; the




33
declination also may be considered accurate; and the only
remaining source of error is in the altitude observed; and
excluding such as may arise from inaccuracy of noting the
times, of adjustment, reading off, &c., which ordinary care
will prevent, this error is usually less than two miles, and is
due to irregularity of refraction, and the occasional unfitness
of the horizon; but we may find, that by the common rules
and with a single altitude, this will make a large error in
the ship's position by chronometer, as the sun approaches
the meridian; because the resulting time is applied directly
to find the longitude. When the sun bears east or west,
the error of time will be exactly equal to the error of altitude turned into time; but as the sun approaches the
meridian, the error of time continually increases, thus
causing continually increasing errors in the position by
chronometer.
But by this method, the error in a bearing of the land, by
a single altitcude, at anzy hozr, is always eyqal to the error of
altitude; as when a meridian observation is observed one
mile too great, the bearing of the land in the parallel of
latitude resulting, is only one mile erroneous. There is no
reason, then, why we may not observe a single altitude of
the sun at any bearing, to find a bearing of the land by
this method.
It was remarked that, as a general rule, any two observations for " double altitudes" are to be preferred, when taken
at right angles to each other in azimuth; and this is because a small error in either or both altitudes observed, or
the times noted, will never occasion, in such cases, but a
trifling error in the position on the earth's surface; indeed
the position will then be determined within a space on the
earth's surface the least possible, having any given error of
altitudes or times; but the same errors of altitudes or times
will always occasion greater errors of position, when the
difference of azimuth between the two observations is
either greater or less than eight points.
But at the same time ";the latitude" may be determined
in many cases, with greater accuracy, if the two observations
have either a less or greater difference of azimuth than
eight points, than when they are taken at right angles;
[s]




34
but such cases are usually most unfavorable for finding
"the longitude," or the apparent time; again, "the longitude" may also be determined, in many cases, in a similar
manner; but such cases are usually most unfavorable for
finding "the latitude."
The considerations which follow will show the degree of
confidence that may be placed in every set of double altitudes that can be observed; and the particular advantages
of each; and, in general, show first, that, towards whatever
point of the compass the mean between both the azimuths
is directed, the same difference of azimuth, and also its supplement, will give the position within the same limits in
space on the earth's surface; but that diferent differences
of azimuth will give different limits in space, and these limits
will be the greater, the more the difference of azimuth
varies from eight points; and secondly, that the value of
any set of double altitudes, to determine "a latitude" or "a
longitude," depends jointly upon the difference of azimuth
and the azimuths themselves, at which the observations were
made; or, for convenience of explanation and reference,
upon the dfference, and upon the mean of the azimuths.
These remarks, it will be distinctly kept in mind, are not
peculiar to this method alone, but apply equally to all methods of double altitudes, even to the common daily method
of a meridian altitude, and an east or west sight for the
longitude, which isonly a particular case of double altitudes.
If all errors were expunged from the declinations, the altitudes, and the times noted and the logarithms in the tables
were carried to a sufficient number of places,*no error would
exist in either "the latitude" or "the longitude," or the
position in space, at whatever azimuths two altitudes were
observed; and it is the liability to error in some of these
quantities which establishes a preference among different
sets of observations. The estimate of the effect of such
errors therefore becomes important; and among the quantities mentioned it will be necessary only to consider the
errors of the altitudes and of the times.
Every altitude observed is liable to three conditions; for
it will be either exact, or too great, or too small; and the
changes of position, resulting from two altitudes, which are




35
each liable to three changes, will be nine, according to the
following
LIST OF CONDITIONS.
Position e1 When both altitudes are exact.
2. When both altitudes are too great.
3. When both altitudes are too small.
4. When the first is exact, but the second too great.
5. When the first is exact, but the second too small.
6. When the second is exact, but the first too great.
7. When the second is exact, but the first too small.
8. When the first is too great, but the second too small.
9. When the second is too great, but the first too small.
The manner in which these conditions affect the position
in space will be seen in the following diagrams, in all of
which the error of the altitudes is one mile. The projected
lines, AA', BB', fig. 1, are supposed to be those resulting
from two altitudes which were exactly observed, when the
Fig. i.   object bore east and south, as in the common
method of finding the ship's place; here
mu the diference of azimuth is eight points, and
6 1       the mean azimuth, S.E., or four points; the
8 5    position in this case is at their intersection
at the numeral 1; but if both altitudes were
&A       observed one mile too great, then the line
AA" must be removed westward one nautical mile to the
position of the line aa'; and the line BB' must be removed
one mile northward to the position of the line bb'; then the
ship's position will be at their intersection at the numeral
2; that is, the error of "latitude" will be one mile, and that
of "longitude" one nautical mile; but the error or limit of
position in space on the earth's surface is the distance
between the intersections at the numerals 1 and 2; or by
the traverse table 141 miles of distance, and on a four
point course.
The numerals at the other intersections correspond with
those of the different positions annexed to the above list of
conditions; some of these affect only "the latitude," as at
4 and 5; some only "the longitude," as at 6 and 7; and
some both, as at 2, 3, 8 and 9. Whence it is seen that the
error of position in space, 1'41 miles, takes place in the
direction of (S.E. and N.W.,) and at right angles to [N.E.




36
and S.W.,] the mean azimuth, [S.E.;] and that these limits
in these directions are represented by the diagonals of the
parallelogram 2, 9, 3, 8; for if we know not whether the
error of the altitudes be too great or too small, we cannot
be sure of the position in space on the earth's surface within
the space enclosed by this parallelogram; the extreme error
in space is then represented by a parallelogram, whose
diagonals are each 2 82 miles in length, and which lie, the
one in the direction of the mean azimuth, and the other at
right angles to it; the parallelogram being a square when
the difference of azimuth is a right angle, the diagonals will
be equal. And we may infer that the bearings of the land
in the directions of the diagonals are not so well determined
by 0 82 miles as in the directions at right angles to each
azimuth respectively; that is, the observation is the most
favorable for determining "the latitude" and "the longitude;" and we shall also see that position in space is as
favorably determined, whatever the mean azimuth may be,
the difference of azimuth being eight points.
Suppose the object to have been observed when bearing
S.E. and S.W., fig. 2. Here the difference of azimuth is
eight points, as before, but the mean azimuth is south, or 0
Fig. 2.  points; but "the latitude" is affected as at
B~ib\ /   2 and 3, instead of at 4 and 5, as in fig. 1;
\X^   / /the longitude at 8 and 9 instead of at 6 and
^^   <'7; and both at 4, 5, 6, 7, instead of at 2, 3, 8,
9; but the extreme error of position on the
a   //8 >    earth's surface is the same as before, 2'82
K; ~ miles; in the direction of the mean azimuth,
N. and S.; and the same quantity, at right angles to it, E.
or W.;' hence we infer that the former case, fig. 1 is more
favorable to find the latitude or the longitude than fig. 2,
by 0 82 miles (extreme;) although the position in space is
equally well determined in both figures; the alteration in
the direction of the mean azimuth having no effect on the
parallelogram, except to turn it about on its centre; and it
is obvious that the bearings of the land at right angles to
each azimuth respectively, that i in N.W. and S.E., and
S.W. and N.E. directions, is more favorably found, by 0-82
miles, than in the direction of, and at right angles to, the
mean azimuth; that is, in this case, than in E. and W., or N.




37
and S. directions; we see here, in some degree, the dependence of the accuracy of the latitude and the longitude
upon the direction of the mean azimuth, the dfference of
azimuth being the same in both figs. 1 and 2.
But an error of one mile in the altitudes will occasion a
greater error of position in space on the earth's surface, and
will affect the latitude and the longitude in various degrees,
when the diyerence of azimuth is greater or less than eight
points,
Let AA', fig. 3, be the line resulting when the object
bears E. I N.; and BB' that resulting when it bears E. I S.;
Fis. 3. which is a dfference of azimuth of one point, and
iA  WEB the mnean azimuth is eight points, or east; then
when both altitudes are exact, the position will be
at the numeral 1; but if both altitudes are too
\   g reat, it will be at 2; being an error of 1-01
nautical miles in the direction of the mean azimuth, or E. and W.; and when the intersection is
at 8 or 9, the error of the position will be 10'20
miles at right angles to the mean azimuth; thus
3 will the extreme error of latitude be equal to one,
21 |and the extreme error of longitude be equal to the
other diagonal of the parallelogram, 9, 2, 8, 3,
o6 I! which includes the position of the ship, the mean
azimuth being east; and the error of position in
space is also increased; here the greatest diagonal
s 8   i ls at right angles to the mean azimuth, and the
least in the direction of it; the longitude is found
EB ~A    within the extreme error of 2-02 nautical miles,
and the latitude within the extreme error of 20'40 miles.
If using the same fig. 3, we suppose BB' to result from the
first altitude, the object bearing E. I S., and AA' to result
when it bore W. I S., that is, with a difference of azimuth
15 points, [the supplement of one point] and a mean azimuth, south, or 0 points, the figures of reference only being
changed in the diagram, the same effect will be produced,
as regards the size and situation of the parallelogram; but
the greatest diagonal is now in the direction of the mean
azimuth, and the least at right angles to it.
It will be easy to see that if fig. 3 be turned about its
centre until the mean azimuth is south, the azimuths being




38
only one point, as before, that the greatest error will be at
right angles to the mean azimuth, and that the latitude will
then be found within an extreme error of 2*02 miles, but
that the position in space, on the earth's surface, is equally
well found as when the mean azimuth is east, or eight points.
Intermediate directions of the mean azimuth will equally
well determine the position in space, having the same
difference of azimuth, but the latitude and longitude will
suffer a change.
Thus, in fig. 4, the object is supposed to be observed,
bearing S.E. j E., and S.E b E. J E, or with a difference of
Fig. 4.       aazimuth of one point, and a
ag  r  mean azimuth, S.E. b E. or
//  5 points; the change in the
mean azimuth does not ifect
the size of the parallelogram,
but only its position, turning
it on a centre, through three
points of the compass, or from
east to S.E. b E.; but the
latitude and the longitude are
both affected; half the extreme
error being represented in the
is/ <s/,4//former, by the line L. 9; and
in the latter by the line L. 1;
*X_ /                   the extreme errors being 16* 96
miles latitude, and 11 34 nautical miles longitude; but we have, notwithstanding, the
bearing of the land in the directions of N.E. b N., S.W. b S.,
or at right angles to the mean azimuth, determined within
the extreme limit of only 2 02 miles; but in the direction
of the mean azimuth, or S.E b E., N.W. b W. with an error
of 20-40 miles. So on the whole, the observation is just
as good as in fig. 3, notwithstanding the errors of latitude
and longitude are so considerable.
Altitudes with other differences of azimuths, and other
mean azimuths, are subject to similar laws. But whatever two
observations are made, a bearing )f the land may always
be found which shall be subject to no greater extreme error
than 2-82 miles for one mile error of the altitudes.
To exhibit at one view the degrees of preference which




39
obtains among any set of double altitudes, the following
tables are constructed upon the above principles. If the
difference of azimuth be found in the LEFT hand column,
[Col. L.] then the mean azimuth must be sought at the
head or foot of the double columns, among those which are
marked with the letter L.; and if found at the top, the
precepts I" Lat. Dep." must be read at the top; otherwise at
the bottom; and similarly with regard to the RIGHT hand
column, [Col. R.;] the tabular numbers show in nautical
miles the error of latitude and of departure, [to be converted into difference of longitude as usual,] consequent
to an error of one mile in the observed altitudes; this gives
the error reckoned from    the central point of the parallelogram formed; or from the intersection given by the observation; but if it be uncertain whether the error of altitude
be too great or too small, then the extreme error will be
double the tabular numbers.     An error of two miles in the
altitudes will be double that of one, and so on.
The tabular numbers which are underscored, show that
one of the altitudes must have been observed either in the
meridian or the prime vertical, the error being equal to the
error of altitude allowed, 1 00 mile.
TABLE A.
Col. L.            MEAN AZIMUTH IN POINTS.             C.
Difference L  R L       R L       R L      E L       R Difference
of azimuth 0.  8. 1.    7. 2.     6. 3.    54.       4. of azimuth
in points.                                  ----      in points.
Lat.  Dep.  Lat..  ep.  Lat.  Dep.  Lat.  Dep.  Lat.  Dep.
1    101 10-20 1-99 10-00 3'90 9-42 5'67 8-48 7-21 7-21  15
2    1-02 5-12 1-00 5-02 1-96 473 2-84 4-26 3 62 3-62  14
3    1-05 3-44 1-033 337 1-32 3-18 1-91 2-86 2'43 2~43  13
4    1-08 2 61 1'06 2-56 1-00 2-41 1-45 2-17 1-85 1-85  12
5    1-13  2-12  1-11  2-081 104  1-96  1-18  1-76  1-50  1-50  11
6    1-20  1-80  1-18  1-77  1-11  1-66  1'00  1-50  1-27  1-27  10
7    1-20  1-57  1-27  1-54  1-19  1-45  1-07  1-31  1-11  1i11  9
8    1-41 1-41 1'39 1-39 1'31 1'31 1'18 18 181-00 1-00  8
Pep.  Lat.  Dep.  Lat.  Dep.  Lat.  Dep.  Lat.  Dep.  Lat.
Difference                                             iference
8.      0.[7      1. 6.    2-:5.     3. 4.    4.
of azimuth L   R     L       R  L           RL RL    R ofazimuth
in points.          MEAN AZIMUTH IN POINTS.             points.
The order of preference which obtains among the different sets of double altitudes, to determine a latitude or a




40
longitude, is evident by a careful inspection of the tabular
numbers of table A; but although the tabular numbers
abreast of any particular difference of azimuth, differ considerably, yet all the numbers in the same line are equally
well suited to determine the position in space as shown by
referring to table B.
Table B. contains the same numbers at the first column
of table A, and the tabular numbers represent, in nautical
miles, half the length of the diagonals of the parallelograms
formed, which include the position in space on the earth's
surace, the errors of altitudes being one mile; one of these
diagonals being in the direction of any mean azimuth, and
the other at right angles to it; the tabular numbers must
of course be doubled to find the extreme error.
The precepts at the top of columns must be read, if the
difference of azimuth be found on the left of the table;
otherwise, at the bottom.
Table C shows the increase of the extreme area within
which the ship is situated, according as the difference of
azimuth varies from   a right angle, the error of altitude
being one mile.
TABLE B.                       TABLE C.
Half the length of the diagonals of the par-   Value of the
allelograms formed, showing, in nautical miles,  nofthe  ~
of aziruth  for one mile error of altitudes, the error of po-   araselo 
of azimuth  sition in space,               paraeo- 
in   grams in 
points,  in the direction of any  at right angles to any  T square nail
mean azimuth.  mean azimuth.        tica miles.
I        1'01         10-20      15     1   20-604  15
2         1 02         5-12      14     2   10-449  14
3         1-05         3-44       13    3    7-224  13
4         1 08         2-61       12    4    5-638  12
5         1-13   2'12             11    5    4-791  11
6         1.20         1080       10    6    4-320  10
7         1-29         1-57        9    7    4-050   9
8         1-41         141        8     8    4-000   8
At right angles to  In the direction of Difference
of azimuth
any mean azimuth.  any mean azimuth.  in points.
The inspection of tables B and C shows that, when the
difference of azimuth is eight points, the position in space
is most correctly determined; and table B      also shows,
that, whatever two observations are made, that one of the
diagonals will always be not greater than twice 1'41 miles;




41
consequently that the bearing of land at right angles to
this diagonal will be found within the extreme error of not
more than 2 82 miles.
Tables B and C show, generally, that the limits of a
ship's position depend wholly on the diference of azimuth;
while table A shows that the accuracy of "a latitude" or
"a longitude," found by double altitudes, depends jointly
on the diference, and on the mean, of the azimuths.
The error of latitude, or of longitude, or of the position
in space consequent to an error in either or both the times
noted, is exactly analogous to the errors just stated; and
the same ground might be gone over, a list of conditions
and similar tables formed, but this will be unnecessary;
it may be stated, however, that the number of changes of
position resulting from two altitudes and tzuo times noted,
which four quantities are each liable to three conditions, will
be 81; but if the effect of an error of the time be equal to the
effect of the error of altitude, this number will be reduced
to 25; for then the errors will frequently cancel each
other; by supposing the error of time changes the position
of a projected line just as much as the error of altitude, we
may represent the various changes, by drawing a line
parallel to each of the sides of any of the parallelograms
in the preceding diagrams, and at a distance from them of
one mile; the parallelograms will then be four times their
former size, and their diagonals double their former length,
and we shall have the combined result of an error of time
and of altitude by doubling the tabular number of tables A
and B, and the extreme error by multiplying them by 4.
But it must be considered that, with a good time-keeper,
the error of noting the times, together with any error of its
performance during the elapsed time, must always be very
small, perhaps not a secend, particularly if attention be
paid to apply the proportional part of its daily rate to the
part of a day which has elapsed between the observations
and an error of so small an amount will make no error of
consequence except in the most extreme cases.
An absolute error in the chronometer will, of course,
occasion an absolute change in the longitude; changing the
whole parallelogram to a more eastern or western position
[6]




42
only; and this error must not be confounded with the
"difference of longitude" occasioned by the error in the
times noted, or altitudes observed; which are relative only
to the central point of the parallelogram, which is at the
intersection given by the observations.
But, after all, we may readily find, by inspection of any
observation projected on a chart, the small limit of error
to which it is liable for by drawing a parallel line on each
side of each projected Pparallel of equal altitude," at the
greatest distance from it that it is possible for the altitude
to be erroneous, we shall have the extreme limit of error;
and thus know what degree of confidence can be placed
in every altitude observed; and hence in the bearing of
the land, by one altitude; and in the latitude, or in the
longitude by chronometer; or in the position, when two
altitudes are observed.
That the navigator may see, at a glance, the errors of
longitude by a stictly accurate chronometer, to which he is
subject, for an error of the latitude used in finding the
apparent time of the ship, when he cannot get sights of
the sun at the time he bears E. or W., the following table
is calculated upon the principles of this method.
This will serve as a guide to know the greatest error of
longitude consequent to any observed altitude; but then
the greatest possible error of latitude must be allowed for
this purpose.
But since the real error of latitude cannot be known by
one altitude, (except it be a meridian observation when the
error of latitude is equal to 0,) it will still be necessary
"to project" the observation as directed, finding thereby
the bearing of the land, in a similar way as by meridian
observation.




43
TABLE D,
Sihowing the errors of longitude by chronometer, in nicadiecdl miles,
[60 to a degree of latitudej] for ONE MILE error of the latitude
used in finding the apparent time at the ship; zwhen the sun
DOES NOT bear true E. or WK at the time of observation.
0'S TRUE AMIPLITUDE" ERROR IN NAUTICAL  QS TRUE AMPLITUDE ERROR IN NAUTICAL
IN POINTS.     MIILESo        IN POINTSo     MILES.
4          0'049              41    1'103
0098              41         1 219
4          0 148              4`    1'348
1          0.199              5           1 497
41         0 -250             5i          1L669:1         09303              5~          1o871
41         0.358             50          2 115
2          0-414              6           2-414
21         0 473              6.          2 795
2'         0.535              6-          3-297
2:         0~599              6i          3 992
3          0'-668             7           5 027
-3,      0-742             71         6C741
31         0'821              7-         10'153
3          o0~907             7         20 356
4           1 e000            8         Infinite.
To know if this error of longitude be to the E. or W. of
the true longitude, the following Rules are given, and will
be found to include every case.
If the ( bear    southerly when in N. latitude,
l northerly when in S. latitude, j
And the time of observation be A.M., then, if
the latitcde, used in finding the apparent time, be    less}}
than the true latitude, the longitude will be too far {     t. }
But if the time of observation -be P.M., then, if
the latitude, used in finding the apparent time, be.less'}
than the true latitude, the longitude will be too far { est. 
P By amplit-ude is here to he understood the angular distance of the sun from the
prime vertical, whether the sun be in the horizon or shove it it would be perhlaps more
proper to call it the co-azimulth,




44
II,
And when the o bears northerly in north latitude,
n  outherly in south latitude, 
And the time of observation is A.M., then, if
the latitude used in finding the apparent time be grleate
than the true latitude, the longitude will be too far { st 
But if the time of observation be P.M., then, if
the latitcdde used in finding the apparent time be {rter }
than the trie latitude, the longitude will be too far X at;  }
N O T E.
After the vernal equinox, when in north latitude, observations may be often taken when the sun bears to the
northward of the E. and W. points; and in south latitude,
after the autumnal equinox, when he bears to the southward of them; also when the latitude and declination are
of the scame name, and the declination is greater than the
latitude in.
The method of using this table will be best shown by
AN   EXAMPLE.
If the sun bore S.W. by S., trule at the time of observation, n latitude 50~ N. by dead reckoning, what will be
the error of longitude by chronometer, if the latitude used
in finding the apparent time at the ship is erroneous
10 miles?
Ans. 23'3 minutes of longitude.
S.W. by S. is five points from W. In the table opposite five points amplitude is the error for one mile, in
nautical miles  -        -    1 497
multiplied by     -    -         10
gives the error for ten miles =  14'97 in nautical miles.
Enter table II, (Bowditch,) with the latitude in, 50~, as a
course, at the bottom of the page; and over it, in the
latitude column, find 14 97; opposite to it, in the distance
column, is 23'3, the error of longitude in minutes of
longitude.
In the foregoing example, if the latitude was erroneous
10 miles to the orth of the trute latitude; we have "the




45
O bearing southerly in north latitude;" and "the time of
observation P.M.;" and the latitude used in finding the
apparent time, "greater" than the "true latitude;" and the
rule gives the longitude too far west; the error by the table
was 23 3 minutes of longitude; which must be subtracted
if in west longitude, from the longitude by chronometer,
but added if in east longitude. When such observations,
however, are projected; these errors are evident by inspection, and the bearing of the land found.




IL
SOME REMARKS UPON THE ADVANTAGES OF THIS
METHOD.
There is no difficulty, by the common methods, in determining the position of places on the earth's surface, when
these places are situated on the land; for then sufficient
time may be taken to select the moment of observation;
and every circumstance can be taken advantage of, which
will conspire to produce a correct result. But when at sea
on board a ship, which is constantly changing her position,
and frequently approaching with rapidity a dangerous
coast, numerous circumstances often render the case one of
considerable difficulty at an important moment.
The common methods of finding the ship's place are
these: a meridian observation of the sun for the latitudes;
an altitude of the sun, bearing as nearly as possible E. or
W. for the longitude by chronometer; lunar observations;
double altitudes; dead reckoning; and occasional meridian
altitudes of the fixed stars, planets, or moon. Of all these
methods, the first two are those which are chiefly relied
on, by a great majority of navigators in all cases; and if
these observations could be had daily at the proper moment,
the ship's position would always be easily known, and no
danger need ensue.
But the proper moment cannot at all times be chosen to
take these observations; for the first must be taken when
the sun is o1 tlie meridian, unless the apparent time is
exactly known; and the second when he bears E. or W.,
except the latitude be previously accurately known; and
the observations will be partially or entirely prevented,
when thick weather prevails, or be liable to error if the
sun does not bear E. or W. at any time during the day.




47
For after the sun crosses the equator, throughout the
whole hemisphere from which he is receding in declination,
he rises and sets, at points daily further removed from E.
or W.; and since an altitude cannot be relied on, which is
not greater than 6~ or 7~ at least, this cause increases the
angle from the E. and W. points at which he is observable.
So throughout a whole hemisphere, for nearly 7 months in
the year, the szm is nzot observable in the proper E. or W.
points; and it may be found, that, when the sun is 23~ 28' S.
of the equator, he will rise and set in latitude 500 N., at an
angle from E. and W. of 38~ 17'; or by compass SE.   B E.
and S.W. 2 W., truez; and when only 7~ above the horizon,
he will bear S.E. 4 S. nearly; and     thus the least error of
longitude by chronometer, to which a ship is liable for an
error of latitude of 30 miles, is 1~ 2'; whenever, then, the
cltitude by account is used to find the apparent time at the
ship; and the sun does not bear exactly E. or TK, the longitude, by a good chronometer, will be always wrogr, unless
the latitude by account is strictly accurate. The necessity
of some toe le-e d of fi nding a slhip's place%, wIhen the latitude
is lucertii    the  app rent.
A meridian altitude shows the ship to be in some point of
a small circle of the sphere, called a parallel of latitude; (or
else on the equator, which is a great circle;) and that the
land, through which this small circle passes, bear E. or W.
The observation of the chronometer shows in a similar
manner, (the latitude being previously correctly obtained,)
that the ship is lilcegvise situated on another circle of the
sphere, called a meridian of longitude; and that the land,
through which this circle passes, bears N. or S.
And at the intersection of these tzvo circles, (allowing for
change of station between the observations,) is the ship's
position on the earth's surface.
But since the accuracy of the position depends upon the
accuracy of the latitude and apparent time, how, when these
are'uncert ain, or even ulI.Aomwn, cain the Ship's position be
fixed upon either of these wtvo circles, by either of these two
methods, as they are usually directed to be used, when
only one altitude of the sun can be observed, and that when
he bore neither E. nor W., nor on the mzeridiaz?




48
No practical rules are laid down which include so important a case; the only resource is, to use the latitude by
dead reckoning. How very erroneous this may prove, is
herein shown.
But these are the cases to which the Method of Projection
is peculiarly adapted. For if it be possible, at any time of
day when the sun is sufficiently high above the horizon, to
observe at all, one altitude of the sun; by noting the chronometer time, and observing roughly his bearing, we shall
find the ship to be, not on a parallel of latitude, it is true,
running E. and W.; nor on a meridian of longitude, running N.
and S.; but on as actual and as simple a circle as either of
these, namely, a parallel of equal altitude; running diagonally
to those circles at an angle which depends on the bearing
of the sun; which parallel, when projected on the chart,
by the rules, will pass through the position of the ship,
and show the bearing of the land, in its course, as it lies
projected on the chart, as truly as by a meridian observation, which can be observed but once in a day; and if two
of these parallels be projected, both the true latitude, and
longitude by chronometer, are evident by inspection.
When approaching the land, (and this is the time when
it is of the most importance to know the true position
of the ship,) it unfortunately happens, that thick weather
frequently prevails at considerable distance seaward, so that
the sun is visible only for a few moments during a run of
several days, and it is certainly important that a single
observation at such times should be rendered available.
There is no part of the seas, that is liable in a greater
degree to fogs and thick weather, than the English channels, north seas, &c.; and there is no part more crowded
by the fleets of all nations; the coast, too, is dangerous;
and the westerly gales are severe, and of long duration;
and ships are often placed in situations there, from uncertainty of their position, which render it dangerous "to
run," and often more dangerous to " lay by," or to " stand
off and on."
Having sailed from Charleston, S. C., 25th November,
1837, bound to Greenock, a series of heavy gales from the
westward promised a quick passage; after passing the
Azores, the wind prevailed from the southward, with thick




49
weather; after passing longitude 21~ W., no observation
was had until near the land; but soundings were had not
far, as was supposed, from  the edge of the bank. The
weather was now more boisterous, and very thick; and the
wind still southerly; arriving about midnight, 17th December, within 40 miles, by dead reckoning, of Tasker light;
the wind hauled S.E., true, making the Irish coast a lee
shore; the ship was then kept close to the wind, and several
tacks made to preserve her position as nearly as possible
until daylight; when nothing being in sight, she was kept
on E.N.E. under short sail, with heavy gales; at about
10, A.M., an altitude of the sun was observed, and the
chronometer time noted; but, having run so far without
any observation, it was plain the latitude by dead reckoning
was liable to error, and could not be entirely relied on.
Using, however, this latitude, in finding the longitude by
chronometer, it was found to put the ship 15' of longitude
E. from her position by dead reckoning; which, in latitude
52~ N., is 9 nautical miles; this seemed to agree tolerably
well with the dead reckoning; but feeling doubtful of the
latitude, the observation was tried with a latitude 10' further
N.; finding this placed the ship E.N.E. 27 znatical miles, of
the former position, it was tried again with a latitude 20' N.
of the dead reckoning; this also placed the ship still further
E.N.E., and still 27 azctical miiles. These three positions
were then seen to lie in the direction of Small's lhlzt. It
then at once appeared, that the observed altitude must have
happened at call the three points, and at Smzall's light, and at
the ship, at the same instant of time; and it followed, that
Small's light must bear E.N.E., if the chronometer was
right. Having been convinced of this truth, the ship was
kept on her course, E.N.E., the wind bearing still S.E., and
in less than an hour, Small's light was made, bearing
E.N.E. 2 E., and close aboard.
The latitude by dead reckoning, was erroneous 8 miles;
and if the longitude by chronometer had been found by
this latitude, the ship's position would have been erroneous
31 I minzutes of lo)gitude, too far W., and 8 m'iles too far S.
The ship had, from current, tide, or error of log, overrun
her reckoning, 1 mile in 20. (See plate III.)
[7]




50
Thus it is seen, that an observation taken at any hour of
the day, and at any angle between the meridian and E. or
W. points, is rendered practically useful, inasmuch as the
chronometer can be depended on.
A great proportion of the chronometers now in use, are
sufficiently accurate to determine the ship's position; and
particularly when they have been out of port only a month
or two; the government of Great Britain has spent thousands to perfect them; but it should be recollected that
the Greenwich time is only one of the quantities which
must be correct, to find the longitude; we must be sure
that the time at the ship is correct also; and it can scarcely
be doubted, that errors of latitucle have caused the loss of as
many ships, as errors of chronometers; while chronometers
have borne the blame, not only of their own occasional
imperfections, but also of these errors of latitude, to which
the navigator is subjected, from the prevalence of thick
weather, gales of wind, and when a ship is under short sail,
wearing, and tacking, and in tide-ways near the land;
making leeway, and changing the rate of sailing with different cargoes on board.
The proverb, that "a seaman always knows his latitude,"
had its origin in those days, when the longitude was the
great point to be determined; for before lunar observations
were used, or chronometers invented, the only observations
which could be relied on, were those for finding the latitude; and thus the latitude was comparatively certain; but
the longitude was estimated by the log, and great errors
were common.
But at present the case is almost completely reversed;
for, with a good chronometer, used with care, it is the
latitude which is the great desideratum; if the latitude is
accurately known, a single altitude is sufficient to find the
ship's place; and if it be uncertain, the Method of Projection
affords the most complete substitute for a meridian observation,
the altitude being observed at any hour.
By this method, if the chronometer is wrong, and the
latitude uncertain, the bearizng of the land would be erroneous,
by a quantity equal to the whole error of chronometer,
when the sun is observed bearing E. or W.; but as the angle
increases, from the E. and W. points at which an observation




51
is taken, the error in the bearing of the land, caused by the
error of the chronometer, constantly diminishes; and when
the sun is on the meridian, va;nishes, when the observation
becomes a real meridianz altitude whence the true latitude
may be found as usual. But by the usual method, no
approximation to correctness can be made.
The case when a second altitude is observed, is deduced
from the manner in which the first is projected; and in
an analogous manner, as a meridian observation, and one
taken when the sun bears east or west, by the usual method,
place the ship at the intersection of a circle of longitude
with a parallel of latitude, allowing for change of station,
so the intersection of the first projected parallel of equal
altitude with the second, is the ship's place.
The extra work to project the second parallel is trifling,
for it will be noticed that many of the logarithms are the
same as in the first.
The,advantages of the method by Projection may be
summed up as follows:
1. When the latitude, &c., are uncertain, one altitude of
the sun, at an, hour, with the chronometer time, is available
in a similar manner as a meridian observation, which can
be taken only once in a day.
2. The errors of longitude by chronometer, consequent
to any error in the latitude, are shown by inspection.
3. The sun's azimuth is found at the same operation.
4. In addition to these results, found by one altitude, two
similar altitudes give the true latitude, and also the longitude by chronometer. By the common methods of double
altitudes, the longitude must be found by a subsequent
calculation; which circumstance renders this method much
the shortest.
5. The usual simple calculation for finding the apparent
time at the ship, is known and daily practiced by every shipmaster who uses a chronometer. No other formula is used.
6. Double altitudes of the sun are therefore within the
reach of all persons who use chronometers, and who are
unacquainted with the various formulas laid dcown in the
books.




IIlI
EXPLANATION      OF THE PRINCIPLES UPON WHICH
THIS METHOD DEPENDS,
To facilitate the understanding of the theory of this
method, a reference to the following common definitions
relative to spherical bodies may be necessary.
A sphere is a uniformly round body, every point on the
surface of which is equally distant from  a certain point
within the body, called the centre.
If any plane or flat surface pass through the sphere, the
intersection of the surface of the sphere by the plane is the
circumference of a circle.
A great circle of a sphere is one whose plane passes
through the centre of the sphere, and so divides the sphere
into two equal parts, called hemispheres.
A s8mall circle of a sphere is one whose plane does not
pass through the centre of the sphere, and consequently
divides the sphere into two tneqtual parts.
The pole of any circle of a sphere is a point on the surface
of the sphere from which every point in the circumference
of the circle is equally distant; thus every circle of the
sphere has two poles, and the straight line joining them is
the diameter of the sphere.
One half   of the spherical surface of the earth being
illuminated by the sun at a given instant, while at the
same time the opposite portion is in the shade, that line,
which is the boundary between the illuminated and dark
hemispheres, is called   by geographers     THE CIRCLE OF
ILLUMINATION."
* The corrections being made by the usual tables, for Parallax, Semidiameter, Refraction, and the Spheroidal Figure of the Earth-and if the eye be elevated, for
Dip also.




53
It is a great circle, and its plane passes through the centre
of the earth, dividing it into two equal parts; in the same
manner as the equator is a great circle of the earth, and
divides it into the northern and southern hemispheres.
But these are also divided by small circles of the sphere,
parallel to the equator, which are called "parallels of latitude;" and by their means the latitude is reckoned, all
places situated on the equator having their latitude equal
to 0~; and proceeding towards the poles, the latitude of
places on these small circles increases regularly, until,
arriving at them, it becomes equal to 90~.
In like manner, all those places which are situated on the
circle of illuZination, since they have the sun's centre in the
horizon, have his altitude equal to 0~, and that point on the
surface of the earth, next towards the sun, and which is the
pole of the circle of illumination, has the sun in the zenith;
consequently, at that point his altitude is equal to 90~. The
intermediate altitudes of the sun may likewise be reckoned
on smacll circles, parallel to the circle of illumination; and
which may be callad PARALLELS OF EQUAL ALTITUDE; since
they serve the purpose of showing all those places on the
earth's surface, which have an eyqal altitude of the sun, at
the samte instant of time; the pole, at which the sun is in the
zenith, may be called the POLE OF ILLUMINATION, and the whole
system of circles, THE SYSTEM OF CIRCLES OF ILLUMINATION.
Thus it appears, that, as all those places on a given small
circle of the earth, called a parallel of latitude, have the
same latitude, and that degree and minute which is the name
of that parallel; so, all those places, on a given small circle,
called a parallel of equal altitude, have the SAME ALTITUDE
of the sun, and the same degree and minute which is the
name of this parallel.
Since the poles of the equator, and of the parallels of
latitude, coincide with the extremities of the earth's axis,
the system of parallels of latitude is not affected by the
daily rotation of the earth; but always remains constant;
indeed, it has been constructed with particular reference to
this subject.
Such, however, is not the case with the system of circles
of illumination; because, the poles of this system  do not




54
coincide with the extremities of the earth's axis; for the
sun being always vertical to one of its poles, this follows
the sun, (at the rate of 15~ per hour,) in his apparent daily
course from east to west, and also in the ecliptic, through
all the degrees of the sun's declination.
But if, at any installt, we can project upon the earth's
surface the position of this moving system, or any parallel
of equal altitude belonging to it, which may correspond to
an observed altitude of the sun, we shall have as sure a
method of determining the position of places on the earth's
surface, as by means of the system of parallels of latitude.
But it may be well to consider in what manner the
system  of circles of illumination intersects the different
parallels of latitude, and the meridians of longitude.
Because the sun is always vertical to the pole of illumination, and the parallels of latitude on the earth's surface are concentric with the parallels of declination on the
celestial sphere, the latitude of this pole is always equal to the
stn's decl;ination.
If, then, about a point, (as Z, plate I,) in any longitude,
but in that parallel of latitude which is equal to the sun's
declination as a centre, we describe small circles of a sphere
at every 10~ distant, after the manner of the parallels of
latitude, and this may be seen to advantage on a terrestrial
globe, we shall observe, that the parallels of equal altitude,
which we are describing, cut the parallels of latitude and
the meridians of longitude at all possible angles; in the
N.W., N.E., S.W., and S.E. directions from the pole of illumination, Z, they will cut them diagonally; in the north,
south, east, and west points from this pole, they will touch
them as tangents; and at intermediate bearings will make
intermediate angles with them.
Each parallel of equal altitude, it will be observed, is
described round the pole of illumination, as a centre, at
a distance, measured on the arc of a great circle passing
through this pole, which is equal to the complement of the
szuz's altitude corresponding to each parallel respectively.
Hence, with a given altitude of the sun, the corresponding
parallel of equal altitude cuts only certain parallels of latitude, not north of a particular latitude in the northern, nor
south of a particular latitude in the southern hemisphere,




55
The distance, then, of the north and south points, in any
parallel of equal altitude from the pole of illumination, is
equal in degrees and minutes to the complement of the
sun's altitude; but these points being on the same meridian
as the pole of illumination, this distance is also equal to
the difference of latitude between the pole and either point;
and because they are on the same meridian as this pole,
the apparent time at these points will be noon, for it is
always noon at the meridian of the point which has the
sun in the zenith.
Now the difference of longitude, between any two places
on the earth's surface, is expressed by the difference of the
apparent times at those places, turned into degrees, 15~ to
the hour.
The difference of longitude of any poiZt in a parallel of
equal altitude, corresponding to any observed altitude of
the sun, from the meridian of its pole of illumination, will
then be expressed by the difference of the apparent times
at those places; that is, by the difference between 0" Om Os,
and the apparent time fronz oon at the point, in the parallel
of equal altitude, which may be given. And the difference
of longitude between any tzuo points given will be expressed
in a similar manner.
But if the altitude, and the declination of the sun be
given, the apparent time will vary only with the different
latitudes which may be used in the calculation.
If, then, we assmze any latitude, not north of the north
point, nor south of the south point, of the parallel of equal
altitude corresponding to an observed altitude of the sun,
the declination being known also; and thence find the
apparent time for noon, (by method 3, Bowditch,) the
dicference between such time, and 0~, Om, 0, turned into
longitude, (15~ per hour,) will be-the dciference of lonlitude
of the point in the parallel, from the meridian of the pole
of illumination, and its latitude is had by the asszumtion: thus
the position of such point is found relatively to the pole of
illumination, which is situated in a certain given latitude.
In the same way, by assuming tzvo latitudes, with the
same restrictions, the difference of their apparent times
will express the difference of longitude of the two points
having the tzuo assumed latitudes; and their positions rela



56
tively to the pole will be expressed by their times from noon,
turned into difference of longitude.
Each point, thus found, has another point corresponding
to it in its parallel of equal altitude, in the same assumed
latitude; namely, one on each side of the meridian of the
pole, and at equal differences of longitude from it; and
they correspond to A.M. and P.M. times of observation; and
to distinguish on which side of the meridian of the pole
of illumination the point lies with which we have to do, we
should notice in practice the bearing of the sun. The required point being westward from the pole, if an observation
be A.M.; and eastward, if P.M.
By assuming as many latitudes as we please, we may
thus project as many points as are necessary; and by
joining all the points determined by a curve line, the whole
parallel of equal altitude corresponding to any observed
altitude of the sun, will be projected, in latitude, relcively
to the pole of illumination.
From the foregoing considerations, the rule for solving
the following problem is deduced.
PROBLEM          I.
The correct altitude of the sun's centre being observed,
and his declination being given, it is required to project,
on Mercator's chart, the corresponding parallel of equal
altitude relatively to its pole of illumination, showing what
parallels of latitude it cuts, and in what manner it cuts
both them and the meridians of longitude.
NOTE.
Owing to the distorted shape of the earth's surface by
Mercator's chart, the curves will not appear as circles, as
they would be if projected on the earth, or on a terrestrial
globe.
R u L E.
1st. Find the complement of the sun's altitude; which
gives the difference of latitude between the pole of illumination and the north and south points of the parallel
of equal altitude.




5 7
2d. Project then on the chart, (plate I,) in a latitude
equal to the sun's declination, the pole of illumination,
the point Z, in any assumed longitude, as 0~; and on the
meridian of Z, in their respective latitudes just found,
project the north and south points N. and S.; at which the
apparent time is noon, or Oh Om 0s.
3d. Asslune a latitude, as that of any point, A, B, or C,
&c., not north of N., nor south of S.; and with the given
altitude and declination, find the apparent time from noon,
(method 3, B,) which, turned into longitude, 15~ per hour,
will be the difference of longitude between Z, and A, B, or
C, &c., and its latitude is had by assumption.
4th. In the latitude assumed, (if that of A,) on each side
of the meridian of Z, project the points A, A', with the
difference of longitude found above; then A and A' will
be two points in the parallel required; in the same manner
find as many points as are necessary.
5th. By joining all the adjacent points with a curve line,
the whole parallel, and the latitudes it cuts, &c., are evident
by inspection.
NOT E.
In this problem, the difference of longitude of the points
have been reckoned from the meridian of Z; but the curve
could be also projected, by taking the meridian of any point,
as A, for a starting point, and thence might be reckoned the
points B, C, &c., and finally the meridian of Z, and then Z
be projected.
EXAMPLE       I.
The sun's central altitude is 60~; the declination 10~ N.,
it is required to project the parallel of equal altitude, &c.
1st. To find the complement of the sun's altitude, and
the latitude of the N. and S. points.
from    90~
sub.    60
remains  30~ =- complement of the sun's altitude.
To    30~ comp. 0 alt.
add   10 dec.
sum. 40~= lat. of N.
ciff.  20 = lat. of S.
[8]




58
2d. Project (plate I) the points Z, N, and S, in their
proper latitudes, and in any one assumed longitude, as 0~.
3d. To find and project any point; as one in latitude
30~ N., which call A.
Lat.  30~ 00 N.     -   -    -      sec. 0-06247
Dec.  10 00 N. -       -   -   -    sec. 0-00665
Diff.  20~ 00     nat. cos. 93969
0 Alt. 60 00      nat. sin. 86603
diff.     7366  log, 3S86723
H.  M.  S.
1 35 57 05     =     log rising,  =    3'93635
1 35 57 05      23~ 59' 15"'75, or the difference of
longitude of A in latitude 30 N., from the meridian of Z,
where it is noon.
4th. On each side of the meridian N Z S with this difference of longitude project two points, A and A', in latitude
30~ N.; which will be two points in the curve required;
the altitude is observed at A, when the time is P.M., and
at A', when A.M.
Find other points B, B', C, C', &c.
5th. Join all the points, and the requisitions of the
problem are evident by inspection.
In the same manner are projected the parallels of equal
altitude, corresponding to sun's altitude, 0~, 10~, 20~, 30~,
&c., when the sun's declination is 10~ N.
After any lapse of time, the sun, and, consequently, the
pole of illumination, to which the sun is always vertical,
and likewise all the parallels of equal altitude, will have
passed, at the rate of 15~ per hour, due west, (no change
of declination having taken place,) and will have arrived
after such time at some point, as Z', (plate I,) whose difference of longitude from Z will be equal to the elapsed time
turned into longitude, 15~ for an hour; because the angular
space passed over is equal to the product of the time by
the angular velocity.
But if, at the expiration of such time, a second altitude be
observed, and its corresponding parallel of equal altitude be
projected as before, it will have for its pole Z'; and it must
intersect the first projected parallel in two points, one to




the northward and one to the southward of the poles Z
and Z'; and one of these two intersected points must be
the position of the observer; because both altitudes are
observed at one station; and if a point is situated at the
same time in two circles, it can only be at one of the zwo
intersected points; which one of the intersections is the
observer's place, obviously depends on the bearing of the
sun at the times of observation; and the two curves being
projected on the chart, the intersection will be situated in
a latitude which is evident by inspection.
The difference of latitude between the two poles of illumination, Z and Z', is equal to the difference in the sun's
declination, which may have taken place during the elapsed
time. This change of declination must not be neglected,
but each pole projected in that latitude which is equal to
the sun's declination at each observation respectively.
Two altitudes of the sun may be observed, either both
A.M., or both P.M.; or one A.M., and the other P.M.; for
as the system of circles of illumination passes westward 
the wtestern arcs of the parallels of equal altitude, which are
interior to the first projected parallel, cut its uestern arc in
two points, and poles being to the eastward of the intersections, the time at which the altitudes are observed at either
intersected point, will be A.M.
So, also, the easternz arcs of the exterior parallels, cut the
eastern arc of the first projected parallel, in two points; but
the poles now being to the zvestward of the intersections,
the time will be P.M.
And, after a greater time has elapsed, eastern arcs of both
exterior and interior parallels will cut the wuestern arc of the
first projected parallel, and the intersected points lie betzveen
the longitudes of the two poles, and show that the time at
which the first altitude was observed, is A.M., since its pole
is eastzard; and the time of the other is P.M., since its pole
is zestzvard of the two intersected points.
We shall now be able, using the rules of problem I, to
project the parallels of equal altitude, corresponding to two
observed altitudes of the sun; the elapsed time giving the
distance between their respective poles of illumination;




60
and intersection which takes place, and which is designated
by the bearing of the sun, giving the true latitude of the
place of observation.
From  these considerations are deduced this problem
and rule.
PROBLEM         II,
Having noted the elapsed time between two observed
altitudes of the sun's centre, the declination at each time,
and the sun's bearing being given, required to project on
Mercator's chart the two corresponding parallels, showing
how they cut the parallels of latitude and meridians of
longitude, within their respective limits of latitude, how
they intersect each other, and the true latitude of the places
of observation.
RU L E.
1st. By problem I, project the parallel corresponding to
the first altitude.
2d. Turn the elapsed time into difference of longitude,
15~ for an hour.
3d. With which difference of longitude, project, wvestward
always from Z, the point Z', in a latitude equal to the sun's
declination at the time of the second observation; and Z'
will be the pole of illumination of the parallel corresponding to the second observed altitude.
4th. Proceed, as in problem I, to project the parallel
corresponding to the second altitude relatively to Z'.
5th. Upon inspection, the curves will be seen to intersect,
as at A and a, (plate I.)
6th. The true latitude is at one of the intersected points;
which point it is depends on the bearing of the sun.
The other requisitions are evident by inspection.
E X A  P LE   II.
The two altitudes are 60~ and 40~; the declination 10~ N..
supposed invariable between the observations; the elapsed




61
time is 11 41m 188 89; the sun bearing between S. and W.,
it is required to project, &c.
1st. The first altitude is the same as in example I, and
is already projected.
2d. The elapsed time, h 41m 18s 89, is equal to 25~ 19'
43'" 4, (15~ for an hour.)
3d. In latitude 10~ N. project Z', 25~ 19' 43'"4 west
from Z, (plate I.)
4th. Project N' and S' as in problem I; and to find any
point, as one in lat. 30~ N., which call X, proceed as under.
Lat.  30~ N.   -        -          sec. 0 06247
Dec.  10 N.           -            sec. 0-00665
Diff.  20         nat. cos. 93969
( Alt. 40         nat. sin. 64279
diff.   29690    log, 4 47261
H.  M.  S.
3 17 15 94          log rising, - 4 54173
3 17 15o94 = 49~ 18' 59"-1, or difference longitude
of X in latitude 30~ N. from the meridian N' Z' S', where it
was noon at time of 2d observation.
On each side of N' Z' S', with this difference of longitude, project two points, X and X', in latitude 30 N;
which are two points in the curve required.
Find the other points Y, Y', &c., in the same way. Join
all the points, and the whole curve will be projected.
5th. It is now evident, how the curves intersect each
other, as at A and a.
6th. The sun having been west of the place of observation for both altitudes, shows that both were taken P.M.;
and having borne southerly also, the northern intersection,
A, is at the true latitude of the place; and it will be found
to be on the chart in 30~ N.
Thus far we have determined the positions of the projected parallels, relatively to their poles, and the distance of
these from each other; and the whole only in relation to
the latitude of the place of observation; consequently, it
is only the bearing of the land in that latitude which has
become known.




62
To determine their actual position, in longitude, also, on
the earth's surface, and project them at any instant, we
must have means to arrest, in their passage, both systems
of circles of illumination, while moving from east to west.
If the longitude of the place of observation be known,
or if the time be noted by chronometer which shows true
mean time at a known meridian; and the altitudes calculated by assumed latitudes, as before, to find the apparent
time, the difference between the apparent times and the
chronometer time will give the longitude of each point,
and the actual position of each curve is fixed in both latitude and longitude.
By such means, having a single altitude of the sun and
chronometer time, we may find a ship at sea to be on some
point of a small circle, which has for its pole, the pole of
illumination, in like manner, as a meridian altitude of the
sun fixes the position of the ship in some point of a small
circle, called a parallel of latitude, and which has for its
pole, the pole of the equator.
Hence, the bearing of any land, through which the curve
passes, is seen by inspection, as we may observe the bearing
of the land, through which the parallel of latitude passes.
Hence, we can solve the following problem.
PROBLEM         III.
The true altitude of the sun being observed at any place,
his declination and bearing, and the true mean time at
Greenwich, by chronometer, being given; it is required to
project the corresponding parallel of equal altitude, showing
what meridians of longitude it cuts, east or west of Greenwich, and what parallels of latitude; and in what directions;
consequently, what land it passes through, and how such
land bears from any point in the curve.
R u L E..
Proceed, as in problem I, to project the curve; only
taking notice, that when "'tle apparent time" is found for
any point, the difference between it and the chronometer




63
time, (corrected for equation of time and rate,) gives the
longitude of such point, east or zest of Greenwicz- (while
its difference of longitude, from the meridian of its pole of
illumination, will still be expressed by the; c1apparent timze
from noon2" as in problem I.) Thus, are all the points projected in latitde and longihtde, and the requisitions of the
problem are obvious on inspection.
EX A M   LE   I   I
Sun's altitude 60~; declination 10~ N.; corrected time
at Greenwich, by chronometer, 011 Om 0s, sun bearing
southwesterly. It is required to project the curve passing
through the position of the observer, showing how the land
bears from any point in the curve.
ist. This example is the same as example I, except the
chronometer time. It will be necessary, then, only to see
how this time affects the position of the curve; (by example I, the curve was projected in any assumed longitude.)
We have Greenwich time correctedcl  i H M..  s 
for rate and equation,     5
The apparent time, example I, in 
assumed latitude 30~, at a point  5   7     P TM
A, (plate I,) sun bearing south3     0    5 Eo.
westerly, was              J
Difference,           1 35 57
The apparent time is greater than the Greenwich time,
showing that the point A, in latitude 30~ N., is P 351' 57s,
or 23~ 59' 15'" 75 east of Greenwich, Project A, then,
in such longitude and latitude, and its achtal position is
determined,
In the same manner are all the other points projected,
and the curve drawn.
The other requisitions of the example will appear if we
refer all the points AA', BB', &c., to a chart, on which
the land is designated; thus A is situated on the north
coast of Africa, in the same meridian as a part of Greece;
B, near the south of Sicily; and the curve passes likewise
near the south of Sardinia; over Majorca, through Spain,
near Madrid; through Portugal, near Lisbon, not far from




64
Madeira; thence to within 6~ west of Cape Verde Islands;
thence 10~ east of Cape St. Roque, S. A.; thence 3~O south
of St. Helena; and thence reaches the continent of Africa
again, in latitude 18. S., near Cape Negro; thence over
the interior of Africa, towards Egypt, to the point A again;
all these places having the same altitude of the sun at the
same instant; the place of the observer being of course
in the curve, having OBSERVED the same altitude at the
same time.
Thus, the land through which the curve passes, is evident
by inspection; and also its bearing from any point in the
curve; and thus, by projecting the curve for any observed
altitude, we have as sure a method of determining the
position of places, as by means of parallels of latitude and
a meridian observation.
From the foregoing three problems, the following one is
directly deduced.
PROBLEM         IV.
Having two altitudes of the sun, the declinations at both
observations, and his bearings, and the Greenwich times
by chronometer; to project the corresponding parallels of
equal altitude, showing what meridians of longitude and
parallels of latitude it cuts, and in what manner; what
land they pass through; how it bears from any point in
either curve; and the true latitude and longitude of the
place of observation.
RULE.
1st. Project both parallels, as in example II, applying
the Greenwich time, as in example III.
2d. The requisitions of the problem will be evident by
inspection.
N O:T E.
The "elapsed time" does not appear, as such, in this
problem; for it enters into the longitudes of the points of
the second parallel, placing them always so much further zvest,
as greater time elapses between the observations.




65
EXAMPLE        IV.
Sun's central altitude was 60~, when chronometer time
corrected for equation was 01 0h OS; and 40~, when chronometer time corrected was 1h 4r1 18s 89; declination
10~ N., invariable between the observations; sun's bearings
southwesterly, required to project, &c.
This example is the same as example II, except chronometer time; project the curves as directed, (plate I,)
applying the chronometer time, as in example III.
This being done, the requisitions are manifest; the latitude sought, being 30~ N., and the longitude 23~ 59' 15'" 75
east; since if, from the apparent time at A, or X, (which
two points coincide, at the time of the second observation,)
namely:
H.  I.  S.
App. time, 3 17  15' 94 P.M. (sun bearing southwesterly,) we
subtract,  1 41  18 89 P.M. the corrected chro. time,
the cliff is 1 35  57 05 =- 23~ 59'" 15"-75 east longitude,
since the chronometer time is least.
The problem of double altitudes admits on these principles, another simple solution, as follows:
PROBLEM V.
Two correct central altitudes of the sun, his declinations
and bearings, at each observation, and the elapsed time
being given; required to project, on a terrestrial globe,
the latitude of the place of observation.
R U L E.
1st. Turn the elapsed time into degrees, 15~ to an hour.
2d. In the latitude equal to the sun's declination, at each
observation, project the two, poles of illumination in two
points, distant a difference of longitude found above.
3d. On the eastern pole, (or that corresponding to the
first altitudes) as a centre, and with a distance measured on
a great circle, equal in degrees and minutes to the conpleL9]




66
ment of the first altitude, describe, with a pair of dividers an
arc in that direction from the pole indicated by the bearing
of the sun, at first observation.
4th. On the western pole, as a centre, with a distance
equal to complement of the second altitude, describe an
arc cutting the former arc.
5th. The intersection will be in the latitude of the place
of observation, if both altitudes be observed at one statio.
i.
In the foregoing problems, it has been supposed, that
both altitudes were taken at one station; but it is necessary
to show how to allow for any change of station between
the observations.
The curve Z M, plate I, is an arc of a great circle, passing
through Z, the pole of illumination, where the sun's altitude
is 90~; and through M, a point in the circle of illutmination
where the sun's altitude is 0~; it is of course perpendicular
to all the parallels of equal altitude.
It is plain, then, that for every mile a ship sails on the
great circle M Z towards Z, that is, at right angles to the
parallels, she increases the sun's altitude one minzte; and
every mile sailed from Z, decreases his altitude one minute.
And if she sails at right angles to M    Z, that is, on a
parallel of equal altitude, she would neither increase nor
diminish it.
So, in sailing at a greater or less angle than 8 points
from  a parallel of equal altitude, she would increase or
diminish the altitude in a proportional manner; and in so
sailing would make a " difference " of altitude, in a similar
manner as, in sailing from  a parallel of latitude, she would
make a "difference" of latitude, according to the course
and distance sailed.                   -
In sailing from the point, then, which was the ship's
place in the parallel of equal altitude corresponding to the
first altitude, she will arrive at a point in another parallel
which belongs to the same system of circles of illbuinzation,
with the first projected parallel, on a certain course and




67
distance; and this new parallel will correspond to an
altitude which would have been observed at the instant
of the first observation, had the ship been at this new point
at the same time; but this new point is that at which the
second altitude is taken, after the time elapsed, as noted by
the watch.
The "difference" of algttude made good, then, proportional
to the course and distance sailed from the first projected
parallel, will be the correction of the first altitude for
change of station, additive or subtractive, as explained
before. This " difference" of altitude will be measured on
the arc of a great circle, passing through Z, (perpendicular
to the parallels of equal altitude,) and this new point to
which the ship may have sailed; and the arc, which is
intercepted between this point and the first projected
parallel, will be the correction in miles. We have, then,
the following rule.
Set off, from  the first projected parallel, the distance
sailed in the direction of the course made good between
the observations. Through this point project a curve line
parallel to the first projected parallel, and the intersection
of this curve line with the second projected parallel, will be
the true latitude corrected for change of station.
II.
Since MV Z, is an arc of a great circle passing through
Z and M, perpendicular to all the parallels; and the sun's
centre, and Z and M, are all in one plane, and the sun is
perpendicular to Z, therefore the arc Z M, lies wholly in
this plane, and the direction of the sun, from any point in
Z M, is projected in the direction M Z, on the chart; but
the angle at any point on the earth's surface, which the
bearing of the sun projected, makes with the meridian
projected at that point, is the sun's true azimuth. Thus,
at M, the sun bears in the direction M n, or about E b
N. ~ N.; and the angle which M n makes with the meridian passing through M, is the sun's true azimuth at M;
now M n is perpendicular to the circle of illumination, and
n o is perpendlicuilar to the parallel of equal altitude for
1O0~ and so of the other portlions of M Z; soE in order to




68
project the sun's azimuth, at the time of observation of his
altitude, at any point in the corresponding parallel of equal
altitude, we have this rule:
Draw a tangent, at the given point, and erect a perpendicular to the tangent at the point given; the perpendicular
will be in the direction of the sun at that point, and the
angle it makes with the meridian, passing through this
point, is the sun's true azimuth.
X IXI,
From the manner in which the parallels of equal altitude
cut the meridians of longitude and parallels of latitude,
may be seen how, with a given altitude of the sun and the
chronometer time, different latitudes, used in finding the
apparent time, give different longitudes by chronometer.
For, if the sun be observed A.M, in north latitude, when
he bears southerly, then latitudes at greater distances from
Z, (plate I,) give smaller differences of apparent time from
noon, and, therefore, a less difference of longitude, if in
wzest longitude, and a greater difference of longitude, if
in east longitude, and analogous differences under other
circumstances.
It may be seen, too, that when the sun bears east or?west,
a considerable difference in the latitudes used in finding
the apparent time at the ship, will occasion but a trifling
error in the longitude by chronometer.
Hence, it is not only because the sun is  rising or falling
faster," when bearing east or west, that this is the best time
to take chronometer sights for the longitude; but because,
also, at such times, an erroneozs latitcde will not much affect
the result.
So, also, when the sun bears north or south, it is easy to
find the latitude by meridian observation; but the longitude
by chronometer is subject to great errors owing to the
difficulty of finding the apparent time at the ship, and not
only because the sun is " rising or falling" sloower, but also
because a slight error of latitude gives a very great error
of longitude by chronometer.
The errors of longitude by chronometer, then, consequent
to an erroneous latitude used in finding the apparent time




69
at the ship, increase regularly, from the time the sun bears
east, until he reaches the meridian, when the error is a
nazxinznm; and thence diminish until he bears wzest, when
they vanish.
By reference to plate I, and from these remarks, it will
be found, that the error of longitude any error of latituzde
produces, at the instant of observing an altitude of the
sun, is equal to the diference of longitude between any two
points in the projected parallel, whose clfference of latitude is
the error of latitude.
Thus, in the winter season, when the sun does not rise or
set either east or west, and is not observable, frequently, until
he bears S. E., or more southerly, it is of the utmost importance to estimate such errors.
IV.
In practice it is not necssary to project the whole curves,
but only such ares as will include the intersected points
required.  Two points in each arc are sufficient, that part
of the curve which is required being indicated by the
bearing of the sun.
It will be found convenient, in finding the apparent
times, for these two points, to assume two latitudes to be
used in the calculation, one on each side of the latitude
by dead reckoning; one being the next degree less and
the other the next degree greater than such latitude, and
without any odd minutes; the position of the four points,
and their arcs being projected, will, in general, be found
to include the intersected point, or be very near to it; for
if we join the points by straight lines, they will either
intersect each other, or converge towards a point beyond
the limits of the assumed latitudes; in this case we have
only to produce the lines to an intersection.
These two straight lines, although they do not lie strictly
in the curves, but may be regarded as chords of their
respective arcs, will not sensibly differ from the arcs themselves, or even from their tangents, at points near the intersection; particularly as the parallels most frequently
used, have a large radius.
Should, however, the altitudes be great, in which case




70
the curves have a smaller radius, we can choose such latitudes as shall be nearer the intersected points and thus
reduce the error, on this account, as much as we please.
Also, if the altitude be observed when near noon, the
assumed latitudes should not be chosen too great, or otherwise, as mentioned in the note to rule 1, practical part;
and they should be such as are much nearer to the supposed
latitude.
Hence, the following practical problem is deduced, when
the times are noted by a common watch.
PROBLEM VI.
Two correct central altitudes of the sun being given,
and also the elapsed time, the declinations and bearings
of the sun at both observations, the course and distance
made good between the observations, and the latitude by
dead reckoning; it is required to project on a "particular"
Mercator's chart, those ARCS of the corresponding parallels
of equal altitude, which are mutually intersected; showing,
lst, the true latitude; 2d, the correction for change of
station; 3d, the sun's true azimuth at each observation;
4th, the errors of longitude consequent to any error of
latitude, when the times are noted by the chronometer.
R     L E.
1st. Turn the elapsed time into degrees, (150 to an hour,)
which will be the difference of longitude between the two
poles of illumination, Z and Z'; the eastern pole, or that
which corresponds to the first altitude, being called Z,
and the zestern pole, which corresponds to the second
altitude, Z'.
2d. Assume two latitudes, one of which is the next degree
less, (without any odd minutes,) and the other the next
degree greater than the latitude by account.
3d. With the first altitude and declination, find (method
3, B) the apparent time fron noon with eacc of the two
assumed latitudes; turn these resulting times into differences of longitude, 15~' to an hour; and, if the altitude was
observed A.M., they will show the differences of longitude




71
of tzuo points in the required arc, WEST of the meridian of
Z; but if the altitude was P.M., EAST of the meridian of Z;
these two points having the assumed latitudes respectively;
name that point which has the least assumed latitude, A,
and the other, A'
Do the same with the second altitude; and name that
point which has the least assumed latitude, B; and the
other, B'. The differences of longitude of B and B', will
be reckoned from Z', and not from Z.
4th. Project on the chart, plate II, the four points A, A',
B, B, in t#eir respective latitudes; and distant their respective differences of longitude, from  the meridians of
their respective poles Z and Z'; (which poles are distant
from each other a difference of longitude equal to the
elapsed time turned into degrees;) and the points must
be projected on that side of these meridians, designated by
A.M. and P.M. times of observation.
E XPLANATORY       NOTE.
Project A in its respective latitude, and in cay assumed
longitude; if A.M., mark it "A, west of Z;" (so zmany
degrees as have been found from the corresponding apparent
time from noon;) but if P.M., mark it " A, east of Z," (so
mazzy degrees.)
Project A' in its respective latitude, but with a difference
of longitude from A, (east or west,) equal to the difference
between the longitudes of A from Z, and A' from Z; and
mark it "A' (east if P.M., or west if A.M.) of Z," (so many
degrees and minutes, as the case may be.)
Now to project B, B', with reference to A, A', we must
first find the difference of longitude of A from Z'; because
their differences of longitude are not reckoned from Z, but
from Z'; this is done by means of the elapsed time, or the
difference of longitude between Z and Z', thus:
I. If both observations be A.M., then difference of longitude of A, west of Z' = difference of longitude of A, west
of Z - difference of longitude of Z', west of Z.
II. If both observations be P.M., then difference of longitude of A, east of Z'= difference of longitude of A, east
of Z + difference of longitude of Z, east of Z'.




72
III. If one observation be A.M., the other P.M., then
difference of longitude of A, east of Z'= difference of
longitude Z', west of Z - difference of longitude of A,
west of Z.
Project B in its respective latitude, with a difference of
longitude from A, (east or west,) equal to the difference
between the longitude of A from Z', (found by one of
these three rules,) and B from Z', (found by method 3, B,)
mark it "B (east or west) of Z', (so many degrees.")
Project B' in its respective latitude, with a difference of
longitude from A, (east or west,) equal to the difference
between the longitudes of A from Z', and B' from Z'; and
mark it c B' (east or west) of Z', (so many degrees.")
5th. Join by straight lines A and A' B and B'; and if they
do not intersect each other, produce them to an intersection;
and if the ship has not changed her station between the
observation, this intersection will be in the true latitude.
But, if the station has been changed, then in the direction
of the course made good, set off the distance from any
point in the straight line A A'; through this point draw a
straight line parallel to A A', and its intersection with B B'
is the true latitude; corrected for change of station.
6th. From this point let fall a perpendicular upon A A',
and the length of this is the correction in miles to the first
altitude; additive, if the course sailed was towards Z, subtractive, if from it.
7th. Produce this perpendicular towards the pole of
illumination Z, and the angle it makes with the meridian
is the sun's true azimuth.
8th. The diference of lozgitude between any tzwo points in
either one of the two straight lines, is the error of longitude
consequent to an error of latitude as great as the difference
of latitude between the same points at their respective
times of observation.
ExAMPL E      V.
[From Bowditch.l
"In a ship running N. by E. 3 E. per compass, 9 miles
per hour, at 10h A.M., per watch the correct altitude of




73
the sun's centre, was 13~ 18', bearing S, 3 E. per compass;
at 1h' 40m P.M. per watch, the altitude of the centre was
14~ 15'; the declination being 23~ 28' S.; the latitude by
account, 48~ 17' N. Required the true latitude."
And also by this method is shown the true correction
of the first altitude for change of station; the sun's true
azimuth and errors of longitude by chronometer, consequent to any error in the latitude, when the time is noted
by chronometer.
1st. The elapsed time, 3h 40m is equal to 55~, or the difference of longitude of Z' west of Z, (plate II.)
2d. The two latitudes less and greater than the latitude
by account, are 48~ N. and 49~ N.
3d. Find the apparent times from noon (method 3, B)
with these latitudes, for each altitude.
FOR THE FIRST ALTITUDE, 13~ 18' A.M.
For a point A, in latitude 48~ gA.
Lat.  48      N.    -   -    -      sec. 0 17449
Dec.  23 28 S.    -    -   -        sec. 0-03749
Sum. 71 28        nat. cos. 31786
o Alt. 13 18      nat. sin. 23005
diff.    8781    log, 3 94354
H.  M. M S.
2   4   6           log rising,  =    415552
2   4   6 == 31~ 11' or diff. longitude (west, because
A.AM.) from Z, of the point A, in latitude 48~ N.
For a point A', in latitude 49"~ N
Lat.  49      N.    -        -      sec. 0-18306
Dec.  23 28 S.    -   -             sec. 0 03749
Sum. 72 28        nat. cos. 30126
o Alt. 13 18      nat. sin. 23005
diff.     7121   log, 3'85254
H.  DI.  So
1  52 37     =     log rising,  =     407309
1 52 37 = 28~ 94' ordiff. longitude (west, because
A.M.) from Z, of the point A', in latitude 49~ N.
[10]




74
FOR   THE   SECOND    ALTITUDE, 14~ 15' P.M.
For a point B, in latitude 480 N.
Lat.   48      N.    -    -    -   - sec. 0 17449
Dec.  23   28  S.  -         -  -  -  sec. 0-03749
Sum. 71    28      nat. cos. 31786
0 Alt. 14  15      nat. sin. 24612
diff.     7174    log, 3-85558
H.  M.  S.
1  51  53             log rising,  =     4-06756
1  51  53 =   27~ 58~' or diff. longitude (east, because
P.M.) from Z' of the point B in latitude 48~ N..For a point B', in latitude 49~ N.
Lat.   49      N.     -      -  -   - sec. 0 18306
Dec.   23  28  S.   -        -  -  -  sec. 0-03749
Sum. 72    28      nat. cos. 30126
o Alt. 14  15      nat. sin. 24615
diff.     5511   log, 3 74123
H.  M.  S.
1  38  51     =      log rising,  =         3-96178
1  38  51 =  24~ 429' or diff. longitude (east, because
P.M.) from Z', of a point B', in latitude 49~ N.
4th. To project A, A', B, B', plate II.
A. In latitude 48~ N., and in any longitude, project a
A   point; mark it "A 31~ 1' west of Z."
(From     31~ 1I'  - long. of A, west of Z,
A.   Subtract 28  9~ =  long. of A, west of Z.
Diff. is   2 521 =  long. of A', east of A.
Project A', then, in latitude 49~ N., 2~ 52i' east from A,
and mark it "A' 28~ 9' west of Z."
Now find the difference of longitude of A from  Z'; one
observation is A.M., the other P.M., (see rule.)
Z' is west of Z =  55~ 00 by the elapsed time.
A is west of Z - 31 012
A, then, is east of Z =  23 58A
Mark A, then, also, "A 23~ 58k' east of Z'."




75
From     27~ 584' _  long. of B, east of Z',
Subtract 23 58I -   long. of A, east of Z',
(Diff. is  3 59i  -  long. of B, east of A.
Project B, then, in latitude 48~ N., and 3~ 593' east
from A, and mark it "B 27~ 58k east of Z'."
(From    24~ 421' =  long. of B', east of Z',
B. Subtract 23  581 =   long. of A, east of Z',
iDiff. is  0 441 =  long. of B', east of A.
Project B', then, in latitude 49~ N., and 0~ 44' east
from A, and mark it "SB' 24~ 42-' east of Z'."
5th. Join A, A',  B B, with straight lines; from any
point in AA', as for instance A, set off a point D, 33 miles
N. by E. | E.; through D, parallel to AA', draw a straight
line DL; its intersection with BB' at the point L, is in
the true latitude at the time of the second observation,
namely, 48~ 51' N.
6th. From the point L, let fall a perpendicular LC on
AA'; and LC is the correction in miles of the first altitude
for change of station: namely, 22 miles subtractive by
the scale.
7th. Produce LC, towards Z, and it will make an angle
CxM, with the meridian; this angle, or its opposite, which
is the same, is the sun's true azimuth, equal to S. 28~ 16'
E., or 2 1 points from S. nearly; and at the first observation
the sun bore S.S.E. I E. nearly.
8th. If the chronometer time had been noted at the
first observation, in order to find the longitude by chronometer, and a second observation had not been observed,
the true latitude being, as we will suppose, 48~ N.; then
the ship would have really been at the point A, in the
straight line AA'; but if the latitude by account had been
erroneous 20' to the N; or 48~ 20' had been used to find
the "apparent time at the ship," instead of 48~ N., then
the error of longitude by chronometer would be equal to
the difference of longitude between A and the point P in
the line AA', in latitude, say 48~ 20' N. or 57' of longitude; that is one mile error of latitude gives nearly three
minuttes error of longitude in this case.




76
N O T E.
The latitude found above is 480 51-' N., the latitude by
Bowditch, old editions, is 48~ 55', by two operations; new
editions, 48~ 54'. The cause of this difference affords a
good opportunity of testing the correctness of this method,
without impeaching the accuracy of his rules.
The reason of this apparent difference is, that he has
accidentally stated the sun's bearing to have been S. i E.,
instead of S.S.E. 2 E., at the time of the first observation.
It is an error of no consequence in itself, as it serves to
exeemplify his rules as well as the true bearing would.
This method gives 48~ 55' for the latitude, supposing
the sun to have borne S. 3 E.; and Bowditch's method
gives 48~ 51~' for the latitude, if the sun bore S.S.E. E,
For the latitude of the place of the first observation by this
method, is 48~ 19'; and by his, 48~ 23', the dec. is 230 28'
S., and the first altitude 13~ 18'; we have, then,
O Alt.  13~ 18' -           -   - sec. 0-01181
P. D.  113 28       -     -   -    se. 0 17717
Lat.    48  19
Sum.   175 05 -                   cos. 8-63238
2 Sum. 87 322
P. D.  113 28
Diff.   25 55       -   -     -   cos. 9-95394
2)18 77530
2  +    75 52                 cos. =  9'38765
151 44 O's azimuth from N.
180 00
28  16 0's azimuth from S.
If calculated with latitude 48~ 23', the O's azimuth will
be 28~ 06', only 10' difference.
Now S.S.E. I E. is S. 28~ 07' E. So it appears that the
sun bore S.S.E. E. nearly, and not S. E.
Let us see, that supposing the sun bore S. | E., the two
methods agree. From L, plate II, th  rue latitude, draw a
line LP, parallel to AD, meeting AA' in P; it will be 33




77
miles in length, on a course S. by W, - W., or N. by E. A
E.; and P will be the place of the first observation.
Upon P, erect a perpendicular Py, which will be the
sun's trute bearing (S.S.E. 2 E.) at first observation; the
angle yPL is the angle of the course sailed with respect
to Py, or the sun's bearing; make the angle yPs equal to
13 points, and Ps will be in the direction S. 3 E.; make
angle LPp, equal to yPs; then yPp will be the angle of
the course sailed with respect to Py, if the sun bore S. 3
E.; from P set off the point, p, on Pp, at the distance of
33 miles; through p, draw a straight line pq, parallel to
AA', and the intersection at q is the true latitude, 48~ 55',
if the sun bore S. 3 E.; which is the same as Bowditch.
From q, let fall a perpendicular on AA', meeting it in
c'; qc', should be the correction in miles to be subtracted
from the first altitude if the sun bore S. 3 E.; this distance,
qc', measured on the chart, is 29 miles; now 29 miles is
the correction in reality used by Bowditch.
Thus it appears the two methods agree, but that the
true azimuth is apparent by this method on inspection;
and we learn, that an error of 13 points in the sun's
bearing, by his method, makes an error of about 4 miles in
the latitude, in this example.
The preceding problem includes the whole theory of
double altitudes by this method; and however useful it
may prove, it becomes much more so, when the times are
noted by chronometer; for then the longitudes of the
points A, A', B, B', become known at once; and thus the
"elapsed time" will not appear as such in the problem;
for it enters into the longitudes of the points B, B', placing
them so much further zest as greater time elapses between
the observations; thus we shall only have to find in the
usual way, the longitude of the points by chronometer, and
project them in their respective longitudes from Greenwich, and assumed latitudes.
And likewise, in accordance with problem III, when the
latitude is uncertain, a SIngle altitude of the sun becomes of




78
great value to determine the ship's position, because it
shows the bearing of the land at any hour; in fact, as
useful as two altitudes by any other method, when the
times are noted by a common watch; add of equal value
with.a meridian observation; because these only show the
bearin#of the land, in the parallel of latitude; this last only
once in day; and the first usually requires two opportunities
for observation.




IV,
APPLICATION OF THESE PRINCIPLES TO THE FIXED
STARS, PLANETS, AND MOON.
Double altitudes may be classified as follows:
CLASS I.
When the same body is observed at two difTerent times.
CASE 1. The sun.
2. A fixed star.
3. A planet.
4. The moon.
CLASS II,
When two diferent bodies are observed at the same time.
CASE 5. Two fixed stars.
6. Two planets.
7. A planet and fixed star.
8. Moon and fixed star.
9. Moon and planet.
10. Moon and sun.
CLASS III.
When one body is observed at one time, and a diferent body at
another time.
CASE 11. Two fixed stars.
12. Two planets.
13. A planet and a fixed star.
14. The sun and a fixed star.
15. The sun and a planet.
16. The moon and a fixed star.
17. The moon and a planet.
18. The moon and the sun.




80
It is plain that the same results will follow, if any celestial body is used, from which the apparent time at the ship
can be known. It will be only necessary to consult the
epitome upon the manner of finding the apparent time by
any object observed; and apply this time in the same
manner as when it is found by the sun's altitude.
The apparent time for the assumed latitudes can be
found from any of the fixed stars, or planets, with accuracy,
with a good horizon, and from the moon, when her right
ascension and declination can be had. In all these cases,
the arcs of the parallels of equal altitude corresponding to
each altitude, are to be found in the same way as before.
If the apparent time at the ship can be found from the
altitude by any of the rules in the epitomne, and the chronometer time is noted, then the latitude and longitude of
each point may be at once projected, as in the case of the
sun; and if the times of observation be noted by a common
watch, then it may be assumed that the "watch" shows
approximate Greenwich time, by allowing its error, if fast
or slow, as in problem III, practical part. The above will,
in general, be the most simple method.
But we may also proceed as follows, the times being
noted by watch.
Assume two latitudes, as before, with which, and the
true declinations of the body at each observation, and the
correct central altitudes, find the hour angles, or apparent
times from noon, (method 3, B,) turn these times into
degrees for their respective differences of longitude, (east
or west, according to which side of the meridian of the
place of observation the body was observed,) from their
respective poles Z and Z'.
The points A and A' can be projected, as before; but to
project B, B', with reference to AA', the value of the arc
ZZ' must be found by one of the following rules; because
the elapsed time without a correction turned into degrees
will not in all cases express the difference of longitude
between Z and Z'.
This being done, BB' can be also projected, and the




81
intersection will show the true latitude as before, allowing
for change of station.
All cases in which the moon is one of the bodies, will
be liable to error, unless the ship's position in longitude
is nearly known beforehand, for her right ascension and
declination must be known with accuracy at the time of
observation, her motion being greater than any other
body; in case 4, however, it is only hourly zotion in right
ascension and the declinations that are required, and these
can be ascertained nearly enough for common use. (See
Appendix, Bowditch.)
The sun's and planets' proper motions are much slower,
and their right ascensions and declinations can be had
accurately.
The arc ZZ' is greater than 180~ (or must be subtracted
from 360~) when the great circle which passes through the
positions in the heavens, in which the bodies were observed, passes also below the elevated pole, the bodies being
also observed on different sides of the meridian. In all
other cases ZZ' is less than 180~.
All bodies situated below a great circle passing through
the east and west points of the horizon and the elevated
pole are below the pole.
Care must be taken not to mistake wahich pole, Z or Z',
belongs to zlhich ob'ect; for this purpose it will be useful to
remember, that the body which has the greatest altitude
has its pole of illumination the nearest to the observer; and
always name the eastern pole Z, the western one Z'.
The nearer the bodies are observed at right angles to
each other, whatever method of calculation is used, the
more likely is the result to be accurate.
Two of these bodies may be observed, either both eastward from the mzeridiaz, or both westward; or one eastward, and
one zvestzard; and in this respect, the times of observation
are similar to the twzo altitudes of the szin, which are observed either both A.Mo, both P.M., or one A.., and one
P.M.; in every case, for uniformity, it will be proper
[11]




82
always to call the eastern pole Z, and the western pole Z';
and having found the value of the arc ZZ', to project the
points A, A'j B, B', eastward or westward of Z and Z', as is
designated by the body's having been observed when east
or west of the meridian.
With the exception of class II, in all cases in which the
sun is not one of the bodies, a correction is to be applied to
the elapsed time, and is that quantity which is called XS
in the following formulas.
The motion of the stars is quicker than that of the sun;
it takes the sun 24 mean solar hours to make an apparent
revolution round the earth; but the stars do the same
thing in about 23h 56m; in any given elapsed time, then,
the stars will be a proportional distance west of the place
where the sun would have been, had the sun been the
body observed, the elapsed time being usually noted by
watch, which shows solar time. The motion of the planets
and moon is compounded of this motion, and their own
proper motions in right ascension and declination.
This correction is 9s 85647 for every hour of elapsed
time. The following table is calculated to make this
allowance in those cases where it is required.
TABLE
For finding the value of XS, in the formulas, during ET
(Elapsed Timze.)
ET      XS       ET    XS
Hours.  min. sec.  min.  sec.
1     009-856    1    0-164      EXAMPLE.
2     0 19-713   2    0'329   The Elapsed Time is
3     0 29-569   3    0493 5h 17m, what is the value of
4     0 39'426   4    0657 " ET+YS."
5     0 49-282   5    0-821
6     0 59-139  10    1-643    5h = 49S'282
7     1 08-995  15    2-464   15m — 2 464
8     1 18-852  20   3-285     2m — 0 -329
9     1 28-708  25   4-107
10     1 38565   30   4-928        52 075   XS
11     1 48-421  35   5-750 
12     1 58-278  40   6-571    5 17 00   -ET
45   7-392
50   8!214 Ans.Sl 17m52 = ET  XS.
55   9-035
60   9-856




83
FORMULAS OR          RULES
To find the value of the Are of ZZ' in hours miinutes, and
seconds, which, being turned into degrees, 15~ to an hour, gives
the dif. long. between Z and Zt
CLASS I.
When the same body is observed at two different times.
Case 1.  The sun
ZZ == ET, or the elapsed time.
Case 2.  A fixed star
ZZ'= ET + XS.
Case 3.  A planet
ZZ' = ET- XS l } planet's notion in R. A.
during ET, if R. A. is { decreasing
Case 4.  The moon
ZZ'=ET -XS —'s motion   in  R. A.
during ET.
In explanation of the rules. In Case 1st it has already
been seen, that ZZ' is equal to ET; but let P, (plate VIII,
fig. 1,) be the elevated pole, ESS'W be a parallel of declination on which the R. A. is reckoned, 1H' a part of
the horizon, O the place of the observer on the earth's surface, and PO the meridian passing through P and 0. The
arrows show the direction of the apparent motion.
Let S be the sun at the first observation. It is vertical
to Z, below the horizon. After an elapsed time, the sun
is again observed at S'; S' is vertical to Z'; the arc SS' is
described by the sun, at the rate of 15~ per hour of ET;
the arc ZZ' is then described in the same time; for the
earth is a sphere, concentric with the celestial sphere;
and difference of longitude between Z and ZA, or the arc
ZZ', is equal to the apparent difference of R. A. of S and
8', and is expressed by the elapsed time turned into degrees, 15~ to an hour; therefore ZZ' = ET.
Case 2d, (lower portion figure 2.) Let * be the position
of a star. After an elapsed time by watch showing solar




84
time it will at *'; if it had been the sun, the second altitude would have been taken when he was at OS, and the
arc * S would be equal to ET = ZZ'; but since the star
moves quicker than the sun, it will have reached the position *' at X, during the elapsed time by watch; being an
excess - the arc XS; * is vertical to Z' and *' is vertical
to Z; ZZ' then is equal to *' or = ET + XS; but since it
is required to reckon the difference of longitudes of Z and
Z' from 0, the place of the observer on the earth's surface, ZZ' must be taken greater than 180~; that is, ZZ' in
degrees, must be subtracted from 360~; because the great
circle GC, which passes through the positions where the
body was observed, and also observed on different sides of
the meridian, passes below P, the elevated pole.
This case is a simple one.
Case 3d (upper portion, figure 2.) Let P be the position
of a planet when first observed; it is vertical to Z; after
an elapsed time by watch, it may be taken at P', vertical
to Z'; therefore ZZ' = PP'. If it had been the sun, it
would have been observed at So the second time; but
PS =ET; and being a star, the correction XS must be
added as before; and x is the place in which it would have
been observed, had it been a fixed star; but during ET, it
has moved from x to P', by its own proper motion in R. A,,
towards the east; * P' then must be subtracted (the R. A.
increasing:) we have then (PX,) or (ET + XS) (XP', or
planet's motion east, in R. A., during ET)= PP'= ZZ'
as in the formula. If the planet's proper motion be to the
west, or R. A. is decreasing, it must be added to ET + XS.
This is an easy case, by this method.
Case 4. The moon, figure 2. This is sufficiently ex-.
plained by the last; the moon's motion in R. A. being
always E., is always subtractive.
CLASS II.
When two different bodies are observed at the same time,
ZZ'- difference of R. A. of the bodies at the instant of
observation, in all the cases of this class.
In figure 3, the objects are observed on dij'cerent sides of




86
the meridian PO  and the great circle GC passes above the
pole P; ZZ' is therefore less than 180~. But in figure 4,
the objects are observed on diGerent sides of the meridian,
and the great circle GC, passing through the position of
the two objects, passes also below the pole P; therefore,
ZZ' will be greater than 180~; then the difference R. A.
must be turned into degrees and subtracted from 360~ to
find ZZ'.
In the two cases in figure 5, because the objects are both
observed on the same side of the meridian PO, ZZ' is less
than 180~ in both cases, although in one case GC is above,
and in the other below the elevated pole.
It is evident ZZ' difference R.A. of the bodies, in all
these cases.
Those cases of this class, in which the moon is not one of
the bodies, are important, as the observations can be frequently taken with great accuracy; particularly in the'
twilight, and they are attended with but little more trouble
than the observations of the sun, because it is not required
to find the "apparent time at the slip9," but only the hour
angle, or that time which is found (method 3, Bowditch,) in
Table XXIII, in column, "log rising."
But, if the apparent time at the ship be found for one
point in each arc, and the chronometer time be noted for
each altitude, then the longitudes of the other points, A',
B', may be found, and the position of the ship in latitude
and longitude.
The R.A. of the fixed stars and planets, is, too, readily
found, in the large edition of the nautical almanac.
CLASS III.
When one body is observed at one time, and a different
body at another time.
This class comprehends a great variety of circumstances,
under which the bodies may be observed. By a few examples, the manner of finding the value of ZZ', in any
case, will be understood.
Let A, figure 6, be the easternv body, and the first observed; and B, the other body, westward from it.




86
The arc AB, is the difference of R.A. of the bodies at
the time of the first observation. After A is observed,
and a time has elapsed by watch, A will have advanced
westward to a point S; and B will have reached T; let B
be now observed at T, and TS is the difference of R.A. of
the bodies at the time of the second observation.
If A be the sun, the arc AS will be equal to the elapsed
time, ET; and ZZ' is equal to the sum of the arcs AS, and
ST; that is,
ZZ'= ET+ (difference of R.A. of the bodies at the tinme
of the second observation.)
And ZZ' is less than 180~; because GC is above the elevated pole.
If A be a fixed star, it will arrive at X, in the same
elapsed time, instead of S; then, TX will be the difference
of R.A. of the bodies at the time of the second observation;
and AX - TX==ZZ', that is,
ZZ'- ET +XS+ difference of R.A., at the time of the
second observation.
If A be a planet, and have no proper motion in R.A., it
will be the same case as the last, and will arrive at X; but
having arrived at X, if it have a proper motion in R.A.,
increasing, (or going E.,) it will during ET, arrive at some
point b; and Tb will be the difference of R.A. at the time of
the second observation; we have then, ZZ' - ATP- ET + XS(Zb, or planet's motion in R.A., during ET,) + difference of
R.A. at the time of the second observation.
But when the planet's motion in R.A. is westward, (or
decreasing,) then it will arrive at some point a, during ET,
and Xa must be added. Ta will then be the difference of
R.A. at the time of the second observation.
If A be the moon, her motion in R.A., during ET, being
always east, (or increasing,) must be always subtracted,
and we have
ZZ' =ET + XS - Xb + R.A. at the time of the second
observation.
Figure 7. If A be the eastern body, and B, the zoestern
body, be first observed, and A be not observed until it
reaches S, having passed the position in the heavens where
B was observed at B, then,




87
If A be the; AS, is the elapsed time, anzd AB, the
difference of REA. at -the time of the first observation;
SB is equal to ZZ, and ZZ    ET - difference of R.A,, at
the time of the first observation.
But if B be the sun, TB is the elapsed time, and TS
the difference of IRoA at the time of the second observation, then
ZZ = ET - difference of R.A. at the time of the second
observation.
Figure 8. If A be the sun, the ecstern body, and first
observed, then after an elapsed time, it will reach S; and
B, the other body, will reach T, which then observe; then,
AS = ET; and TS = difference of R.EA, at the time of the
second observation; and AT = ZZ'; then
ZZ' = difference of R.A. at the time of the second observation - ET.
The bodies being observed on the same side of the
meridian, ZZ' is less than 180~o
In figure 6, A, the sun, was also the eastern body, and
first observed, but the rule for finding ZZ', is different,
because both objects were above the pole, and the apparent
motion was from east to west; but in this case, both bodies
being belozv the pole, the apparent motion is reversed, or
from west to east, and in this case we might with propriety
call A zvest of B, although it appears to be east of it, and
will appear so, when both are above the pole.
In this manner, by attention to the circl.umstanc.ees of the
observation, a rough diagram may be made, and the value
of ZZ' may be readily found; and it will be necessary, for
this purpose, for one of the cases of " Class II,"9 to notice
1st. What tvo bodies are used.
2do Which was first observed.
3d.o Which was the eastern.
4tho W'hich side of the meridian was each observed.
5th. Did the great circle passing through the positions
in the heavens where the bodies were observed, pass also
beloev the elevated pole?
6th. Was the second body observed before, or after it
passed the position in the heavens, where the other body
was previously observed?




88
These two last considerations will seldom be required,
but to include all possible circumstances they are given.
By using discrimination in the bodies used, these observations will be found useful, and, when familiar with the
principles, they will be found quite simple.
The motion of the planets in R.A. is generally small
during ET, and can, for the most part, be neglected; a
glance at the Ephemeris of the planets in the nautical
almanac, is sufficient to determine this. The planets will
be found exceedingly useful for these observations; or for
meridian observations for the latitude; but the moon cannot
always be relied on for a meridian altitude, because her
mneridian altitude is not always her greatest altitude; and
unless the ship's position in longitude, or the Greenwich
time is well known beforehand, it will be impossible to be
accurate in finding her right ascension and declination.




Ve
CURRENTS IN THE GULF OF MEXICO AND THE
FLORIDA STREAM.
By plate IX, are shown the currents experienced in the
Gulf, in June, 1840.
The position of the ship was found by the method of
projection; and the accuracy of the work was tested by
the usual mneridian observations, and chronometer sights, the
sun bearing E. or W.
The land was not seen, neither any light, after leaving
the Mississippi.  Soundings were had on the Tortugas
Bank, on the evening of 21st June, but the ship was some
20 miles further E. than marked on the chart for the noon
of that day; for only the distances, made good from noon to
noon, are shown by the chart.
The full black line — _   shows the true course and
distance daily, by chronometer, &c.
The light black line         shows the course and
distance by log, carefully attended to; and is carried out
without correction for the observations, for the whole distance from the 10th to the 25th inclusive.
The dotted lines...... show the daily courses and distances by log, reckoned from the position of the ship at
each preceding noon determined by observation; and the
daily differences caused by currents.
The arrows point in the direction of the current, and
the velocity is marked in miles and parts. To prevent
confusion on the chart, they are placed at the ends of the
dotted lines. They should be referred to the full black
lines, in order to show the place where the currents existed;
thus, between 18th and 19th, on the full black line, the
current was S. by W ~ W., 2I miles per hour.
The figure of the ship on the 10th, shows which way she
headed, while the full black line shows which way she was
going over the ground.
[121




90
Extract from the Journal.
IN THE GULF OF MEXICO.
Sea act. Dis-            s  Clear weather  Under  Err2o Velocity. Direction.
Dis-  Course.  Winds,               in 24     Direction.
1840.  tance.               throughout. what sail. hours.
JunelO 105 SE 2 S  ENE ~ E  strong br'zs s. reefs 75 3 1-8 W  - N
11 96 SE by S   ENE   E     do.     do.   34 1 1-2 NE b N
12 95 SSE - E   E j N       do.     do.   43 1 3-4 NE I N
13 83 SSE   E  E~N         fresh  reefsout 42 1 3-4 NN 3WW
14 35 SEbE      EENE        light  all sail 53 2 1-5 WbN~N
15 84 SEbE4ENE              fresh   do.   86 3 1-12 WhN1N
16 98 ENE       SE  S       do.     do.   50 2 1-12 W  N
17 82 NEbE    ESE   E       do.     do.   17   3-4 NW|N
18 70 NEbE4ESE            moderate  do.   54 2 3-4 ShbW-W
19 65 S b E  E E     S      do.     do.   20   5-6 WNW
20 71 E  N      SSE' E      do.     do.   26 I 1-12 WNW-W
21 95S   W     SEbEEE      fresh   do.   25 1     NbE      E
IN  THE   FLORIDA     STREAM.
22 53   ESE     ESE        strong   do.   70 2 1-12 E ) N
two tacks s
23150 NNE       ESE         fresh   do.   70 211-12 NE  N
24156 N 4 W     ESE       moderate  do.   68 25-6 N   E
Whole amount of current in every direction in 15 days-= 735 miles.
It will be seen by the plate, that if no observation had
been taken, that the dead reckoning placed the ship in the
fair way of the stream, on the 19th of June.         But, if a
course had then been shaped to the northward, a run of
40 hours would have put the ship ashore, in about longitude 84, latitude 30~.
The various directions and velocities of the currents, and
the uncertainty of their duration, show how little dependence is to be placed in any " reckoning " when thick weather prevents observations, and that a good loo/k-out and
attention to the lead are essentially necessary.




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betl-v',:  Z  &  Z'.7 i
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?A 0a1IV?
L P~    LyOITTUDE WEW ST 800 or GRERERWIe CH.
ILA -WT110~s^^^a117^^0^- 0 OM-oX
\\ s ~ ^ $\ -?   - -  -" l0J   -3e 1
4 OuetAlituie & Ciro- A, olerTnnIe giwewu
t mW;  to fmil-the tue~1Bear of fiellani......^-s^yP'd*'rJ o/ft
Thetime mooteilby.Ikroiioiioeter.
Jo corredtioiLiS require[ for alunge of Statiomthe Siap
b ein g Me arll e valmeLbetwe ei the ob ervations.
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The'imes -notealiyChroinommetera
ICorre dtiolL for olange of Stati4.,o4''~'.....
37N  B   6,,   lo. ist of _  _ ____ 6 
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The times.moted l~ a well regilated
-Mfapse Time -is 16~ 5. I't' 3lb; or Ote lliT
be twe. & Z.
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The timnei botei ly a welL zregulate&i
Ila(e&TA Time i0 Ii6mi$.]~ 31'4 orfhle Diff
IbetweeIL f.& -Z'.
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