343 MONTHLY WEATHER REVIEW. JUNE, 1906
afternoon, though apparently diminishing slowly. The nest day
i t was found to be positive again and did not depart from the nor-
mal conditions. Another peculiarity of the record of May 9 is
the vacillating character of the charge observed, the larger es-
cursions of the curve having superposed upon them a number
of small and rather rapid fluctuations, whereas, in general, the
variation of pnt,ential occurs gradually and rather ~lowly.
f0
+5
These results appear to indicate an abnormal electrical con-
dition of the atmosphere, and the influence of some energetic
dist,urbing agency. It seems not at all improbable that the
immense volumes of gases and other eruptive products ejected
from the crater of the volcano, and which were manifestly the
seat of intense electrical charges, may have made their influ-
ence felt, gradually and progressively, at great distances from
their point of origin, and SO have produced a widespread dis-
turbance of the electrical condition of the atmosphere. This
would naturally accord with the nbsence of any imuediate
and radical departure from the normal in the observation of
May S, inacle within about two hours after the great eruption,
the minor peculiarities observed being such as might be due
simply to inductive effects at a distance.
IMPROVED METHODS FOR FINDING ALTITUDE AND AZIMUTH, GEOGRAPHICAL POSITION, AND THE VARI- ATION OF THE COMPASS.
According to St. Hilaires method, having found the altitude
and azimuth of a celestial body for an estimated geographical
position of the ship, a navigator can at once obtain a Sumner
line or locus of position. He simply draws a line through the
estimated position on the chart in the direction of the azi-
muth or true bearing of the body and lays off along this line
of bearing an intercept from the estimated position equal to
the difference between the altitude as deduced from instru-
mental measurement and the altitucle that the observed body
woulcl have if the observer stood in the estimated geographical
position. This intercept is clrawn toward the direction of the
observed body or away from it, according as the measured
altitude is higher or lower than the altitude due to the esti-
mated position. A straight line clrawn at right-angle8 to the
line of bearing, through the point thus obtained, will be the
Sumner line required.
This method rids the observer at once of the trammeling
process of selecting a celestial body on or near the prime
vertical for tlie computation of the longitude, and a celestial
body on or near the meridian for the computation of the lati-
tude, because i t has the indisputable superiority of giving a
Sriniiier line from the observation of a celestial body in any
azimuth. But the drawing of this line requires the finding of
the altitude and the corresponding azimuth at the estimated
geograpliical position of the observer, and the deterinination
of these elements by the ordinary formulas necessitates as
much computation as navigators and geographers have been
accustomed to perform in the calculation of time and longitude.
Purposing further to recoinmend to navigators the advan-
tages of the nietliod of St. Hilaire, Professor Souillagouet,
of the French Navy, in the year 1900, published estensive
tables, intended to shorten the work of calculating the altitide
ani1 azimuth, iu a volume entitled < Tables du point ausiliaire
pour trouver rapidement la hauteur et lazimut estimbs.
ALTITUDES.
The table for finding the altitudes is based upon the follow-
iiig conditions. I n fig. 1, let P be the pole of the celestial
sphere, 2 the zenith of the observer, and A the pobition of the
observed celestial body. Then, drawing a Rpherical perpen-
The Sumuer line is $0 called from its inventor, Cdpt. Thomas H.
8umner, an American shipmaster, who seeins to h a w accidentally die-
covered it5 aliplication to owan navigation in 1837. If when at sea the
navigator iiieasures the apparent altitude of the sun, or any other celes-
tial liotly whose right asrension, hour angle, and declination are known,
he thrn knowh that he iiiust be somrwlrere on a sindl circle on the globe
wliokr center is vertically heneath the s u n or star. That portion of this
circle passing through his a9s~inied ayproriiuate pobition on the globe is
calleil a Siiioorr line. IP he can observe another oelrstial body, or i f he
can wait a while and otwerve the same body in a different position, he
can draw a second Sumner line, and the intersection of the two will give
him tlie loratiou of his vessel with all the accuracy that is practicable.
A full description oP Sumners method is given in works on navigation,
and esproially in the American Practical Navigator, Chapter XV, as pub-
lished by the U. s. Hydrographic Offi~e.-E~.
~~ ~~~ ~~ ~~~~
JUNE, 1905. MONTHLY WEATHER REVIEW. 243
dicular from 2 to AT, and thus divid-
ing thet riangle P Zd into two right-
angled spherical triangles, these rela-
tions exist:
tan y = cot L cos P (1 )
z sin H = (---)cos(~-p). sin L (3 )
cos $P
P
As the accompanying specimen
page-heading (see Table 1) will show
ble are P, the angle a t the pole, and
L, the latitude; and the tabulated
the arguments for entering the ta-
-
Lnt
-
P
~~
OhOU
1
2
4
S
6
7
8
9
IO
11
12
13
14
1s
16
l i IS
19
'LO
21 r2
2.3
24
25
26
27 2s
29
30
31 3 2
33
.w 35
36
37
38
39
40
41
42
43
44
45
46
47 4s
49
50
51
52
#5s M
55
56
57
58
59
60'
-
values are p aucl log (<fO: :), which v v\,
a
FIG. 1. are used in connection with J the
TABLE l.-Frotii pnye 61 of S O ~l i ~~~g 0 1 1 1 ' f ' d Table.
__
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0 .0
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59 31)
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57 20 56 L5
51; 30
56 IU
55 50
55 ?o 54 54
54 20
53 XI
33 20
52 50
52 1:
51 40
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49 50
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46 I O
45 25
44 40
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41 11)
40 15
39 211 3s 2u
37 20
36 20 35 21)
34 IS
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32 5
31 0
29 50
27 30
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25 5
23 50
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21 10
2s 4n
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Lat.
P
60"
59
53
57
51;
55
54
53
5.'
51
50
49
4s
47
46
45
44
43
42
41
40
39
33
37
36
:is
34
33 32
31
30
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2s
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24 23
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20
19
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IS
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L(
: 5
4
3
2
1
t l l m
-
23X+6 I , 172+7~,116+81;065+9
codeclination or polar clistance of the observed body, in the
logarithmic computation of H, the altitude, in accordance
with equation (3). Although the intervals between the values
of the arguments for entering the table are set for 15' of
latitude and 1 minute of time, yet in problems in which the
Sumner line is sought interpolation ilqentering the table may
be avoided by an appropriate choice of the estimated geo-
graphical position.
AZIMUTHS.
In order to construct within a limited volume tables giving
the exact azimuth with all the geuerality required in St.
Hilaire's method, that is to say, for all declinations and all values
of the hour-angle of the celestial body. i t is found necessary
to have recourse to two successive entries. It' the astronomi-
cal triangle PZ.4 (fig. 2 ) be clecomnosed into two right-angled
spherical triangles by drawing the spherical perpendicular A R
from A to the opposite side PZ, the following relations exist:
tan y = tan Pd cos P = tan 3 cos P
A specimen page of Souil1agoui;t's azimuth table is also intro-
duced here (see Table 3 ) in order that the following rules for
using the talde mag be more readily esplninecl:
1. Find in the table the augle a t the
pole (vertical nrgunieut), and t,he decli-
nation (upper horizontal argument)
This first entry gives the two ausiliar)
arcs y and y ' . The arc 9 is to be reclt-
onecl positive if the. angle a t the pole
a d the polar distance are of the same
name, ancl negztive if of a contrary
name.
2. Find the algebraic sum of L and p
(regnrding the latitude of the observer
as always positive), a d y posiitive or
negative as determined by the foregoing
rule.
3. Enter a second time in the table with $0' as upper horizontal argument, FIG. 2.
and L + y (on the right or left) in column P. I n column p
will be foiind an arc equal to the amplitude d (the comple-
ment of the aziiuuth).
Although the altitude and azimuth, which are conjointly
needed in laying down the locus of geographical position by
Mt. Hilaire's method, must be found separately, and each indi-
rectly, by Professor Souillagouet's tables, his work has never-
theless, by reducing the tedium of numerical and logarithmic
computation, nccoiuplisherl a great advantage in oceanic navi-
gation, and has been miclely sought after and highly com-
mended.
(3)
(4) cot Z = cot p' sin KZ = cot y' cos (L + y ).
P
A further important advance, involving the accomplishment
of a heavy and laborious task, has now been performed by
Mr. Littlehales, Hydrographic Eugineer of the United States
Hydrographic Oltice, mho has prepared for publication n vol-
unie in which the coorcliuates of the celestial sphere are in
effect charted from minute to minute of arc for the whole
circuit of the heavens in such a inanner that, the charts being
entered with the latitude of the observer and the declination
and hour-angle of the observed celestial body as arguments,
the altitude and azimuth may be siiuiiltaneonsly found mith-
out any computation whatever.
His plan of solution employs a stereographic projection of
the celestial sphere on the plane of the 06.srr1.rr's meridian such
as that represented on a reduced scale iu fig. 3.
If the latitude of the observer be laid off along the bound-
ing meridian of the projection as a t L, and the declination of
244 MONTHLY WEATHER REVIEW. JUNE, 1905
I /
14 42
56
15 Y 23
36
49 16 3
16
251
42
55
17 6
21
34
47
1R 0
12
25
38
50
19 2
15
2 i
31
51
20 3
15
27
3Y
50
21 2
13
25
36
47
59 22 9
20
31
42
53
23 3
14
24
34
45
55
24 5
15
24
34
43
53
25 2
11
21
30
3s
47
56
26 4
the observed celestial body be laid off a t Jf along a meridian
making an angle with the bounding meridian equal to the
hour-angle of the observed celestial body, an astronomical
triangle will be formed in which the known parts are the
two sides PL and PM, representing, respectively, 90 minus
the latitude and 90' minus the declination, and their in-
cluded angle LPM, which is the hour-angle of the observecl
celestial body. Two of the unknown parts of this triangle
are the azimuth PLM and the coaltitude LM of the observed
celestial body. If the triangle P L N mere revolved about the
central point of the projection, with the side P L kept in coin-
cidence with the bounding meridian until the point L is brought
to the position of the point Y, the latter would then occupy
the position P', and the point dl woulcl fall a t iU', so that the
unknown side of the triangle, representing the coaltitucle,
would lie along some meridian, and the altitude could be nieas-
ured from the graduation of the projection. Thus the unknown
angle, representing the azimuth, would become an included
angle between two meridians, which could likewise be meas-
ured from the graduations of the projection. Thus the alti-
tude and azimuth of nnj- observed celestial body could be
simultaneously determined from the diagram with auy degree
of precision that the wale of the projection might permit.
TABLE 2.- From prcqe Y l of Soui1laqou;~t'e fnbles.
160' 30
149
311'
148' 30'
147' 30'
146 30'
146' 'W
144 XI'
143' 30'
142(
YO'
141' 311'
140' 30'
139 RU'
138 311'
157' 3w
136' 31)'
13SC 31)'
134 :W
133 31)'
138' 90'
131' 30'
130 3U'
lagc 30'
128' 30'
127' 30'
126[ 30'
1 2 6 30'
124' 30'
123' 30'
122' 30'
121. 30'
120'
To obviate the necessity for actual revolution of the triangle,
as described above, a series of equally spaced concentric cir-
cumferences and a series of equally spaced radial lines have
been drawn over the projection, as shown by the lines of
dashes in fig. 3. For the purpose of identification, the over-
laid system of concentric circumferences is numbered serially
from the center of the projection outward to the bounding
meridian. The radials are also marked by numbers indi-
cating their angular distance in minutes of arc counted in a
clockwise direction from the line OS. After having plotted
the declination ancl hour-angle of the observed celestial body
at 111, i t is only necessary to note the number of the circum-
ference ancl the number of the radial which pass through this
position;-add to the number of the radial 90" minus the lati-
tude, expressed in minutes; find the intersection, di', with the
noted circumference, of the radial whose number is the ~u m
just found, and read off from the graduated arcs of the pro-
jection the altitude and azimuth of this point of intersection.
P
30'
38" 30'
390 w
% 41 30'
49
39
30
20
I I
25 5:
41
30
64O 30'
66"
dV, 66
30'
67O
30'
68 3'
69O 30'
600
32
18
4
19 50
36
22
8
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39
25
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30 30' Z I'
-
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5s
56
54
52
50
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46
44
42
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36
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52
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44
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48
50
52
51
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58
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4
6
8
10
12
14
16
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26
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FIG. 3.
In order that the required results may be found to the
nearest minute of arc, the stereographic projection has been
constructed for a sphere whose diameter is twelve feet: but
this being too large for ordinary use when kept in one continu-
ous sheet, Mr. Littlehales cuts each of the four quadrants into
92 overlapping sections, all conreniently indexed and arranged
to form a volume of 368 pages. I n this form the projection can
be used with the same facility as if presei-ved in one continuous
sheet, for it has been pointed out that in effecting the required
solution only those parts of the projection are involved which
lie in the immecliate vicinity of the points whose coordinates
are to be plotted or read off.
The simplicity and directness of this method will be more
clearly seen by examining an example of the solution of a
problem in navigation.
E.mniple.-At sea, April 2, 1902, about Gh 35" p. m., in lati-
tude 39" 16' north and lougitude GO" 00' west by estimation.
Observed #L Aurigtr, bearing north GOo west per compass, to be
in altitude 66" 22' when the Greenwich mean time, as shown
by the chronometer, was loh 31" 03.2'. Required the Sumner
line of geographical position and the total error of the
compass.
From the statement of the problem and from the Nautical
Almanac, the hour-angle of u Aurigw is found to be 2h 02"
51s or 30' 43' 30", and the declination to be 45" 53' 56" north.
Nofe.-Plotting the declination and hour-angle roughly with reference
to the parallels and meridians (counted fmm the left-hand bounding
meridian) of the Index to Plates, (fig 4), we find that the position of the
observed body falls on plate 51, (flg. 5), approximately at the intersection
of circumference 17.2 with radial 8400. Then adding 8400 to 900 minus
JUNE, 1905. MONTHLY WEATHER REVIEW. 245
MONTHLY TmATHER REVIEW. JUNE, 1005
R" 22" 5 6 24' 26" 2ff 52" 30"
26 28" 30" 42" 32" 34" 36" 44"
FI~:. 5.-Reduced for purposes of illustration from the full-+izetl drawiug tlo-,iguetl to accompany the comlsl4.e work.
JUNE, 1905. MONTHLY WEATHER REVIEW. 2-17
248 MONTHLY WEATHER REVIEW. JUNE, 1905
the latitude expressed in minutes, which is 3044, we flnd the approximate
place of the revolved position to be at the intersection of circumference
17.2 with radial 8400+ 3044= 11444, which intersection falls within the
limits of plate 136, (flg. 6). Turning now to plate 51 (flg. 5), and plottirig
the hour-angle and the declination to the nearest minute, we find the
position of the observed body to fall at the intersection of circumference
495.6 with radial 8411. Adding 900 minus the latitude expressed in
minutes t o the number of this radial we obtain 8411+30444=11455 as the
number of the radial, at whose intersection with circumference 495.6,
on plate 136, (flg 6), the solution is to be found by reading off the alti-
tude with rererence to the parallels and the azimuth with reference to
the meridians, counting from the right-hand bounding meridian.
Solrltim.
Hour-angle 30 43.5' / ....... 1 Circ. 495.6
Declination 45O 54 0' i / R:%T811C
Lat. N. 39O 16': colat. SOo 4 4 ' . .... .3U44
Rad. 11455 1 \ 660 36' altitude.
Circ. 495.6 -1 N. 630 32' W. azimuth. True altitude by observation.. ......................... 660 23'
Altitude due to estimated position.. .................... 66 36
..................................................... 14
True bearing.. ........................................ N353 32' TV.
Compass bearing by observat,ion.. ...................... N. GO 110 TI'.
Total error of compass for heading of ship when observa-
tion was made.. ..................................... 30 33' TV.
It should not escape attention that the whole work, which we
have now performed, of finding the star's altitude and azimuth
due to the estimated geographical position of the ship may be
done before the observer goes on deck to measure the actual
altitude a t the prearranged instant of Greenwich mean time,
and that no more time need subsequently elapse in drawing
the Sumner line than is required for taking the difference of
the two altitudes -I, and hying it off as an intercept along the
line of true bearing of the observed body in the proper direction
from 0 the estimated geographical position of the observer, as
has been done on the sniall chart, fig. 7.
~~~
~~~ ~
Ah
FIO. 7.
Frequently a star that is favorably placed for observation
can not be identified because clouds obscure the surrounding
parts of the sky. If, when the altitude of such a star is
measured, its compass bearing be observed, and the approsi-
mate true azimuth be obtained by correcting the benring for
the variation and deviation of the compass, then the identity of
the star may a t once be ascertained by reversing the order of
proceeding that has just been described for finding the
altitude and azimuth from the declination and hour-angle.
Having plotted the corrected altitude of the star on the meri-
dian of the projection which makes an angle with the right-
hand bounding meridian equal to the star's azimuth counted
from the North Pole, note the number of the radial and the
number of the circumference that pass through the point so
plotted. Subtract 90" minus the latitude of the place of
observation, expressed in minutes, from the number of the
radial; find the intersection of the noted circumference with
a second radial whose number is t,he remninder thus found by
subtraction, and read from the graduations of the projection
the declination of this point and its hour-angle from the left-
hand bounding meridian. The hour-angle of the observed
star thus found must be converted into right ascension. Then
the star tables of the Nautical Almanac may be scanned to
find the name of the star whose tabulated right ascension and
declination come nearest to the values of the right ascension
and declination that have been cleducecl. The stars that are
of a sufficient magnitude to be observed by navigators are so
widely separated that there will be no difficulty in making the
selection from the tables, even when we proceed no further
than the use of the intles to plates in effecting the required
solution.
It will be found npon examination that these graphical
tables are also adapted to find, with very great facility, the
course and distance in great circle sailing, and that they pro-
vide a sure and simple solution, with a degree of precision
limited only by the scale of construction, for all those problems
of trigonometry and nautical astronomy that depend upon
solving a spherical triangle in which two sides and the in-
cluded angle are given.-X.
The Editor would commend Mr. Littlehale's methods and
his charts to the serious.attention of all who have occasion
to solve spherical triangles to the nearest minute of arc-
whether in geodesy, navigation, astronomy, or general mathe-
matical w 0 r k -E ~.
RECENT PAPERS BEARING ON METEOROLOaY.
R. A. EDWARDS, Acting Librarian.
The subjoined titles have been selected from the contents
of the periodicnls and serials recently received in the Library
of the Weather Bureau. The titles selected are of papers or
other communications bearing on meteorology or cognate
branches of science. This is not a complete index of the
meteorological contents of all the journals from which i t has
been compiled; it shows only the articles that appear to the
compiler likely to be of particular interest in connection with
the work of the Weather Bureau. Unsigned articles are indi-
cated by a -
Science. New Ilbrk. *New Series. Vol. 22.
Bauer, L. A. Work of the department of terrestrial magnetism of
the Carnegie Institution. Pp. 25-37.
Ward, R. DeC. Barometer and weather. [Note on article by
vaii B~bber.] P. 54.
Ward, R. DeC. Marine nieteorvlogical service of Chile. [Note.] P. 55.
Ward, R. DeC. Cliniate of *Jerusalem. [Note on work by G.
- Extended explorations of the atmosphere by the Rlue Hill Ob-
Arvanitakis.] P. 55.
servatory. Pp. 57- 58.
Science Abstrtccts. London. Vol. 8.
B[orns], H. Measuring the duration of rainfall. [Abstract of pa-
per by T. Ukacla.] p. 387.
B[orns], H. Metvorology of the eqnator. Observations at Par6 in the Museum Goelcli. P. 387.
Blorns], H. Annual variation of the height of sea-level and of the barometer in Japan. [Abstract of paper by F. Omori.] Pp. 387-388.
Butler, C. P. Solar origin of terrestrial magnetic disturbances. [Abstract of artirle by E. TV. Maunder.]
[Abstract o f article by J. Hann.J
P. 388.