NAVY SCIENTIFIC PAPERS, No. 1.
. US
(3 33257 (. A

ASTRONOMY:
COMPRISING SUGGESTIONS TO U, S, NAVAL OFFICERS,
BEARING UPON POINTs connecTED witH
NAUTICAL ASTRONOMY, ASTRONOMICAL GEOGRAPHY,
AND GENERAL ASTRONOMY.
BY
PROFESSOR: WILLIAM CHAUVENET,
Chancellor of Washington University, St. Louis, Missouri.
B U R E A U O F NAVIG ATION.
N A WY DEPARTMENT.
WA. S. HIN GT ON:
GOVE R N MENT PRINT IN G OFFIC E.
1868.
ASTR0 N () MY.
BY PROFESSOR WILLIAM CHAUVENET,
Chancellor of Washington University, St. Louis, Missouri.
I. NAUTICAL ASTRONOMY.—The navigator who is disposed to contribute to
the advancement of science will naturally direct his efforts first to the branches
with which he is most familiar in the daily practice of his profession. His
attention will therefore be first directed to the improvement of methods of find-
ing a ship's place at sea by astronomical observation; for much as this subject
has been studied, it is far from being exhausted. The researches of mathemat-
ical navigators, such as our distinguished Bowditch, during the first half of this
century, were directed chiefly to the abridgement of the processes of navigation,
on the ground that the practical seaman has little leisure, even if he have the
taste, for long numerical computations. But to satisfy the demands of the nav-
igator of the present day, we must furnish him with methods not only brief
but accurate; and it is therefore to be hoped that those who attempt to make
improvements will keep in view both these essential qualities of a good nautical
method. The following remarks upon some of the leading methods in actual
use, and upon some proposed for adoption, are not to be regarded as absolute
dicta, but as suggestive of the course of inquiry which may head to improve-
ment. -
1. Latitude by meridian observations.—The meridian observation of the
altitude of the sun is so simple, the subsequent computation of the latitude so
brief, and the result so certain, that no improvement of this method seems pos-
sible. It may be worth while, however, to direct attention to the precautions
to be observed in those special cases where extreme accuracy is desired. In the
open sea, indeed, where the determination of the ship's place within one minute
is even more than sufficient, the usual mode of observing the maximum altitude,
by beginning before noon and following the sun until it begins to descend or
“dip,” may be regarded as quite accurate. Time, however, is often wasted in
this way, by beginning the observation unnecessarily soon, (as early even as
11”. 30”.,) the observer being uncertain as to the ship's time. This uncertainty
arises from the usual practice of determining the time by an altitude of the sun
near the prime vertical or as far from the meridian as possible, after which the
ship's change of position, known only from the dead reckoning, introduces more
or less error by the time the meridian observation arrives. This defect will be
obviated in many cases by finding the time from circum-meridian altitudes, of
which we shall speak more particularly hereafter.
But when the greatest accuracy is sought, the observer must no longer regard
the maximum altitude of the sun as the meridian altitude, but must remember
that since the sun's declination is continually changing, (excepting when exactly
4 ASTRONOMY.
at the solstices,) the greatest altitude is reached either before or after noon. If
the ship is not moving during the observation, or at least does not change her
latitude, this source of error is insignificant, (never reaching 1';) but if the
ship's change of latitude is rapid—say from 10' to 15' per hour, as may often
happen—this change not only produces a sensible error itself, but also causes
the error arising from the change of declination to become sensible. For exam-
ple, if the ship is in latitude of 70°, and is running south so as to decrease her
latitude 12 per hour, the sun's declination being 0°, and the change of declina-
tion 1’ per hour, the greatest altitude differs from the meridian altitude” by
(12-1-1)*-i- (0.7 × 4)=60". If we neglected the change of declination, the dif-
ference would be (12)”--(0.7 × 4)=51". If the ship did not change her lati-
tude, the difference would be 1--(0.7× 4)=0".3, which is altogether insignifi-
cant. Hence the change of declination which produces no sensible error when
the ship is at rest, has produced an error of 9", the ship being in motion at the
supposed rate.
Again, the method of observing the greatest altitude by following the sun's
increasing altitude with the sextant until the sun “dips,” is subject to another
kind of inaccuracy when the ship is in a heavy sea; for the height of the eye
and consequently also the dip of the horizon are then variable, and it is impos-
sible to cause the sextant reading to follow the sun's altitude exactly. For ex-
ample, with the eye 16 feet above the ship's water-line, the dip of the horizon
with a smooth sea will be 3/56!/; but if the waves cause the ship to rise and
fall, say 4 feet, above and below the mean height, the dip will vary from 2'47"
to 3' 56”.f The remedy in this case appears to be to observe a succession of
altitudes in the vicinity of noon, as rapidly as possible, making an independent
contact of the sun's limb and reading off the sextant at each observation. After
it is apparent (from the altitudes thus read off) that the sun is declining, we
may either select the greatest recorded altitude as the meridian altitude, and
allow the dip for the height of the eye above the ship's water-line, (which
amounts to assuming that this greatest altitude was obtained when the ship was
on the top of a wave,) or we may select this greatest and the two adjacent ones,
take the mean of the three, and allow a mean dip corresponding to the height
of the eye above the water-line diminished by one-half the height of a wave.
2. Latitude and time by circum-meridian altitudes of the sun.—When clouds
have prevented the observation of the meridian altitude of the sun, and yet
altitudes near the meridian have been observed, the method of Bowditch's
Navigator, (p. 201,) by means of his Tables XXXII and XXXIII, is the most
convenient for use at sea, and is sufficiently exact when the local time is toler-
ably well known. But the occasions when circum-meridian altitudes have to
be resorted to are mostly on those days when no observations for the local time
* See rule in Bowditch's Navigator, p 169.
f This is upon the supposition that the sea horizon is then determined by the tops of the
distant waves; so that when the ship is on the top of a wave, the observer's eye is to be
considered as 16 feet above the horizontal surface, and when the ship is in the trough his
eye is but 8 feet above this surface; and therefore the proper dip to be deducted from an
average of a number of altitudes is that corresponding to 12 feet in this case, or, in general,
that which corresponds to the height of the eye above the ship's water-line diminished by
one-half the height of the wave reckoned from the bottom of the trough.
ASTRONOMY. 5
have been possible, and the reduction by Bowditch's method (or by any of the
methods now in use) is then liable to such uncertainty as often to be abandoned
as wholly impracticable. A simple method of using such circum-meridian alti-
tudes of the sun, when the time is not known, is the following,” which is ex-
plained here because it has not yet found its way into our practical works.
First, to find the latitude.—The chronometer time of each altitude of the sun’s
limb near noon is noted, as we wish to use the chronometer intervals, although
the correction of the chronometer on the local time is unknown. Select any two
altitudes thus observed that are not more than 26 minutes apart by the chrono-
meter. Correct these altitudes for refraction, and find their mean and their
difference. Also find the difference of the corresponding chronometer times, or
the chronometer interval. With the sun's declination and the approximate lati-
tude find the change of altitude in one minute from noon by Bowditch's Table
XXXII. With one-half the chronometer interval take the number from Table
XXXIII, with which multiply the change in one minute; the product is the
first reduction. Take the square of one-fourth the difference of the altitudes,
(reduced to seconds,) and divide this by the first reduction; the quotient is the
second reduction. Add these two reductions to the mean of the two altitudes;
the sum is the meridian altitude of the sun’s limb, which is then (having already
been corrected for refraction) to be further corrected for the dip of the horizon,
parallax, and semidiameter, and the latitude deduced as in the case of a meridian
altitude directly observed.
Calling h and h' the two observed altitudes of the sun's limb corrected for refraction,
t one-half the chronometer interval, a the change of altitude in one minute from noon,
and hi the meridian altitude of the limb, the above rule is expressed by the formula—
h1 = }(h+ h') + at? -- [{{h-h')].
- at?
When the difference of the altitudes is small and the refraction therefore
(sensibly) the same for both, it will not be worth while to correct the observa-
tions for refraction before reducing to the meridian. All the reductions (includ-
ing also the index correction of the sextant) will then be applied to the reduced
meridian altitude of the limb, as in the following example:
ExAMPLE.—The approximate latitude being 38° N., the declination at noon
1° 48' 9" S., the height of the eye above the sea 19 feet, suppose the following
observations taken :
Chronometer. G)
Sh Om 225.5 } = 50° 10' 0"
8 10 13.5 h/ = 50 11 40
2) 9 51 h—// = — 1 40
cºsmºmºmºsºs #(h—h') = — 25
t = 4 55.5
B. XXXIII. tº — 24.2 #(h+h/) = 50 10 50
B. XXXII. a = 2//.4 at” = 1st reduction = 58
[#(h—h')]* = 625 * = 2d reduction = 11
Meridian altitude G) = 50 11 59
* The investigation of this method will be found in Chauvenet's Spherical and Practical
Astronomy, Vol. I, Art. 204.—ED.
6 n ASTRONOMY.
Dip = — 4 16
Semidiameter = + 16 6
Refraction and parallax = — 42
h1 = 50 23 7
Zenith distance = 39 36 53 N.
Declination = 1 48 9 S.
Latitude = 37 48 44 N.
The accuracy of the result depends in a great degree upon the accuracy with
which the difference of altitude is obtained. If in the above example this differ-
ence had been 2' 40", or 1' too great, we should have found #(h—h') = 40",
and the second correction = 1349 = 28"; consequently the resulting latitude
would have been only 17" too small. Since the same causes of error, such as
displacement of the sea horizon by extraordinary refraction, unknown instru-
mental errors, &c., affect both altitudes alike, the difference will usually be
obtained, even at sea, within less than 1'. The most favorable case is that in
which the altitudes are equal, and the second correction, consequently, zero. It
will be well, therefore, always to endeavor to obtain altitudes on opposite sides
of the meridian. -
When several circum-meridian altitudes have been observed, they may be
reduced in pairs as above, and the mean of all the resulting meridian altitudes
will be taken.
Secondly, to find the time.—Employing the same two circum-meridian alti-
tudes as have just been used to find the latitude, divide one-fourth the difference
of altitudes (in seconds) by the product of the change of altitude in one minute
and the chronometer interval (in minutes;) the quotient is the number of min-
utes from noon of the mean instant between the two observations, and it will
represent a time before or after noon according as the second altitude is greater
or less than the first altitude.
Thus, in the preceding example we have
*T*) = F^* = -2.1
at T 2.4× 4.9 T • L. 9
and therefore the middle instant between the two observations was (approxi-
mately) 2* 6* before noon. Suppose now that the equation of time was H-
8* 55° (i. e., additive to apparent time,) and that the chronometer correction to
Greenwich mean time was -- 27” 21°, we shall then proceed as follows:
- Mean of chron, times = 8h 5m 185 Appar. time at mean inst. = — 0" 2" 6°
Chron. correction- - = +27 21 Equation of time - - - = + 8 55
Mean Gr, time - - - = 8 32 39 Mean local time - - - 0 6 49
Mean local time - - - = 0 6 49
Longitude - - - - - = 8 25 50 = 126° 27' 30" W.
ASTRONOMY. 7
If the ship has changed her latitude between the two observations, it will be
proper to correct the first altitude by adding to it the ship's change of latitude
in the chronometer interval if the ship has sailed towards the sun, or subtracting
the change if the ship has sailed from the sun.
3. Littrow's method of finding the time by circum-meridian altitudes.—The
method just given is adapted to the reduction of observations which may be
called strictly circum-meridian, namely, those so near to the meridian as to admit
of reduction on the principle that the differences between the observed altitudes
and the meridian altitude vary as the squares of the hour angles. In the
Comptes Rendus of the French Academy for March 7, 1864, M. Faye discusses
at great length a method of using circum-meridian altitudes for finding the time,
suggested by Littrow, the present director of the observatory of Vienna; and
as this method is adapted to observations at somewhat greater distances from
the meridian than the above, it may often be very useful. We do not think
that it admits of the high degree of precision ascribed to it by M. Faye, but
nevertheless it will be found accurate enough for the wants of the navigator in
the open sea, when employed in proper circumstances (especially in low lati-
tudes) and with suitable limitations. It is as follows:
Two altitudes of the sun's limb are taken near the meridian with a chro-
nometer interval which may vary from 5 minutes to 30 or 40 minutes according
to the latitude, the interval being made longer in high latitudes. Deduce the
true altitudes of the sun’s centre and form their half sum and half difference;
also form the half sum and half difference of the chronometer times of the two
observations. Take the sun's declination for the Greenwich time corresponding
to the half sum of the chronometer times, i. e. for the middle instant between
the two observations, and bring up the latitude by account to the same instant.
Add together the logarithmic cosecant of the half chronometer interval, the log.
sine of the half difference of the altitudes, the log. cosine of the half sum of the
altitudes, the log. secant of the declination and the log. secant of the latitude :
the sum is the log. sine of the apparent time from noon at the mean of the
chronometer times, from which the longitude will be deduced in the usual way.*
This time is before or after noon, according as the second altitude is greater or
less than the first.
Calling the true altitudes h and h', the apparent hour angles of these altitudes T and
T', the latitude j and the sun's declination 6, this rule is expressed by the formula—
sin #(h—h') cos #(h+h')
sin #(T-HT')= sin #(T—T") cos @ cos Ó
EXAMPLE.—(We give M. Faye's example, adapted to our own Ephemeris and
* This rule is very easily used with the aid of Bowditch's Table XXVII; for to find the
log. cosecant of the half chronometer interval, we have only to enter that table, in the col-
umn P. M., with the whole chronometer interval: and by entering the column sine with the
above sum of the logarithms we find in the P. M. column a number of hours, minutes, and
Seconds, one-half of which is the required apparent time from moon.
8 ASTRONOMY.
Bowditch's Tables.) In latitude 20° N. by account, August 25, 1864, the fol-
lowing observations were taken : -
Chronometer. True alts. G)
3h 18m 435 h = 780 6' 1977
3 33 43 h/ = 79 52 49
Middle chron. time, = 3 26 13
Chrom. slow - - = 3 40
Middle Gr. time - = 3 29 53
for which time we find from the Ephemeris 6–10° 32' 57" N., and the equation
of time = +1 ºn 45°.
The computation is then as follows:
TV, T = Oh 15m 0s
#(T/–T) = 0 7 30 Log. cosec 1.48520
#(h—h/) = 0° 53' 15" “ Sin 8.19001
#(h+h/) = 78 59 34 * COS 9.28090
6 = 10 32 57 ** Sec 0.00740
= 20 0 0 “ Sec 0.02701
4(T-HT')= Oh 22m 275 (A. M.) “ Sin 8.99052
Hence we have
Local apparent time - - - - - =23° 37* 33°
Equation of time - - - - - - = + 1 45
Local mean time - - - - - - =23 39 18
Greenwich do. - - - - - - - = 3 29 53
Longitude from Greenwich - - - = 3 50 35 W.
The correction for any run of the ship between the two observations, when
necessary, will be made as in the method of the preceding article.*
4. Sumner's method.—As this method, though published long since and well
known to some of our navigators, is not sufficiently explained in our practical
works, it may not be out of place here to direct the attention of navigators to
its merits. It not unfrequently happens that after a long period of cloudy
weather, the navigator sees the sun for the first time at a considerable distance
from the meridian, and yet wishes to use an observed altitude for the determi-
nation of his position immediately, that is, without waiting for a meridian alti-
tude which would give him his latitude with certainty. The only use commonly
made of an altitude remote from the meridian is to find the longitude by chro-
* The examples cited by M. Faye, from actual observations on board the Novare, under the
command of the Austrian Admiral Wüllerstorf, seem to us to be calculated to give an exag-
gerated opinion of both the importance and accuracy of the new method. The first series
of observations, the results of which present the most surprising agreement with each other,
was taken in the low latitude of 12° N., when the sun was within less than 69 from the
zenith. The second series in which the sun was nearly 40° from the zenith, the latitude
being 34° S., though evidently taken with the greatest care and combined so as to make the
chronometer intervals nearly forty minutes, give results varying nearly 5' in longitude.
Ordinary observers under the same circumstances would probably find that their results
could not be depended upon within 10'. Sumner's method we think will be found to be
more generally useful as well as more accurate, though we recommend Littrow's method for
a full trial by our navigators, as the object of this article is to direct attention to all possible
means of improvement.
ASTRONOMY. 9
mometer; and to do this we must know the latitude with more or less accuracy
according to the distance of the sun from the prime vertical. But in the case
supposed the latitude is almost always very uncertain, and the longitude found
will therefore also be erroneous; consequently the navigator will not be able to
determine either latitude or longitude with any precision. Now Sumner's method,
while it furnishes neither latitude nor longitude, gives what is just as valuable
as either, namely, a line on the globe on which the ship is situated at the time
of the observed altitude. When we have found the latitude, we have only
placed the ship on a certain east and west line; when we have found the longi-
tude, we have only placed the ship in a certain north and south line ; and in
neither case do we locate the ship at any particular point of the line. A Sum-
ner line (or circle of position) in general makes an oblique angle with both par-
allels and meridians, but it is a line on which the ship is certainly situated, and
it can be easily laid down on the chart with all the precision necessary. The
importance of being able to determine such a line when approaching a danger-
ous coast must be obvious to every navigator of experience.
As it may not be otherwise accessible to the reader, it will not be superfluous
to state here the rule for using any single altitude of the sun (or indeed of any
celestial body) according to this method, when the time has been noted by a
chronometer regulated to Greenwich time. Find the true altitude, and for the
Greenwich time given by the chronometer take the sun's declination from the
Ephemeris. Assume two latitudes, embracing between them the ship's probable
position as nearly as can be estimated by the dead reckoning; they may differ
10, 20', or even 30'. With each latitude find the corresponding “longitude by
chronometer” by the usual method, employing in each computation the same
altitude and declination. Mark the two positions of the ship thus found on the
chart and join them by a straight line; this is the required Sumner line, and
although the ship is at neither of the two points computed, she must be some-
where on this line or on the line produced.
Strictly speaking, a Sumner line is curved, (being in fact a circle,) and two
points are not sufficient for its entirely accurate projection on the chart; but the
curvature will not be of importance unless a very long line has to be drawn in
consequence of the assumed latitudes differing very widely, which will of course
be the case when very great uncertainty exists as to the latitude. In any such
case we have only to assume three, or more, latitudes, and find as many corre-
sponding longitudes; then, putting down all the positions thus found on the chart,
a curve traced through them all will be the required circle of position or Sumner
line.
The same method may be applied to the determination of a second line of the
same kind from a second altitude observed after a suitable interval on the same
day, and the intersection of this with the first line will be the ship's actual posi-
tion; but it is not necessary to occupy our space with these details, especially
as the common method of “double altitudes” will give the same result. We
have given above all that is peculiar and distinctive in Sumner's process, which
is the determination of a circle on which the ship is situated, having all the
10 ASTRONOMY.
practical advantages of either a circle of latitude alone or a circle of longitude
alone. - -
5. Latitude by the rate of change of the sun's altitude near the prime vertical–
This is a method suggested by Prestel”, which confessedly affords but a rude
approximation, yet may be of great value in an emergency, because it can be
applied when other methods of finding the latitude fail, namely, when the Sun
is near the prime vertical. We think that, taken in connection with Sumner's
method, its value will prove even greater than was supposed by its author; for
while Sumner's method will determine a line on which the ship is situated,
Prestel’s method will, from the same observation, if near the prime vertical, give
an approximaton to the latitude and thus confine the position of the ship to a
certain limited portion of the Summer line. -
The simplest mode of observing for this method is as follows: In the morning,
when the sun is near the prime vertical, bring the image of the sun's upper limb
reflected from the sextant mirrors into contact with the sea horizon and note the
time by the chronometer or a good watch; let the sextant reading remain un-
changed, and note the time when the contact of the lower limb with the horizon
occurs. In the afternoon begin with the lower limb.
Then take the sun's semi-diameter from the Ephemeris and reduce it to seconds;
also find the chronometer interval between the two contacts in seconds. To the
constant log, 9.12494 add the log. of the sun's semi-diameter, and the arithmeti-
cal complement of the log. of the chronometer interval. The sum is the log.
cosine of the latitude, if the sun is very near the prime vertical. But if the sun
is not very near the prime vertical, observe its bearing as nearly as possible with
the compass, allowing for the variation; and to the above logs. add also the log.
secant of the Sun's amplitude: the sum will be the log. cosine of the latitude.
ExAMPLE.—Suppose that on June 10, 1865, in latitude 42° N., longitude 126°
west by account, about 8" 20" A. M., the sextant being set at 41° 0' 0", the
two contacts of the upper and lower limbs of the sun with the horizon are ob-
served as follows:
Chronometer.
Upper limb - - - - - - - 4” 40” 32°
Lower limb - - - - - - - 4 43 23
the amplitude of the sun at the middle time being about 5° S. Then the
computation will stand as below :
Constant log. - - 9.12494
Sun’s semidiam - - - = 15' 47"=947// Log. - - - - - 2.97635
Chron. interval - - - = 2* 515 =1718 Ar. co. log. - - 7.76700
Sun's amplitude - - - = 5° Log. Sec. - - - 0.00166
Latitude - - - - - =42° 10' Log. cos - - - 9.86995
This result must be admitted to be but a rough approximation, since an error
of one second in the chronometer interval would change the computed latitude
in this case about 20'. In higher latitudes the method gives more accurate
results. However, a knowledge of the latitude within about 20' may often be of
* Astronomische Nachrichten, vol. xxxvii, p. 281.
||||||II
ASTRONOMY. 11
the greatest value; for suppose, in the present case, the navigator, knowing he
is near the coast of Oregon, but being in consequence of continued bad weather
very uncertain as to his position, is anxious to determine at once the probable
bearing of the land; he will use the altitude at which the sextant was set in
the preceding observation as an ordinary altitude for time and proceed to find
his line of position by Sumner's method, employing in the two computations
two latitudes, each differing from the one just computed by 20', namely 41°
50' and 42° 30'; and finding the corresponding longitudes by chronometer,
will put down the two resulting positions on his chart. He may then safely
assume that he is on the line joining these two positions, and (not on this line
produced but) somewhere between them ; since the computation of the latitude
puts him midway between them with a probable error of not more than one
half their difference. If the reader wishes to carry through this example, let
him assume that the dip, refraction, and parallax amounted to —5' 0", and that
the index correction of the sextant was -- 5' 0": the observed altitude will be
that of the sun’s centre at the middle instant, so that 41° 0' will be the true
altitude of the sun’s centre at the chronometer time, 4” 41* 575.5. The chro.
mometer being fast, say 2" 0°, we shall have the Greenwich mean time 4h 39”
57°.5, for which the sun's declination is 23° 3' 26" N., and the equation of
time — 0" 48*. With these data, he will find that the latitude 41° 50' gives
the longitude 125° 1'.5 W., and the latitude 42° 30' gives the longitude 124°
57' W.; and the ship must be assumed to be on the line between these two
positions. It must not be forgotten, however, that this result is reliable only
so far as dependence can be placed upon the chronometer; but if the chronom-
eter has gone wrong, the line of position thus determined will still have the same
direction on the chart, but will be shifted in longitude east or west by an amount
equal to the deviation of the chronometer from its supposed error.
We must not omit to notice one remarkable feature of Prestel’s method,
which is that it requires no knowledge of the sun's declination, and may there-
fore be resorted to in cases where there is no copy of the Ephemeris on board
ship. In such cases we may also dispense with a knowledge of the sun's semi-
diameter, by setting the sextant successively at two readings 30' apart and ob-
serving the times of contact of the same limb with the horizon; then in the
computation by the above rule we use 15' in the place of the sun's sémidiameter.
6. Lunar distances.—While we have various methods of determining a ship's
latitude which are both simple and sure, we have but two methods of finding
the longitude which are sufficiently simple for use at sea, and of these but one
can be pronounced sure. Nothing more simple can be desired than the method
“by chronometer,” but unfortunately it cannot be regarded as sure. To use
the words of Sir John Herschel,” “the chronometer, though greatly and indeed
wonderfully improved by the skill of modern artists, is yet far too imperfect
an instrument to be relied upon implicitly. However such an instrument may
preserve its uniformity of rate for a few hours, or even days, yet in long ab-
sences from home the chances of error and accident become so multiplied as to
destroy all security of reliance on even the best. To a certain extent this may,
* Outlines of Astronomy, Art. 258.
12 ASTRONOMY.
indeed, be remedied by carrying out several, and using them as checks on each
other; but besides expense and trouble, this is only a palliation of the evil.”
On the other hand, the method of lunar distances, by which the Greenwich
time is read from the face of the heavens and not from a frail piece of human
mechanism, is (within assignable limits of precision) an absolutely certain method,
in the hands of any navigator of average skill in observation and computation.
This being incontrovertibly true, we are led to inquire, why is it that both in
our navy and the mercantile marine the method has of late years so nearly gone
out of use altogether ? We shall attempt to answer this question, and in doing
so give reasons why the method should be rescued from the neglect into which
it seems to have fallen.
(1.) The observation of a lunar distance is subject to an unavoidable error,
greater or less according to the skill of the observer, but always of considera-
ble magnitude.
Even with the greatest skill, the probable error of observation cannot be re-
duced much below 10" in the distance. For, let us consider the components
of this error, understanding here by error of observation the combined effect of
all causes that render the recorded apparent distance of the moon’s limb from
the sun or a star different from its actual value. We have (a) the error of the
pointing, or actual error of the observer in making the contact of the images;
an error whose mean value in each single contact, observed on board ship at
sea, appears to be seldom less than 10", according to the experience of good
observers. Assuming this value as the average one for good observers, let us
suppose a complete observation to consist of sia: observed contacts, (which is
perhaps as great a number as it is judicious to combine in one group;) then the
mean error of pointing is reduced to 10" divided by the square root of 6, or to
4". But this is upon the supposition that the limb of the moon observed is so
perfectly defined in the sextant telescope that the point where the contact is ob-
served is strictly the point whose distance from the star is given in the ephem-
cris. Now we know, from the most accurate series of transits of the moon’s
limb observed at Greenwich and elsewhere, that the precise position of the
limb cannot be observed even with the largest and best telescopes within 1/.5,
an error which is constant during a transit, but may vary at different transits.
In the small telescope of the sextant, it is probable that the error thus constant
during any number of successive contacts on the same day would be much
greater; but that we may not be supposed to exaggerate the effect of errors, let
us assume it to be only twice as great as in the transit, and that we have as
our second error (b) the constant 3". We have also (c) the error of the index
correction, which, with the greatest care, using a number of observations, will
seldom be less than 3”. Then we have (d) the error of division, the errors of
the mirrors, and the eccentricity (usually unknown) of the sextant, the com-
bined effect of which may be estimated as not less than 4" in the best instru-
ments. The mean error of observation representing all these effects combined
will therefore be about
V [(4)” + (3) + (3)” + (4)* } = 7”.
ASTRONOMY. 13
This is even less than the estimate of experienced observers in regard to the
total effect of these errors, as found on shore by comparing the results by lu-
nar distances with those obtained by more exact methods, so that we are justi-
fied in assuming the total mean error of observation of a set of lunar distances
(carefully taken by the best observer) to be not less than 7".
(2.) The methods of computation commonly used at sea have been inaccurate.
We say commonly used, for although methods have long been known which are
sufficiently accurate, designated in nautical works as “rigorous” methods, they
are so tedious that navigators will not (and usually cannot) employ them.
These (so called) rigorous methods require the use of tables of six or seven
figure logarithms, and even with these are far from being in a strict sense rig-
orous, for they do not consider the moon’s parallax in a strict manner, since they
neglect the parallax in azimuth ; nor do they take into account with absolute
rigor the effect of the spheroidal figure of the earth and the true figure of the
disks of the sun and moon as affected by refraction ; still, with the aid of sup-
plementary computations, embarrassing to any but the most skilful computers,
they are rendered accurate enough, and therefore do practically fulfil the condi-
tions required in a rigorous or perfect method.* But the labor attending them
and the doubt attaching to any result which is obtained by long and intricate
computations, made amid the distractions of a ship, must ever prevent such
methods from being generally adopted. The great majority of navigators will
therefore continue to rely, as they have heretofore done, upon what are called
approacimative methods. Of these Bowditch’s “ First Method '' is as accurate
as any now in general use, so far as the effects of the moon's parallax and the
mean atmospheric refraction are concerned. But even this method furnishes
only a rude approximation to these effects, inasmuch as cases are quite com-
mon in which an error of 10% in the resulting true distance remains, in conse-
quence of the inaccuracy of the formula on which the rule is based. Again,
this inaccurate formula is tabulated upon the supposition of a mean state of
the air, so that when we wish to take into account its actual state as shown by
the barometer and thermometer at the time of the observation, we have to re-
sort to troublesome supplementary computations; and as the necessity for these
is not insisted upon in connection with the rule, most navigators have
been in the habit of neglecting them altogether. Now, this neglect entails
even greater error than the inaccuracy of the formula, for it will often produce
errors of 10", 15", 20" in the distance, and may in extreme (though indeed
unusual) cases produce errors of 40" or even more We are, therefore, quite
within bounds when we assert (what we could prove satisfactorily, if our space
permitted) that the mean error of computation of a lunar distance by the meth-
ods heretofore most commonly used at sea is at least 15".
(3.) The ephemeris of the moon has until within a few years been inaccurate.
It is well known that for a number of years previous to the first publication of
... “The objections here stated do not apply to Bessel's method of lunar distances, which is
in fact the one really rigorous method. It requires, however, a different ephemeris from any
now published, and even with this peculiar ephemeris is still too long and intricate for com
IO, OIl llS6.
14 ASTRONOMY.
the American Ephemeris and Nautical Almanac in 1855, the ephemeris of the
moon in the British nautical almanae (of which the American almanacs at that
time were only reprints,) and in other European publications of the same kind,
was greatly in error. For example, the solar eclipse of July 28, 1851, proved
that the moon's right ascension as given in the British almanac was on that
date in excess 28” (of arc,) and consequently the distance of the moon’s limb
from any star in her path on that date, could it have been used at that time,
would have been given in the almanac with an error of about the same amount.
The error of the ephemeris, however, was not constant, but varied during each
lunation, so that the error of the lunar distances was usually less than 20", and
sometimes even under 5", but it never disappeared entirely. Hence, even if
the navigator had taken a perfect observation and had computed it by a perfect
method, he was still liable to this unavoidable error of the ephemeris, at that
time unknown and by most navigators not even suspected, as they were in the
habit of relying upon the almanac as an absolutely perfect publication.
Let us then suppose what was at that time a common case at Sea. The navi-
gator having found his longitude “by chronometer,” wishes to verify it by
“lunars.” He takes a set of distances with the utmost care and reduces them
by Bowditch’s First Method or some equivalent reputed reliable method. His
result is liable to the three sources of error above named, i. e., mean error of
observation =7", mean error of computation=15", and mean error of epheme-
ris = (say) 10"; and thus he was liable to the total mean error,
v[(7,” + (15)” + (10)”] = 19",
and this error of 19" in the distance would produce in the longitude a mean
error of 38 seconds of time. Now his chronometer must have performed very
badly to be in error 38 seconds; yet this discrepancy is only the mean error to
which his result is liable. He takes another set of distances and finds a still
greater discrepancy; again a set which (in consequence of a happy balancing
of errors) happens to agree very nearly with his chronometer; then a set in
which the discrepancy (in consequence of all the errors happening to fall in
the same direction) runs up to double the mean, or nearly 80 seconds; and after
considerable experience of this kind finds that his lunars differ from his chro-
nometer on the average as much as 38 or 40 seconds of time, and upon making
port finds that his chronometer is right. Occasionally, indeed, he will find that
his chronometer is totally wrong and his lunars, rude as they are, may have
saved his ship ; but the frequent experience of the discrepancies between his
lunars and his chronometer, with the subsequent proof of the greater accuracy
of the latter, must inevitably destroy his confidence in the value of lunar obser-
vations. The constant improvement in the manufacture of the chronometer has
also contributed to this result; and certainly if this instrument were liable to
no accident which could derange it, no more simple and perfect method of find-
ing the longitude could be desired than the chronometric. But it may go wrong,
totally wrong , and this possibility, however remote, makes it the imperative
duty of every navigator to whom the safety of his crew and passengers is in-
trusted, to resort to the only available check upon the chronometer's result
ASTRONOMY. 15
namely, the lunar method. And this was his duty even when the lunar method
afforded no better check than we have shown it to have been before the year
1855, when it was liable to a mean error of about 40 seconds of time and to
possible errors of more than twice that amount; for the experienced navigator
would always allow a large margin to his results and make the land with cor-
responding caution. But if we turn to a consideration of the present condition
of the methods of observation and computation and of the ephemeris, we shall
find that now more than ever the lunar method is available as a check on the
chronometer.
First. As to the error of observation, it is not probable that its amount can be
reduced much below that above estimated. The attention of the navigator
should, however, be directed to the means of reducing this error. The eccen-
tricity of the sextant ought to be eliminated by the use of a complete circle
reading with two opposite vernieºs in the prismatic circles of Pistor and
Martins. Distinctness of vision should be sought by the use of the best tele-
Scopes and most perfect mirrors, and in these respects also the prismatic instru-
ments of Pistor and Martins are very excellent. It may also be found possible
to devise some mode of suspending a chair so as to afford a steady seat to the
observer, as suggested by Sir John Herschel, who recommends for the purpose,
not a perfectly free suspension, “which tends to prolong and perpetuate oscil-
lations once impressed,” but a “stiff suspension, as, for example, by a rigid
rope or cable, or by a Hook's joint, purposely made to work stiffly (and that
more or less at pleasure) by tightening collars,” &c. For the present, we assume
the mean error of observation with the best instruments to be 7” as above given.
Secondly. The new method of computation proposed by the writer of this
article and published in the American Almanac for 1855* belongs to the class.
of approximative methods and is very little longer than the briefest methods
heretofore used, while it takes into account all the corrections in so simple a
manner that any navigator of average dexterity can work the true distance in
a few minutes with great accuracy and certainty. The method is based upon
a rigorous formula, but in consequence of the use of only four figure logarithms
and the neglect of fractions of seconds, it is in fact only approximative; but its
mean error is not more than 2", and thefore it is practically perfect. Since
this method is now available and can only be supplanted by one which gives as
accurate a result with less labor, we shall assume that the mean error of com-
putation is now reduced to 2".
Thirdly. The improvement in the lunar ephemeris, first introduced in the
American Almanac, by the use of Peirce's Tables of the Moon, and subsequently
in the British Almanac and other European Ephemerides, by the use of Hansen's
tables, reduces the mean error of the lunar distance to about 5".
Hence, the result obtained from an observation, consisting of six individual
contacts of the moon's limb with the sun or a star, computed by the most accurate
method and with the new ephemeris, is subject to the mean error
V[(7)” + (2)” -- (5)”] = 9 /;
[* Chauvenet's New Method of Lunar Distances is also separately published, and can be
obtained at the nautical booksellers.-ED.]
16 - ASTRONOMY.
that is, about one-half its former amount. The mean error of the longitude de-
duced will thereforébe only 20 seconds of time, or 5 minutes of arc, a degree of
accuracy that must be regarded as quite satisfactory, and nearly fulfilling all the
requirements of modern navigation.
The object of the preceding discussion will have been gained if our naviga-
tors will resume the habitual practice of observing lunar distances as a check
upon the chronometer; and if at the same time they will study the sources of error
we have indicated, with the view of discovering means of reducing their effects to
still smaller amounts, and also endeavor to shorten the method of computation,
without impairing its accuracy, they will contribute at once to the security of
navigation and to the facility of its practice.
II. AstroNOMICAL GEography.—The navigator frequently has opportuni-
ties of contributing to the imp. 7&ment of this department, when visiting places
whose positions have not been well determined.
1. If the only astronomical instrument he possesses is the sextant, he will
endeavor to determine the latitude of the place by a good series of circum-meridian
altitudes of the sun, or of stars, with the artificial horizon. The observation of
the sun for this purpose is usually to be preferred, because most observers with
the sextant can make a contact of the limbs of the two images of the sun with
more accuracy than that of the images of a star, and, moreover, any error arising
from a faulty habit of observing this contact will be eliminated by taking the
mean of the results of two distinct series, in one of which the lower limb is used,
and in the other the upper limb. The observation of the sun is also to be preferred
on account of the greater certainty in the sextant readings by daylight. How-
ever, when the observer is experienced in night observations and has proper facil-
§ties, there are some advantages in their use; for the sextant is at night less liable
to the errors produced by changes of temperature, and constant instrumental
errors and errors in the refraction may be eliminated to a great extent by com-
bining results from stars north and south of the zenith.
In all cases a set of circum-meridian altitudes should consist of a number of pairs
of nearly equal zenith distances, east and west of the meridian, so that any small
error arising from imperfect determination of the chronometer correction will be
eliminated in the mean. This condition is irr practice sufficiently well satisfied
when the same number of altitudes are taken on each side of the meridian, the in-
tervals between them being made nearly equal by taking the successive observa-
tions as rapidly as may be consistent with accuracy. “The observer, whether he
reduces his own observations or not, should be careful to report all his data just
as they were observed; including, also, the state of the barometer and thermom-
eter, as noted both before and after each set of altitudes, the observations for
determining the index correction of his sextant and the chronometer correction,
together with notes of any circumstances which may affect the character of the
observations, either favorably or unfavorably. He must remember that in order
that any determination of position should have authority and be received with
confidence, it must be communicated to the scientific world in such a shape that
it will carry with it the evidence of its accuracy.
ASTRONOMY 17
In low latitudes, when the meridian altitude of the sun is too great to be ob-
served with the artificial horizon, and it is not convenient to observe stars, resort
may be had, with advantage, to equal altitudes of the sun, from which both the
time and the latitude may be deduced with great accuracy. The most favorable
case for determining both these elements with precision, from equal latitudes, is
that in which the azimuth of the sun is about 45°; for determining the latitude
only, the nearer to noon the better.
2. For finding the longitude, the observer having only the sextant must em-
ploy lunar distances, which, as we have shown, may with proper care give a result
subject to a mean error of 20 seconds of time, by a single set of distances; and by
taking a large number of observations of the sun, in both the first and second half
of a lunation, and forming two separate means from the observations in each half,
the final mean of these two means will be free from constant unknown instru-
mental errors. The errors of the ephemeris will be finally eliminated by obtain-
ing from the observations at some standard observatory the corrections of the
moon's right ascension and declination on the date of the observation, and cor-
recting the longitude according to known methods.
But if the observer has the means of observing occultations of stars, he will be
able to deduce a more precise result from two or three such observations than
from the most extended series of lunar distances.
3. The scientific explorer who wishes to determine the positions of fixed points
with the greatest accuracy should be furnished with the zenith telescope, as now
constructed and used in our Coast Survey and the interior surveys of the United
States. This instrument is specially designed to furnish the latitude, which it
does with a degree of precision unequalled by any other portable instrument.
It may also be used for determining the longitude by observing with it equal
altitudes of the moon and a neighboring star, according to the method first pro-
posed by Professor Kaiser, of the Netherlands.”
4. A portable transit instrument is also one of the most useful auxiliaries. It
may be arranged so as to answer the purposes of the zenith telescope, as sug-
gested by Professor C. S. Lyman;f and then it will serve to determine both longi-
tude and latitude by observations in the meridian—the longitude, by moon cul-
minations in the usual manner; and the latitude, by using the instrument as a
Zenith telescope.
If not adapted as a zenith telescope, the transit instrument may still be used to
find the latitude with great precision by mounting it in the prime vertical, and
observing according to Bessel's method. For this purpose, it is desirable to have
the instrument provided with a horizontal circle, so that after it has been adjusted
in the meridian it may be revolved through precisely 90° by this circle, and thus
brought at once into the prime vertical. When used in this manner, marks should
be set up at suitable distances in both the meridian and prime vertical, and the in-
strument should be set upon one of these marks in all observa.ions; or at least if
... “This method, with the modifications adapting it to practice with the zenith telescope, is
$xplained at length in Chauvenet's Spherical and Practical Astronomy.—ED.
:*:Amerſºn: Jºurnal of Science, Vol. XXX, p. 52.
* * s 2
18 ASTRONOMY.
it is provided (as it always should be) with a micrometer, its position with respect
to each mark should be measured and recorded with each set of observations.
5. When the navigator has opportunities of making several runs between two
important stations, he should carefully determine the error of his chronometer at
each place, (by equal altitudes or transits,) and thus determine their difference of
longitude. The observations themselves, as well as the results, should be com-
municated in full to the Bureau of Navigation.
III. GENERAL ASTRONOMY.—The navigator will seldom find opportunities
for contributing to general astronomy not possessed, in a superior degree, by
observers on land and in fixed observatories. Occasionally, however, he will be
able to make observations which may prove of great importance.
1. A comet may suddenly appear in high south declination, and be observed
on board a ship in south latitude, when it may wholly escape the attention of
observers on land, and even (by reason of clouds) those at the observatory at
the Cape of Good Hope; or the comet may be seen just after sunset or before
sunrise, very near the horizon, and can be observed at sea when it could not be
seen on land. The position of the comet, as determined by a careful observer at
sea, will sometimes be of great importance in determining its orbit. The only
observations practicable with the sextant will be those of the distance of the comet
from at least two, but always when possible three, conspicuous fixed stars. The
three stars should be selected, when the position of the comet permits, so that
great circles drawn from them to the comet will make angles at the comet of about
120° with each other. If but two stars are used, one should lie east or west of
the comet, the other north or south of it; or, if that is not practicable, the circles
drawn from them to the comet should make an angle of about 90°. Care must be
taken to record the approximate position of the comet with respect to the stars,
(east, north, &c.,) as judged by the eye, as a check against gross errors. Each
star should be identified by means of a star-chart or good globe; and if any uncer-
tainty exists, its position with respect to some known stars should be recorded.
The observations of the distances of the comet from the several stars should be
made as nearly simultaneous as possible. If one person takes all the observa-
tions, he should, after measuring the three distances, repeat the measure of the
first two—taking them in reverse order; and it will then be easy to reduce all the
measures to the instant at which the third was observed. -
The altitude of at least one of the stars should be carefully observed at the same
time with the distances; indeed, it will always be well to observe the altitudes of
all the stars and of the comet at the same time, as this will serve the purpose of
a final check and facilitate the computation of the effect of refraction. The state
of the barometer and thermometer must be recorded, together with the index cor-
rections of the sextants used and the height of the eye above the level of the sea.
The navigator will not attempt to reduce such observations as these at sea, but
should communicate them to the Bureau of Navigation precisely as they have
been taken; and to guard against misapplication of the index correction, he should
give both the distance, as read from the sextant, and the distance corrected. The
same method should be followed in reporting the altitudes, applying.bothº
index correction and the dip of the horizon. The chronometer timé, ºf back obº.
ASTRONOMY. 19
vation must be stated, together with the supposed correction of the chronometer
on Greenwich time. Finally, the latitude and longitude of the ship should be given
with as great accuracy as possible. If the time at the ship has been determined
not long before the observations on the comet, and the latitude of the ship is well
known, the observation of the altitudes may be dispensed with ; in that case, the
observations by which the time was found should also be reported in full.
In addition to these observations, it will be useful to make a drawing of the
whole outline of the comet, putting down upon the drawing the stars near the
outline, or within it, as accurately as possible. A drawing on a star-chart will
be the best.
2. When an eclipse of the sun occurs while the ship is lying in port, the local
time of the instants of first and last contact should be observed; if the eclipse is
total or annular, also the instants of beginning and end of totality or annularity.
If the latitude of the port has not been well determined, observations should be
taken for the purpose.
During the eclipse, any attending phenomena should be carefully observed and
described. At the beginning, observe whether the indentation of the sun's disk
is a regular curve, or whether any indentations appear in advance of the moon's
general outline. As the eclipse progresses, observe the contacts of the moon's limb
with prominent spots on the sun's disk, and record any special phenomena that
may present themselves. The distance between the two cusps may also be ob-
served with the sextant, especially near the beginning and end. The phenomena
occurring just before totality deserve special attention, and particularly the
mode in which the last trace of the sun's light disappears—whether breaking up
into points, “bead-like,” or vanishing in a well defined, crescent-shaped line, &c.
During totality, observe any colored projections that appear around the moon's
dark edge, and make an accurate drawing of them in situ, if possible. In estima-
ting their dimensions, compare them with the apparent diameter of the sun; state
their color and successive changes of color and form. Observe the luminous
corona that always surrounds the moon during the total phase, noticing specially
its apparent breadth in various directions, compared with the sun’s apparent diam-
eter, and whether it is concentric with the sun or with the moon. At the end of
totality, observe the phenomena attending the reappearance of the sun's light; and
as the moon’s disk passes away, make similar observations to those made when
it was coming on.
If the observer is at sea, his attention may be directed exclusively to these
attending phenomena, as the observation of the precise instants of the constants
will then be of no special value.
3. Occultations of stars and planets by the moon, observed in a port whose
latitude and longitude are well known, will also be useful. In observing an oc-
cultation of Jupiter and Saturn, a telescope should be used of sufficient power
to show a well-defined disk of the planet, and the two instants of contact of the
two opposite limbs of the planet with the moon's limb should be observed if
::Rºle; In the occultation of Saturn it will occasionally happen that the
20 ASTRONOMY.
instants of contact of the moon's limb with the outer and inner edges of the
planet's ring can be observed, as well as with the body of the planet.
4. Eclipses of Jupiter's satellites may also be useful, especially if both disap-
pearance and reappearance of the same satellite are observed. The transits of
the satellites and their shadows over the disk of Jupiter, and the occultations of
the satellites by the body of the planet, may also be observed. But in order
that these observations may be of real value to science they should be carefully
observed with a telescope of not less than six inches aperture.
5. Observâtions of the dip of the horizon can be made with a dip-sector or
Pistor and Martins's prismatic sextant. Such observations will be of use in
perfecting the theory of terrestrial refraction and the common nautical tables of
the dip. Every observation should be accompanied by a record of the temper-
ature of the air and of the water, as well as the height of the barometer. It is
quite possible that from a good series of such observatious a formula may here-
after be constructed by means of which the dip may be computed for any given
case with accuracy, even where the horizon is much displaced by extraordinary
refraction, and thus a frequent source of error in observations at sea will be
removed.
6. Observations of the apparent magnitude of variable stars may be made;
but most of these stars are now so assiduously observed on land that the navi-
gator will not be able to add anything of value to the knowledge already pos-
sessed, unless he devotes much time to the subject and pursues it with special
interest.
7. The navigator may contribute to the advancement of the theory of atmo-
spheric refraction by observing the rising and setting of the sun and moon. The
ship time being well determined, let the instants of the contact of the upper and
lower limbs of the sun and moon with the sea horizon be observed with the
sextant telescope or any small telescope which will sharply define the limb and
is furnished with suitable colored glasses. The latitude of the ship must be
accurately known, and the approximate longitude, so that the true zenith distance
of the observed body may be computed for the time of the observations. Special
observations of the actual dip of the horizon should also be made at the same
time if possible; but whether this is practicable or not let each observation be
accompanied by a record of the temperature of the water, the temperature of the
air, and the height of the barometer. It is also desirable that the state of the
air as to moisture should be examined with some hygrometric instrument; the
wet bulb thermometer will usually be most convenient.
8. The navigator often has excellent opportunities for observing the zodiacal
light. The plan of observation followed by Chaplain George Jones,” of the
United States Navy, may be recommended. It may be well to draw on the star
chart not only the outline of the light, but the line of maacimum intensity of the
come of light from the vertex to the base.
*United States Japan Expedition, vol. 3.
C :... ... : : " :