G. M. T.
Longitude at
Run between (b) and (a) ...
Lonsritude at noon
Index error,. .,
g
Correction Table I
7
C% 2 or 1"'28 (54m. c
Table IV
23
10
6
278 (b)
29
R
B
18 59
12
23
23
3
51
588
34
71
oq 9
48
26 S.
47
35 W.
26 W.
Latitude at max. alt
47 58
22
5
N.
N.
Run in 5 min
1
48
1?,
1 W.

0''2 W.
Latitude at noon
or....
47 59
27
___
__ _____
UNEQUAL ALTITUDES.
Chron. times.
h. m. s.
7 33 11
9 17 8
THE IST AND 3RD OBSERVATIONS.
Altitudes.
, i it
a x 17 45 10
Green, date of 1st obs.
h. in. s.
7 33 11
Error 4 18 23 slow.
a a 17 58
7 1
g (in Ih. 44m.)
43 57
54 W.
23 51
34
12 50 = 770*
/  a
1 43 3
i(7 g) 51 315
19
2 C (7  g)
= 51m. 31'5s. +
SB 54m. 26 '7 sec.
770"
2 x 1" 28 x 103 min.
THE EXMERIDIAN ALTITUDE PROBLEM.
17
S. A. T. of mer. alt o' (a)
t (approachment) 4 57 '5
S. A. T. of max. alt 4 57'5
A! (always subtract! ve) 54 267
S. A. T. of 1st observation 23 10 30 8 (b)
Equation of time 6 29
S. M. T. of 1st observation 23 4 18
G.M.T.
Longitude at,, ,,
Run between (b) and (a) ...
23 51 34
47 32 W.
26 W.
Longitude at noon 47 58 W.
or..., ...11 59 5 W.
Observed altitude 17
Indexerror
Correction Table I
Ch* or 1"'28 (54m.
Table IV.
45
2
7
49
9
3
10
10
58
30
54
58
32
18 59 12
71 48
Declination 23 2 26 S.
Latitude at max. alt 47 58 22 N.
Run in 5 min 1 5
Latitudeat noon 47 59 27 N.
" Degree of Dependance." Assuming an error of 30" to have been
committed on the difference of the altitudes. The error in the longitude
will be
For the equal altitudes 15 x 30 "
2 x l"28 x 109m.
For the unequal altitudes 2
, Q3m>
i1
EXAMPLE III. July 12th, 1894, at dawn, in lat. D.R. 47 50' N., long.
D.R. 11 45' W., the following sights were taken near the meridian to
determine the longitude and latitude at time of transit :
Chron. times.
h. in. a.
6 57 7
Obs. alts. Markab.
O ' //
56 o
7 57 45
800
56 5
55 59 30
Indexerror + 1' 30", height of the eye 18 feet. The chronometer being
fast on G.M.T. 3 h  2 m  46 8 , and the ship steaming N. 45 E. (true) at 18
knots.
To FIND S.M.T. OF MERIDIAN TRANSIT.
Green, dateapprox.
d. h. m.
S. A. T. approx. July. ..11 15 31 (Table V.]
Equation of time ... 5
S. M. T. approx 11 15 36
Long. D.R ... 47 W.
G. M. T. approx 11 16 23
R.A. M.Sun.
h. m. s.
7 17 317
2 377
38
7 20 132
R.A. Markab... 22 59 30'9
R.A. M.Sun .. 7 20 13'2
S.M.T. of transit ... 15 39 177
R.A. and declin. Markab.
h. m. s.
22 59 309
N. 14 38' 113"
18
THE EXMERIDIAN ALTITUDE PROBLEM.
From Traverse Table.
N. 45 E. 18' d.lat. 12'83 dep. 12' 73 = d.long. 19' = 76 seconds.
t = K_ E=Ct*
2 (7 = 2 " 3 (2m. 46s. ) (Table IV. )
12 "73 = 19"
2 x 2"'3
= 2m. 4f)S.
The ship's run in longitude in 2m. 46s. is about 3'5s.
Therefore the corrected hour angle is 2m. 42  5s. (See Rule A. under Max. and Mm. Alts.)
EQUAL ALTITUDES. IST AND 2ND OBSERVATIONS.
Chron. times. Green, date.
h. in. s.
6 57 7
7 57 45 Eiror
h. in. s.
6 57 7
3 2 46 fast.
/ 1
38
1 17 E.
g in lh. Om. 38s.
/ + g 1 1 55
AI = i (/ + ff) 30 575
15 54 21
S. M. T of transit
h. m. s.
15 39 177 (a)
Observed altitude
56 5
t (separation)
2 425
Indexerror
1 30
S. M. T. of max. alt. .
h l
15 36 352
30 57*5
Dip and refr
56 6 30
4 50
S. M. T. of 1st obsn. .
15 5 377 (b)
56 1 40
G. M. T ...
. ... 15 54 21
Chi or 2 "'3 x (30m. 57s ) 2
Tablp IV
f 32 2
( 4 48
Longitude at ,
48 43 '3 W
R
19
Run between (b) and (a) .
Longitude at transit ....
or
427 E.
48 06 W.
12 O'l W.
Declination
56 38 49
33 21 11
14 38 11 N.
Latitude at max. alt
Run in 2m. 42s
Latitude at transit ...
47 59 22 N.
35 N.
47 59 57 N.
UNEQUAL ALTITUDES.
Chron. times.
h. m.
6 57
8
/in
Acceleration tor lh. 3m
9
2 53
10
1 20 E.
1 4 23
32 1T5
THE IST AND 3RD OBSERVATIONS.
A Ititudes.
56 5 6
55 59 30
_ 5 30 =  330"
330'
32m. 11 5s.  1m. 7s.
31m. 4 '5s. in sidereal time.
30m. 59 '5s. in mean time.
THE EXMERIDIAN ALTITUDE PROBLEM.
19
S. M. T. of transit
h. in.
15 39
s.
17 "7 (a)
Observed altitude . ..
56
1
5
n
o
t (separation)
2
42'5
Index error
1
30
S. M. T. of max. alt
A,
15 36
30
352
59 '5
Refraction and dip
56
6
4
30
50
S. M. T. of Istobs
GMT
15 5
15 54
357 (b)
9]
C h* or f? "*3 x (31m) 2 (
56
1
JO
40
1 Tahlp TV
4R
Longitude at
48
45'3 W.
R
19
liun between (6) and (a)
4 9 '7 E
Cf*
OQ
Longitude at transit
48
26 W.
00
oo
4tf
or
iog
O'fi W
Decimation
33
. 14
21
38
11
11 N.
Latitude at max. alt
47
59
22 N
Run in 2m. 42s
35 N
TiQtitnrlo of. tronoif
A1
X.Q
K7 XT
" Degree of Dependance." Assuming an error of 30" to have been
committed on the difference of the altitudes. The error in the longitude
will be
15 x 30"
For the equal altitudes
For the unequal altitudes
2 x 2"3 x 62m.
15 x 30"
= r58.
2 x 2"'3 x 65m.
The following observations, selected from several procured for the
authors by W. H. Bolt, Esq., were taken on the S.S. " Ophir," Captain
Ruthven, when in the Mediterranean :
Date. G. M. T. Chron. Obs. alls. Sun's LI. True Course.
28 Sept., 1894.
h. m. h. m. s. * , n ,
11 30 A.M. 22 50 1 48 35 55 N 74 E. 13'5
11 41
23 P.M.
38 ,
23 38
23 45 3
23 58 59
48 35 55
48 57 30
48 45 15
48 8 40
Height of the eye 36 feet.
The results obtained are :
1st and 4th obs.
1st 3rd
2nd 4th
2nd 3rd
Hear*
Longitude 7 30'2 E.
7 309
7 315
7 322
Position at Noon.
Lat. 38 34' N.
Long. 7 31 E.
7 312
Latitude 38 32'2 N.
38 324
38 325
38 320
38 323
20
THE SUMNER PEOBLEM.
GENERAL PRINCIPLES.
Let P, P l be the poles of the
terrestrial sphere, C its cen
tre, QQ 1 the equator ; and
let the straight line joining
the centres of the heavenly
body and the earth (con
sidered spherical) cut the
surface of the latter at the
point S.
This point, termed the
geographical position of the
heavenly body, is readily
determined in position by
taking the heavenly body's
declination as its latitude,
and the Greenwich hour
angle (expressed in arc) as its longitude.
Let S C T represent the observed zenith distance of the heavenly body,
T being the point where C T meets the surface of the earth.
Now, if with S as pole, and with spherical radius arc S T, which
measures the observed zenith distance, a small circle T t be described,
at all places situate on this circle the altitude of the heavenly body must
be the same.
Such a circle is called a "Parallel of Equal Altitude," and it is clear
that the ship must be somewhere on its circumference.
If two parallels of equal altitude be obtained from observations of
the same, or of two different heavenly bodies, since the ship must lie on
both of them, she must be on one of the two points of their intersection,
the latitude by account limiting the position of the ship to one only of
these points.
For a successful application of the above principle, certain limitations
of bearing are necessary, as is explained in the usual text books on
Navigation.
Parallels of equal altitude on the sphere, when projected on a Mer
cator's chart, become what are termed " Curves of equal altitude," being
closed or open curves, according to the position on the surface of the
sphere of the pole relatively to the circumference of the parallel of
equal altitude.
THE SUMNER PROBLEM. 21
A tangent to a curve of equal altitude at the point representing the
true position of the ship, or at a point on the curve very near to this,
may be considered for a short distance to coalesce with the portion of
the curve in the neighbourhood of that point, and is called a " Line
of Position."
When the heavenly body is on or near the prime vertical, a greater
length of the curve of equal altitude in the neighbourhood of the point
representing the ship's position, may, without any great error, be replaced
by a straight line on the Mercator's chart than is the case when the body
is near the meridian.
It is evident, therefore, that the point on the chart through which a
line of position is drawn should be as near as possible to the point which
would represent the true place of the ship.
Since the bearing of a heavenly body is always at right angles to the
tangent to the parallel of equal altitude, at any point, and from the
nature of the construction of Mercator's chart all angles upon the sphere
remain the same in magnitude when projected on the chart, it is easy to
draw a line of position on a Mercator's chart.
The use of what are commonly known as " Sumner Lines," or " Lines
of Position," has of late years attracted the attention of navigators to a
much greater degree than formerly, and a growing interest seems to be
awakened as to the advantages to be derived from the practice of draw
ing on the chart a single " line of position," or from the determination of
the ship's position by the intersection of two such lines.
For example Simultaneous altitudes of two stars (differing sufficiently
in bearing) taken during twilight or moonlight, will give a good position
of the ship without the element of error arising from " run," and in the
case of a line of position obtained from a single observation, it should be
borne in mind that such a line can frequently be made use of for making
the land (should it run towards it), or as a clearing mark for it should it
run parallel to the trend of the coast. It is also occasionally useful, in
conjunction with a sounding, in giving an approximate position, if at the
time of observation a cast of the lead be taken.
LINE OF POSITION FROM ONE OBSERVATION ONLY.
In endeavouring to arrive at some conclusion as to the best method
of obtaining a reliable line of position from one observation only, it must
be fully recognized that the unwitting possession of a good latitude or
longitude, drifting due to current, error of course, &c., &c., may combine
in favour of one method more than another.
With an estimated latitude, a longitude may be computed from an
observation of a heavenly body (not near the meridian). The position of
22 THE SUMNER PROBLEM.
the ship thus determined being marked on the chart, and a straight line
drawn through it at right angles to the true bearing of the heavenly
body at the time of observation, a line of position is obtained. This
mode of proceeding is the one most generally practised, and is known as
the " Chronometer method."
It is evident that the point on the chart thus determined is the pro
jection of the point on the surface of the terrestrial sphere, in which the
parallel of equal altitude corresponding to the observation is cut by a
parallel of estimated latitude.
To use the chronometer method with advantage necessitates a favour
able condition, viz., that the heavenly body observed should be on or near
the prime vertical, the method practically failing to give a good line of
position when the body is near the meridian, particularly so should the
estimated latitude be very much in error.
It may here be remembered that a heavenly body can be on the
prime vertical only when the latitude and declination are of the same
name, and the latitude is greater than the declination.
Thus the most favourable opportunity for calculation by the " Chrono
meter method " does not, for the sun at any rate, present itself with such
frequency as might at first sight appear.
The " New Navigation," first introduced by Commandant Marcq
SaintHilaire, affords a means of obtaining a line of position by a method
which is not more difficult either in computation or in execution on the
chart than is the " Chronometer," and is of special value when, from
circumstances of weather, &c., sights have not been obtained for some
days, and consequently the ship's position cannot be estimated with any
degree of certainty.
In the accompanying figure let S represent the geographical position
of a heavenly body, the altitude of which has been observed, y T t the
corresponding parallel of equal altitude, E the estimated position of the
ship, and P the pole of the terrestrial sphere.
Now the distance between E and the true place of the ship being
quite unknown, assume this distance to be at most E T. With E as pole,
and arc E T &s spherical radius, describe a small circle T t x.
The ship's position must be somewhere on the earth's surface com
prised within this small circle, and as she must be on the parallel of
equal altitude T t y, her true place must be at some point on the arc
T E t between T and t.
Now, as it is impossible to say at what point of the arc T R t the ship
is situate, the point jR, the middle point of the arc T R t, is the best,
which can be selected as representing her position, since it is the mean of
all the positions which she may occupy on that arc.
The point R will always be nearer to the true position of the ship
THE SUMNER PROBLEM.
23
than will the estimated position E, for E R being at right angles to the
arc T R t, it follows that R T is always less than E T or E t.
R is the point in the " New Navigation " through which a line of
position is drawn.
The data used in the calculation are the estimated latitude and longi
tude of the ship.
Now E and 8 being the poles of T t x, and T t y respectively, E R
and S R cut the latter circle at right angles, and therefore form part of
the same great circle, since S R is an arc of a vertical circle.
In the spherical triangle P S E, with polar distance P S, colatitude
P E, and hour angle 8 P E (two sides and the included angle), compute
the zenith distance E S for the estimated position of the ship. This
compared with R S, the true zenith distance (obtained by observation),
gives E R, the difference between the computed and true altitudes.
This difference, though really an arc of a great circle, is so small that
it may, without any great error, be considered as an arc of a loxodrome
or rhumb line, and will therefore, in accordance with the principles upon
which Mercator's chart is constructed, be projected on the chart as a
straight line.
21 THE SUMNER PROBLEM.
The difference E R, projected on the chart, measured (from the gradu
ated meridian at the side of the chart) from E in the direction of the
azimuth E S, if the observed altitude (corrected) be greater than that
calculated, and in an opposite direction if less, will determine the position
of R, which, as has been shewn, is the most probable position of the ship.
A straight line drawn through R, at right angles to E R, will be the
" line of position " as determined by the " New Navigation Method."
Properties of the point R. The following important advantages
possessed by the point R are worthy of notice.
Should the true place of the ship be at t, and an error be committed
in the estimated latitude, which causes the Chronometer method to place
the ship on the parallel of latitude passing" through T, the distance of the
ship's position, thus determined, from her true place t, would be equal to
the whole distance T i.
Again, the true place of the ship being at t, the Chronometer method,
owing to error in the estimated latitude, may place the ship at G beyond
T and t, in which case the distance between the point given by the
" Chronometer " and the true place of the ship at t, would exceed T t.
Whereas it is clear that the " New Navigation " gives a point R,
through which a line of position may be drawn, which can never be at a
greater distance from the true place of the ship than one half of the dist
ance T t.
Again, it must be evident that an error in altitude observed produces
at all times its whole effect, and no more, in displacing a point as R, and
therefore the line of position drawn through that point.
The point of intersection of two lines of position, obtained from
observations of the same or of two different heavenly bodies, will deter
mine the position of the ship with great accuracy.
THE DOUBLE ALTITUDE PROBLEM.
The Double Altitude Problem is one of the most important in
Navigation, and it is hoped that the solution given in these pages may
be of use to those engaged in navigation.
The result may be worked out either with or without having recourse
to the chart.
Should there be any change in the ship's position between the observa
tions, the 1st observation must be brought up to the place of the 2nd by
the " run " in the interval in the usual way.
For the sake of simplicity, " run " has been left out of consideration
in the explanation.
THE SUMNER PROBLEM.
25
Suppose the usual observations for the Sumner or Double Altitude
problem to have been taken :
Let A and B be the geographical positions of the same, or of different
heavenly bodies, A having the greater bearing ; C T and H T the parallels
of equal altitude corresponding to the observations, the intersecting point
T giving the true position of the ship.
Now, suppose the altitude of A to be worked with an assumed latitude,
the parallel of which is represented by H D, and the resulting longitude
to give a point such as G. The azimuth S C A, or its equal T C D which
call a must be either calculated, taken from tables, or from Weir's
Azimuth Diagram.
O
With latitude and longitude of C, calculate the altitude of B, and take
the difference (p) between this and the corrected observed altitude. Com
pute, or obtain as before the azimuth S C B, or its equal NCR, which
call j3.
R C, though really an arc of a great circle, is so small that it may,
without any great error, be considered as an arc of a loxodrome, or
rhumb line, and may therefore be projected on the chart as a straight
line.
Since B has been observed nearer the meridian than A, the triangle
R C T may be represented on the chart as a plane rightangled triangle
with Less error than would be the case with triangle H C T.
26 THE SUMNEB PROBLEM.
Now, TD = TC. Sin. T C D.
TD = C R. Sec. R C T. Sin. TOD.
o a T j P Sine a
or Cor n> for Lat. =   
Sin. (a/3.)
Again, C D = C T. Cos. T C D.
CD = G'.R. SQC.RCT. Cos.
f or Long. Co, Lat. =
^ . ,. T P' Cos. a
Cor n  for Long. = ^
^
Cos. Lat. sin. (a/3)
(a /3) being the angle between the greater and lesser bearings.
The following might be considered more simple to some navigators :
Mark on the chart, or on tracing paper pinned down to it, the point C t
determined from the assumed latitude, and the longitude deduced fi oin
the altitude of A, and the line of position C T at right angles to true
bearing of A. Through C draw the line of bearing of B, and from C
measure off the difference (p) between the calculated and corrected
observed altitudes, either towards B, or in the opposite direction, accord
ing as the computed altitude is less or greater than the corrected
observed altitude. Then R T drawn at right angles to R B will give the
second line of position.
Now, it is evident from the figure, and from all similar figures that
can be drawn, that the length of the arm R T must always be shorter than
H T the one obtained from the observed altitude of B by the " Chro
nometer method/'
Also, since TCD is greater than THD, C must be nearer to T than
isH.
Therefore, the line of position derived from the azimuth of B at C is
less in error than one obtained from the azimuth at H.
From the remarks, page 21, it follows that the tangent to the curve of
Equal Altitude representing the parallel HRT, at the point of projection
of R on the chart, will give a more reliable line of position, and a more
accurate intersection, than will a tangent to the curve at the point
representing H.
It is to be remembered that the observation corresponding to the
greater bearing should be worked by the " Chronometer" method, and the
one corresponding to the lesser bearing by the " New Navigation "
method.
The ship apparent time will naturally be obtained from the observa
tion corresponding to the greater bearing, and will, therefore, be less
liable to error than if obtained from the other.
The following examples taken on board H.M.S. " Orlando " illus
trate the solution of the problem :
o
c i
~ H
o> S
1 I
O

" 2
02 ^ o
J=s J
sStf S
1 1 2
S ^ o
SS 8
cnfe'
as <n
1 t
<j <;
S S^J W
n s.2^
i ;
i S
S a
111
ill sSli
 C >
g 1 P  3
5 1' 2
+ 5
*s '3
8 ri
la
 ^ J s
a 95
S 2
8?
II 22
2^
^S
2S
s as
,ln
i!
^^ tsj
bcS bi'i
^ 73 .
f^ < . CQCC CO
t"S Oco ooo
1 ftf id 1 J ifiil
; Issa i
S 5
Q HM
^551 1
29
EXAMPLE L PROJECTED OK A CHART.
A Position from 1st observation with D.R. Lat. Chronometer Methoi
A B Run between observations.
B R Difference between true and computed altitudes,
@ Position of ship at, 2nd observation
30
o ^
be a
II
00 OS 01
s l iij
*odfc
S2
SU 3"
CSH OSM
III 3"
?! rf
+ .2
*s Q
3 N
J
^ 05
31
d.g
I
OH
i *
.u
S g> .ix'ji
ja a .sajss
5 g> "8*8
^ o 5 'fcc'tr* ^ ^i w CN i
ggh^ cdC^ co^vc^ SJt
i
1
111 a
*? *'i'i* sTa 9?
o i s^j^ gfeg
:!
*
^
^
III 1
^ 5 fc* JS w .
** ^11 si
1
X H S
1C Iftl if
43 2 S a : s
a
H 3
SS S^ ^^ SI] VI
grf l :S
1 " ^ 1 1 5
?n la*
J 5 a 5
(to
Ml "*
fl^KM Wrl vflO < *'*I
a^ N c,^ nU. 1 g.
M ^SS S S SBS1 
: :: : : 1 e:
00 Cf) TO
;s?sh'^^l^
!2S 5 V V
^a ^ ^ ^&
j
t?
: i i i J i ?
 1 '
~ s , s ^.g l^ 1
H s  6
D
4
p
It *
1^4
" *
S * s
oJ w
I *
Sg S
IB "
!* 4. *.
fc fe^
<j o  j=g  os,
W  ^S 8 '
S e = al^. l as
glsl
5 S
"all
J
^
1 1 a 1^ 
^ If 1 **' 1
2 g H^ H 
ao o
?.sl # 1
mo M j ia
S ^
31
S *as
< &s ** ^ c ii
CO SS 
1 = 1 Id J J j's
1 I
l
1* Hi
^ :
<'*
S
<! S fe ^
M 22 S ^ e
pH
as
w i
g 6 6 3c
N3 K
s
QQ
TABLE I
HEIGHT
OF THE EYE.
Observed
8 FEET.
9 FEET
10 FEET.
11 FEET.
12 FEET.
13 FEET.
Observed
Altitude.
Sun's
Correc
tion.
Star's
Correc
tion.
Sun's
Correc
tion.
Star's
Correc
tion.
Sun's
Correc
tion.
Star's
Correc
tion.
Sun'*,
Correc
tion.
Star's
Correc
tion.
Sun's
Correc
tion.
Star's
Correc
tion.
Sun's
Correc
tion.
Star's
Correc
tion.
Altitude.
*
*
*
*
*
*
add.
subtract.
add.
subtract.
add.
subtract.
add.
subtract.
add.
subtract.
add.
ubtrnct
6 30
5 21
1048
5 "
10 58
c
I
11
8
452
II
17
443
II
26
435
II
35
630
40
5 3 2
1037
5 22
1047
5 12
1057
5
3
11 6
454
II 15
446
II
24
40
5
542
1027
532
1037
522
1047
5 13
1056
5
4
I 1
5
456
II
14
50
7 o
t
! 5 2
10 17
542
10 27
532
1037
5
23
10 46
5 14
1055
5 5
II
4
7 o
IO
(
) 2
10 7
10 17
542
10 27
5
33
10 36
524
10 45
5 J 5
1054
IO
20
6 ii
958
6 5 i
10 8
551
10 18
5
42
10 27
533
1036
524
1045
20
7 3
6 20
949
6 10
959
6 o
zc
> 9
c
y
10 18
5
42
10 27
5
33
1036
730
40
6 28
940
6 18
6 8
IO O
i
59
10 9
550
10 18
5
42
10 27
40
5
636
932
626
942
6 16
952
I
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10 I
558
10 10
5 5
10 19
5
8 o
6 44
924
634
934
6 24
944
615
953
6 6 10 2
5
58
10 II
8 o
IO
652
9 17
6 42
927
632