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1
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S
6
PREFACE,
This manual has been drawn up for the use of the
Cadets of the Royal Military College of Canada. The
first five chapters on Practical Astronomy embrace that
portion of the subject with which all Land Surveyors in
this country ought to be familiar. The remaining chap-
ters, together with the part of the work which treats of
Geodesy, touch on the more important parts of the ad-
ditional course, as regards those subjects, laid down by
Government for candidates for the degree of Dominion
Topographical Surveyor. It has become absolutely
necessary to draw up some compilation of this kir' be-
cause, while many of the Cadets are anxious to v. ify
themselves as far as possible in the above-mentio,.od
course, the number of different books they would have
had to refer to in order to obtain the requisite knowledge
would have entailed on them a heavy expense. In order
to make the work as cheap as possible the number of
diagrams has been cut down to a minimum, it being in-
tended to supply the place of expensive plates of instru-
ments ct cetera by lecture illustrations. The author has
also made the higher portion of the Astronomical course
03 lU'f
IV
Preface.
as brief as possible. It will be found treated in the
fullest manner in Chauvenet's Astronomy.
Geodesy being both a difficult and a very extensive sub-
ject no attempt has been made to write anythinj,' like a
treatise on it. All that has been aimed at has been to
give a sketchy account of its most salient points, adding
a few details here and there. The student who wishes to
pursue the subject further is referred to standard works,
such as Clarke's Geodesy.
The author has to acknowledge having made more or
less use of the following:
Chauvenet's Astronomy, Puissant's Gdodt<sie, Clarke's
Geodesy, Frome's Trigonometrical Surveying, Loomis'
Practical Astronomy, Gillespie's Higher Surveying,
Deville's Examples of Astronomic and Geodetic Calcula-
tions, the U. S. Naval Text P>ook on Surveying, and
Jeffers' Nautical Surveying. He has also to thank Lieut.-
Colonel Kensington, R.A., for valuable assistance in in-
vestigating some doubtful formulas.
I
Kingston, Cadada,)
January, 1883. [
I
I
CONTENTS.
PART I.
PRACTICAL ASTRONOMY.
CHAPTER I.
Idea of the great sphere
"declination circle,'
tude." "declination
"sensible horizon
General view of the universe. The fixed stars. Their classification
magnitudes, and distances, The sun. The planets. Their rela-
tive sizes and distances from the sun. Apparent motions of the
heavenly bodies. Their real motions. Motion of the earth with
reference to the sun. The solar and sidereal dav. Mean and
apparent solar time. The equation of time. Sidereal time.
The sidereal clock
CHAPTER n.
Meaning of the terms "pole," "meridian "
"hour circle," "zenith," "latitude," "longi-
' "right ascension," "altitude," "azimuth "
, , ,. . •, "rational horizon," "parallels of latitude'"
declination parallels," "circumpolar star," "transit," "paral-
lax. Refraction The Nautical Almanac. Sidereal time. The
celestial globe. Illustration of the different co-ordinates on the
great sphere
CHAPTER III.
Uses of practical astronomy to the surveyor. Instruments emploved
in the field^ Their particular uses. Corrections to be applied to
an observed altitude. Cause of the equation of time Given the
sidereal time at a certain instant to find the mean time To find
the mean time at which a given star will he on the meridian
Given the local mean time at any instant to find the sidere.il time
Illustrations of sidereal time. To find the li<,iir angle of a "iven'
star at a given meridian. To find the ine,-.n time by .•-niaralti-
tudes of a fixed star, To find the local m..an tinu. by an !.bserved
altitude of a heavenly body. To find the time by a meridian
transit of a heavenly body _ ,
PAGE.
II
22
VI
Contents.
CHAPTER IV.
CHAPTER V
Sun dials. Horizontal dials. Vertical dials
CHAPTER VI. ''^
iS''TL'^lniH?P?' ?^ M'"onieter. The Reading Micro-
scope. The Spirit Level. The Chronometer. The Electro
Chronograph. The Sextant. The Simple Reflecting cTrcie"
Jircle !r""^ '"^'^'"""S ^''■'='«- The^PrismaS R^flectilTg
36
66
77
Tt, . u, ^ CHAPTER VII.
anerto,„fde,m,onon ,h= ialilude, V p=r™al e,;S.
CHAPTEH VIII,
f SI/ £te,. ^^.r o","," as'of °,L :sii;
=;. '^is'.rc'o^r:'.'!'.''.";. . '!!■'. r *.'!r .'! '"•™;
CHAPTER IX.
'^'"''Jlrn' .•"^'^°'^^ °f finding the latitude.-By a single altitude
h^ ^ "^ -H- """"''S "'"^' ^y observations of the pole star ou"of
the meridian. By circum-meridian altitudes. °. g^
CHAPTER X.
Interpolation by .second difterences. E.xamples. To find the Green-
moon'!!n': *■'"" ^corresponding to a givL right ascension of the
To finH .h^7'" ^,^y,- ^"t^rpolation by diflerences of any order
IoLS. H /°"g""^« by moon-culminating stars. To find the
longitude by lunar distances
T fi j.u ,. , CHAPTER XI.
on /h/if^P '"'^V^"^ ''?"'■ ^"^'^^ Of a given heavenly body when
heav/nlv°hT- T° '?"d the equatorial horizontal parallax of a
Tn finJ^.h ^^'i;" g'.^«" distance from the centre of the earth.
10 find the parallax in altitude, the earth being regarded as a
(^nZ\ ^'f 'c^'^'op!'--^- Differential variationsof co-ordinates
thP H^ K '"]^'\ '."«'j"^l't''^s in the altitudes when finding
hP tT ^ ^'^."''J altitudes. Effect of errors in the data upon
lutl computed from an altitude. Effect of errors of zenith
r,i,l '•^^''''"f'°"i ^"d time upon the latitude found by cir-
cum-mendian altitudes. The probable error joj
33
89
Contents.
Vil
PART II.
GEODESY.
_ »
III
_,. . ^ CHAPTER I.
cfr^'rcs ,!? V""," °Wate spheroid-proved by measurements
,, , . CHAPTER H.
aSiZ'''}-'- ^^'^'"J^'^dop.ed for mapping country Tri-
„ . , CHAPTER HI.
cal excess. Correcting the a ■• , ","' . V"' P'^ne. The spheri-
sides of the triaSs The n- '; , .A !^"^i^- Calculating the
CHAPTER IV.
the earth. F^/rueH^seS «K'd"°ul oHhti^'"'^ °'
sphere described with radiusequaltoX normal of *hV?rf»,^'"''-7
Reduction of a difference of lafit ,ri» ^^ .u ^ ■, spheroid.
responding difference of Lf.H the spheroid to the cor-
ili^on Tj^iS^i'^^^A ins
.he la,ii„d= o onfj^'i'n,, S aziZh of'?;/,'"""' '"«'"'''«
133
-■l*fete«_„
vrii
^'ontents.
'n fho North A
'"" "10 Uffariia .,.
I , ['"'J Hie area
""sets to a parallel
''hodsofclelineatin ^"AI'TEK V N5
J'>nom,.(r,v,i . ... <^'"APTl<:i< VI »C0
''>iK"nomeincal lev..ir *^'"'^'"I'KK VI
'■■I"' «M of ,1,5 . CHAM-g,, v,V
'.'"-• Pen,i„l,„„ ninsri ■■"'■ ''" '""y ascemin ^ ""'' "''-' «'me of
182
'75
Devilles
"if area
parallels
parallel
'45
actions,
cator's,
i66
above
Ke-
; used
di/ler-
i. and
d tile
To
erva-
»7J
the
»rce
idu-
e of
1 of
ven
!0f
ir's
er-
.. 182
NOTE TO PAGE 52.
By drawing a figure it can be easily shown that, in the
case of a horizontal dial, ii <p is the latitude, P the hour
angle, and a the angle the corresponding hour line makes
with the meridian line, then :
sin y)=cot P tan a
or tan a -siq f tan P.
Similarly, in the case of a dial on a vertical wall facing
south, ^
tan a=cos <p tan P.
In the latter case the angle a is measured from a ver-
tical line on the wall. The stile is, of course, set parallel
to the polar axis.
We can thus find the hour lines for each hour, for any
given latitude, by solving these equations.
L
,4r
fA-t/-<^-<C ' \r c
V
P a A
lie
<iL.r^
^<^ yv.
^^:
Part I.
PRACTICAL ASTRONOMY.
CHAPTER I.
GENERAL VIEW OF THE VISIBLE UNIVERSE. THE FIXED
STARS. THE SOLAR SYSTEM. APPARENT AND REAL
MOTIONS OF THE HEAVENLY BODIES. DIFFERENT
METHODS OF RECKONING TIME.
The visible universe, outside our earth, comprises the
sun, rnoon. planets, fixed stars, milky way, nebute. shoot-
mg^stars, and the zodiacal light, besides an occasional
The comets shooting stars, and zodiacal light will not
tlT 7 1"'^' 'u '"" ^'^ '""^y ^-y - 'the nebut
(white cloudy patches) when examined with powerful tele-
scopes generally resolve themselves into clusters of
separate stars; a few nebulae, however, still retaining their
cloud-hke appearance. ^
The fixed stars, as they are called, are doubtless suns,
scattered irregularly (or more properly in clusters) through
The Fixed Stars.
space. They are classified by astronomers into magni-
tudes, the bnghtest being those of the ist magnitude.
Those of the 6th magnitude are about the smallest visible
to the naked eye, those below that si^e being only visible
through telescopes. Although differing so much in bright'
ness, the most powerful telescope fails to show theni of
any measurable size, and they all appear mere points of
lifi:h . Their brightness, as seen by us, depends, probabh-
partly on their distances, partly on their si.e, and partiv'
on their natural brilliancy, while that of a few
of them vanes at regular intervals. The colour of the
stars also varies, inclining in some to white, in others
to red, blue, or green. Some stars are connected in pairs
tamed by the spectroscope that the elements present in
the sun and stars are identical with those composing
our earth ; at least no new ones have yet been discovered
tJJrT"^ ^Tir^^'^ ^y '^'' '"^'^"^^ '"^^ •^^"stella-
tions, of which the Great Bear and Orion are instances;
and a number of the most remarkable stars received
special Arabic names, e.g., Arcturus and Aldebaran. The
stars composing a constellation are catalogued according
to their brightness, the Greek letters being used to dis
tinguish them. Thus Aldebaran is a Tauri, and the two
stars of that constellation next in brightness are /? and r
Taun. When the Greek alphabet is exhausted English
etters are used, and finally numbers. Thus we have
h Virginis and 51 Cephei. The stars are numbered
not according to their brightness, but in the order of thei^
right ascension.
The distances of the fixed stars from the earth and from
each other are so great as to be almost beyond human
conception. It was for long believed that they could not
be measured. It was, however, eventually found that in
the case of some of them, by taking a line through space
Their distance.
joining opposite points of the earth's orbit as a base, and
the star as the apex of a very acute-angled triangle, the angles
adjacent to the base could be measured and the acute angle
thus determined. The length of the base being known
gives the star's distance. To give an idea how far off the
nearest star is it may be mentioned that a ray of light
would pass round the earth (about 24,900 miles) in a
quarter of a second ; it takes ^ minutes to traverse the
93 millions of miles from the sun to the earth aid \i
years to reach us from the star. And yet, could we be
transported to that star, we should still see all the other
familiar constellations and stars apparently in exactly the
same positions as we see them here. So vast are the dis-
tances that the change of position of the observer would
have about as much effect on that of the stars as would
an interchange of two adjoining grains of sand on a laree
table covered with them.
The nearest star, as at present known, is a Centauri
which IS 200,000 times farther off than the sun. The ap-
proximate distances of a few others, in terms of the num-
ber of years it takes their light to reach us, are as follows :
i? Centauri ' 6+ v^arc
61 cygni ::::;: fy^^^^-
Sirius 'jg
^^^^yo'^ '•■i-;:;;;:::::::::::::.i6
Arcturus j^
Vega ..!.!!!!!".'.!!;!; 16
PoleStar ZZ^:^
About 100.000 stars have been catalogued altogether.
The number visible with the naked eye is about 15,000
In latitude 50^^ north only about 2,000 can be thus seen
at any one time.
Our sun is only one of the stars, and the latter, though
called fixed," are in reality all moving according to the
laws of dynamics. What these motions are we cannot
tell, as we do not yet know ihe manner in which the
The Planets.
masses are distributed through space If h..\
been ascertained, not only .ha^ Z^lr. 1^^ ITZZ
'„!: *"= ^""^s are, as a rule, the centres of planetarv
o tshaS^h r ' "'" °f d-velopmen, permitting
oi Its habitation by hvmg creatures. Our own solar «..
lupUer P!i"f .'■'™'""« between the orbits of Mars and
is a ocus c P'^7'= ""^'^ '■■ ="'P^-^. of which the sun
■s a focus Several of them have moons or satellites and
all, .nclud.ng the sun, revolve on their own axes
an airof 7- TZ' ^^ °'^'' '' '"=""^<' '» o-"^ af
angle ot 7 . Looking down on the plane of the «=
em from its northern side, the direction^f .he Lotfolf
Xetrdran'it"' '7 V' "'^""- -»""=
pmnets, and of all its members including the sun) rn.,nH
from it about 60 .tdii of te earth "MarT" *f' '"'
four, Saturn eight. Uranus fouV:"' 'iept^ rne^"-^;:
dtances of the planets from the sun are near^ i„ the
foUowins proportion: Mercury i, Venus ., the Ear^
a 6 Mars4, Jup.ter 13, Saturn .5, Uranus 50, Neptune 80
Tnd thaH V ^•''^"'^^-"•y of Mercury! a littLmore'
and that of Venus a little less, than that of the earth
Mars Ath. Saturn ith, Uranus ith, and Neptune 1th.
To compare their relative sizes ; if we took a globe four
fee, in diameter ,0 represent the sun, the moo„lo„u ^'
about the size of a grain of shot, Mercury of a bTcksho^
Their relative size. -
Venus and theEartha small spherical riflebuHet, Mars^ s^l
revolver bullet Jupiter an x8.pounder round iTsatrn
a 9-pounder round shot, and Uranus and Neptuie lar^e
grape shot; th^ latter the largest. The mass'of Jup'^r
.s 300 times that of the Earth, while that of Mercury is
only about ,Vth, Mars ^Vth. and the Moon ^th. ^
The planets are easily recognized by their changing
he.r places m the sky relatively to the fixed stars-henc!
their name, which means "wanderer." Thev mav also Hp
loTt ^' T ^y ^h'"^ °"Jy>y the sunlight reflected
fiom then- surfaces, and when viewed through a good
telescope, look like small moons, instead of m'ere points
of hght, as m the case of the fixed stars, and may be
noticed also to pass through phases like the moon
especially in the case of the two that are inside the eTrth^
orb. The variability of their brightness is caused pr^y'
by this, partly by change in their distances from the earth
The well-known rings of Saturn are now supposed to con
sistofa shower ofmeteorites revolving round him
Supposing us to be situated in the northern hemisphere
and not too far north, if we watch the apparent motions of
he heave^y bodies in the sky we shall Zee tCfToX
list ^f n u "''f '"*''' ""^ ^^^^ ^^^J'^^t about th?
the ax t oTr ' n'"' *'^ ^^^P^^'*^ '^ *^^ -- about
the 2ist of June. During the winter half of the year his
nsing and setting is south of the east and west pdn s of
the nonzon, and during the summer half they are north
of It; while at two intermediate periods, known as the
equinoxes he rises due east, remains in sight for la hours
and se 3 due west. At midwinter the afc he descXe^
ZZt ''' '' ''' '°"^^^' ^"^ ^* -^^— er tt
When the moon is first seen as a young moon she is a
I'ttle to the east of the sun. She rapidly'move" th^.h
i
The Moon.
the sky towards the east, so that about full moon she
rises as the sun sets, and later on is seen as a crescent
rising before the sun in the early morning. T4ie^ight ) L.U^
to^whichsh€nses^in4h<iskywillbeobservedtobe (unlilca{'^' -^
the-^ase^f the sun) quite independeixtu^._tlie_tinie oF
y«ar. The interval between two new moons— that is the
time she takes to make an apparent circuit of the sky-is
about 2$ days ; and she rises each day alA '1)1^00 mmyl '"
terc of an bow later than the day before.
The stars, if carefully observed, will be noticed to rise
each night a little less than four minutes earlierthan they
did the night before, so that at any given hour a certain
portion of the sky which was visible at the same hour the
night before will have disappeared in the west, and a
similar portion will have come into view in the east In
fact the whole mass of the stars appears to be slowly over-
taking the sun (or rather the sun to be moving through
the stars); and, as a consequence, if the stars
were visible in the day time this motion could be
plain y seen. The points of rising and setting of the stars
are always the same. The sun and all the stars reach
their greatest height in the sky-or culminate, as it is
termed-at a point where they are due north or south of
the spectator.
The stars in the northern portion of the sky. from the
horizon up to a certain point depending on the position
ot the observer, never rise or set, but describe in the
twenty-four hours concentric circles round an imaginary
point called the pole, and in a direction contrary to that
of the hands of a watch.
The different planets, {{ carefully observed will be
noticed, not only to chan;,^e their positions among the
hxed stars, but to vary in brightness from time to time.
So much for the apparent motions of the heavenly
bodies. We have now to consider their real ones.
The Earth'' & Motion.
she
;ent
ight ) iLcU^
The earth describes an elliptic orbit round the sun in
about 365 j days. It also revolves on its own axis in about
a day. This axis remains parallel to itself and is in-
clined to the plane of the orbit at an anj^le of about 23"
27'. Hence the phenomena of the seasons, and of the
varying positions of the sun from day to day.
Fifr. I.
Figure i shows the position of the earth with reference
to the sun at the different seasons. N is the north pole,
S the south pole, and A a point in the northern hemis-
phere. The left hand sphere shows the earth's position
when it is midwinter at A, and the right hand sphere
when it is midsummer.
The motion of the earth round the sun causes the latter
to continually change its apparent position amongst the
stars. Its path through them is called the ecliptic, and
lies, of course, in the plane of the earth's orbit. The
earth's revolution round its own axis, although on an
average 24 hours if taken with reference to the sun, really
takes place in space in about 3 minutes and 56 seconds
less than 24 hours, the difference being, in fact, the same
as that between two successive risings or settings of the
same star. It should also be noted that, owing to the
enormous distances of the fixed stars from us, all lines
drawn from the earth, no matter what *ts position, to
any star, are sensibly parallel.
Mean time.
th po", A ="^"=7!,<'''^^ "'- i' - apparent „oo„ a
ine pomt A. O, a .d O, are the earth's centre, and S
a'o S 'C"' '°l!'r^ "' * ""=■■" =•- - --"fronl
th'e°arfl^.H^5''T '''°°^^'- ^<'' ^^ 0» '"'-=<^ct
me eartJi s circumference at B. It is PviH^nf ♦» * i.
the earth's revolution on-itsixis^rSh^'ft^il?^
pos.l,on B in E, that it will have descr bed a coranlet
revo:nt,on with reference to space and that the Tr w ,
agara be on the meridian-in other words tha a sTder"!
day w,ll have elapsed since it left the position Elnd
that to bnng the san on to the meridian at A ink h
w. I have to describe an additional arc B A This ate
^he same as the angle B O, A, which is e,nal to te a"g le
s°defea?dav" Kth"' """'f ^ ''^*"^^" » -'- and'a
aereal da) . If there are n days in the year the value of
th,s arc will be l^. Reduced to time it is about 3m. 56s.
^.y^"'^'^ "''^''^ ^''' ^"^"^^'^ '" l«i^P S"ch a rate that
24 0flhe,rho„rs give the average interval between tv^o
successive culminations of the sun. The real inTerval I
however, sometimes more, sometimes less than ^'n"'
consequently, the sun does not culminate or pass tife
mer^tan at noon, but sometimes before it, somet n et
Thai ^t'a^fbrf '"■''"?"''''"'' "'>""• "-'■! ""•""-
mstant the sun ,s on the meridian is called ".ipparent
1
r
The equation of time.
^
r
noon. Noon as shown by a perfect clock is called
mean noon." The interval between the two is called the
"equation of time." Its greatest amount is about the ist
of November, when the sun culminates about iih. 4jm.
41S. A.M. The equation then diminishes till about the 24th
December, when mean and apparent noon coincide. After
that the equation increases (the sun culminating after
noon) till It attains a maximum of 14I. minutes about the
nth February, and then continues to decrease, becoming
;^t.ro again about the 14th of April. It attains a maximum
"t 3m. 50S. about 14th May, becomes zero about 14th
June, 6i mmutes about 25th July and 2ero 31st August.
The cause of the equation of time is as follows. If the
earth moved round the sun in a circle and at a unifprm
rate, and if the axis on which it itself turns were perpen-
dicular to the plane of its orbit, the sun would culminate
each dr.y at noon exactly. But the earth moves in an
ellipse and at a variable rate, and its axis is inclined to
the plane of the ecliptic at a considerable angle, the com-
bined effect being that we have the equation of time.
The great circle on the earth whose plane passes
through the centre and is at right angles to the axis is
called the "equator," and the projection of its plane in the
heavens is also called the equator, and sometimes the
equinoctial. If the sun, in its apparent annual path
moved at a uniform rate and traversed the equinoctial in-
stead of the ecliptic we should have no equation of time
An imaginary sun moving in this way is called the "mean
sun."
In addition to the time kept by an ordinary clock and
that kept by the sun— in other words "mean time" and
apparent solar time"-we have a third kind called
sidereal time," that is, the time kept by the stars. It
has been already mentioned that the interval between two
successive culminations of the same star is a little less
than 24 hours ; the time it takes, in fact, for the earth
lo
Sidereal time.
to make a single revolution on its axis. If we divide this
interval into 24 equal parts we have 24 sidereal hours; and
if vve construct a clock with its hours numbered up to 24
instead of 12, and rate it to keep time with the stars, it is
easy to see that the hour it shows at any instant will give
the exact position of the stars in their apparent diurnal
revolution round the earth, ('locks and chronometers of
thisdescription are used — the former in fixed observatories,
the latter for surveying purposes.
The subject of sidereal time will be referred to later on.
Before proceeding further it will be necessary to explain
the meaning of the various astronomical terms in ordinary
use.
de this
rs; and
I to 24
rs, it is
'ill give
diurnal
iters of
itories,
ter on.
explain
■dinary
I.
CHAPTER II.
EXPLANATION Ol' CERTAIN ASTRONOMICAL TERMS.
NAUTICAL ALMANAC.
THE
For practical purposes the earth may be considered as a
stationary globe situated at the centre of a vast transparent
sphere at an infinite distance to which are attached the
fixed stars, and which revolves round it in a little less than
24 hours. The sun, moon, and planets appear to move on
the surface of this great sphere, the sun in the ecliptic,
the rest in their respective orbits.
The extremities of the earth's axis are called the poles ;
and the poles of the great sphere are the points where the
axis produced meets it.
Great circles passing through the poles are called
"meridians." This term applies both to the earth and
the great sphere. In the case of the latter they are also
called "declination circles." Meridians are also called
"hour circles," and the angle contained between the
planes of any two meridians is called an "hour angle," be-
cause it is a measure of the time the sphere takes to
revolve through that angle. It follows that the hour
angle is the angle formed by two meridians at the poles.
In speaking of the meridian of a place we mean the
great circle pa ung through the place and the poles; and
a great circle passing through the poles of the great
sphere and the zenith (or point in the sky immediately
12
Latitude and longitude.
ridian for the instant,
ki^§t the observers head) is the
as regai ds the great sphere.
To fix the relative position of points on the earth's sur-
face we employ certain co-ordinates, called "latitude" and
"longitude." The former is the angular distance of any
point from the equator, and is measured along a meridian
north or south as the case may be. The latitude thus
varies from zero at the equator to 90° at the poles.
Longitude is the angular distance of the meridian of the
place from a certain fixed initial meridian, and is measured
either by the intercepted arc of the equator or by the
angle contained by the two meridians. Longitude is
measured east for 180° and west for 180°. Different
countries reckon from different initial meridians. The
English use that of Greenwich. The present system has
many inconveniences, and it is to W. hoped that someday
the world will unite in adopting some fixed meridian and
will reckon longitude through the whole j6o degrees in-
stead of as at present.
The position of the heavenly bodies on the great sphere
is determined by similar co-ordinates, but the latter are
called "declination" and "right ascension," the former
corresponding to latitude and the latter to longitude.
Declination is measured from the equinoctial towards
the poles, and right ascension eastward from a certain
meridian. The latter is, however, reckoned through llir
whole 36o",and is counted byhour^, minutes, and sec ids
instead of by degrees, i hour corresponding to isdegiees.
The point where the ;{ero or 24-hour meridian cuts the
equator is called the "first point of Aries," and is desig-
nated -V the symbol /". It is also one of the intersections
of theeq. ^'o'- -ith theecliptic. On referringtothe Nautical
Almarn. ^ viii hf ; een that the co-ordinates of the stars
are co.uim;>' lly changing. The fact is that, owing to the
slow cci.ical .; orion of the earth's axis known as the "pre-
%
Altitude and azimuth.
n
cession of the equinoxes," the planes of reference are
changing,'. This, however, causes no practical iticonveni-
ence. as the relative positions of the stars remain the same.
It should be noticed here that the terms "latitude" and
"longitude" are also used with reference to the heavenly
budies, and are liable to cause confusion. These co-
ordmates are measured from and along the ecliptic, and
are not required for the problems here treated of.
Hesides the above-mentioned co-ordinates which relate
to the relative position of points on a sphere another set
IS necessary to fix the position of a heavenly body with
reference to the observer at any instant. They are called
"altitude" and "a;jimuth." The first scarcely needs ex-
planation. The second is the angle formed by the verti-
ca' plane passing through the observer and the object
with the plane of the observer's meridian. The altitude
and azimuth of a star at any instant are, in fact, the angles
read by the vertical and horizontal arcs of a theodolite
respectively when the latter has been clamped with its
Aovo due north, and the telescope has been directed on the
star. Azimuth is generally reckoned from the north
round by the east, south, and west; but it is sometimes
reckoned from the south.
The plane of the "sensible horizon" is the horizontal
plane passing through the observer's position, and there-
lore tangential to the earth's surface at that point. The
"rational horizon" is a plane parallel to that of the sen-
sible horizon and passing through the centre of the earth
The projections of these two planes on the great sphere
coincide, being at an infinite distance.
It is easy to see that about half the great sphere is in
sight at any instant. The portion that is visible depends
generally on the latitude of the place and the sidereal
time of the instant. At the north pole the whole north-
ern hemisphere would be always in sight and no other
%
14
Definitions.
part At the south pole the view would be limited to the
southern hemisphere. At the equator both poles would
be on the hor,.on, and every point on the j^reat sphere
would come m sight in succession. At intermediate
places a certam portion round one pole would always be
above the hon.on, while another portion round the other
pole would never be visible.
"Parallels of latitude" are small circles made by the in-
tersection with the earth's surface of planes parallel to the
equator. Similar circles on the great sphere are called
declination parallels." A little consideration will show
that w.thin a certain distance of the equator at each side
at mid / '" t^k' /"''' '" '^' y'^'' P^'' °v-head
at mid-day. The belt enclosed between the two parallels
within which this takes place is known as the "tropics."
sutlZT.^ ''<,I'"'''^ '''""' '"^"'^^ explanation. When
speaking of the "hour angle" of a heavenly body at any
instant we mean the angle formed at the pole by the
meridian circle of the instant and the declination circle
passing through the body.
By the term "circumpolar star" is meant a star which
never sets but appears to describe a complete circle round
-he pole. These stars cross the merdian twice in the
twenty- our hours. One crossing is called the "upper
wTn 'th : '''"" ''' ''°"^'" *^^"^'^ •" A* ^he points be-
tween the transits at which the stars have the greatest
azimuth from the meridian they are said to be at the
greatest elongation," either east or west.
The words "transit" and "culminate" have the same
meaning when used with reference to stars which rise and
tio7Tv ''; '' '^' ''''"^' ^" '^' "PP^^^"* '•^'-tive posi-
lon of objects owing to a change in the observer's posi-
lon^ Astronomically it generally signifies the difference
in the apparent position of a heavenly body as seen by an
}
Parallax.
J5
observer from what it would be if viewed from the centre
of the earth. Parallax is greatest when the object is on
the horizon, and nothing when it is in the zenith. The
moon, from being near the earth, has a considerable
parallax. That of the sun does not exceed 9". The
positions of the sun, moon, and planets given in the Nauti-
cal Almanac are those which they would have as seen
from the earth's centre, and it is therefore necessary to
correct all observations on those bodies for parallax.
Parallax causes the object to have less than its true
altitude. Refraction has the opposite effect. The latter,
like the former, diminishes with the altitude. Near the
horizon— say within 10 degrees of it— its effect is very un-
certain, and observations of objects in that position are
therefore unreliable. At an altitude of 45° the refraction
is about i'. As it varies with the temperature and atmos-
pheric pressure the barometer and the thermometer must
be read if very exact results are required. '"^
The corrections for refraction and parallax are not to be
found in the Nautical Almanac, but are given in all sets of
mathematical tables. The N. A., as a rule, gives only
variable quantities — such as declination, right ascen-
sion, equation of time, etc. It is rather a bulky volume, but
the portions of it in general use by the practical surveyor
could be comprised in a small pamphlet. The most use-
ful are the sun's declination and right ascension, the equa-
tion of time, the sun's semi-diameter, and the sidereal
time of mean noon — all given for every day in the year ;
the declinations and right ascensions of the principal
fixed stars, taken in regular order according to their right
ascensions ; and the tables for converting intervals of mean
time into sidereal time and 7ricc versa. To these may be
added tables of moon-culminating stars, and tables for
finding the latitude from the altitude of the pole star when
off the meridian.
I 'i
i6
The Nautical Almanac.
riven r\ ^u""" ^"' P^"?" °f ">» ')'■='"'««
g van for each month ,„ the Nautical Almanac, and of the
da a for fi.ed stars, are reprinted below. All he quanti-
t on tT" °' "T "' ''■■"""'^h on the day in' ques-
.on They must, therefore, be corrected by a propor-
t.on for any other hour o, longitude. Thus.'^when ^ s
noon at a place in 90' west longitude, or si.v h^urs we,, of
Greenwrch, ,. ,s 6 p.m. at the latter. ThereforeTt an
observafon were taken at the western station at noon the
quan.,t,es requ.red would have ,0 be corrected for "heir
change m six hours.
Owing to the earth's uniform revolution round its axis
Zfh7 a' , ""'^^ ''"^" "* Greenwich was 3 p.m
and the sidereal time iih., they would be g a m and ^h'
respectively at a place in longitude 90" west ^
In the Nautical Almanac the day is supposed to com
mence at noon and to last for 34 hours. Thus ga mTn'
1st ot January. This astronomical method of reckoning
mean time must not be confounded with sidereal t^.
which is quite a different thing. *™''
fhl^^Tl""" *^' ^''' °^ ""^^ "^°"th are given also at
the end of the preceding month. Thus, weLd sTlyl
ZZVfFr'''''''Z'''''^'' ^^ ''^ 3ad being real^
at ng '' "^'^ " '°^ convenience in interpo-
Nautical A Imanac.
17
JUNE, 1880.
AT APPARENT NOON.
THE SUNS
Apparent
Right
Var.
Apparent
AscensionUour Declination. hoVr.MeSL,
Sidereal Eflaation
time of of Time
the to be
Semi- i *"*<./rom
, diameter j «rf,/,,rf to Var
Var. passmg U/„..„ J i^"^,-
in I I, the II Time, jhour.
:h m s s
I i4 39 071 110241
10-258 j
2j'4 43 671
3 4 47 1310
I0'274 '[ 22
|4 51 i9'87 10-289
|4 55 26-gg 10-304
i4 59 34 44 10-317
N.22 8 59-6 19-79
22 16 42-9I18-82
' 24 2-9|i7-85
3 4220
5 7 5024
5 II 58-54
10-329
10-340 ;
10351
5 16 708 10-360
5 20 15-82 10-368
'5 24 2474 10-375
1315 28 33-8r 10-381
'4115 32 4302 10-386 j
i5||5 36 52-34110-3801
22 30 59-4116-86
22 Z7 32-3I15-87
22 43 41-3 14-88
22 49 26-5|i3-88l I 8-72
22 54 47-612-88 ;i 8-7S
22 59 44-6.-II-87 ,1 8-79
23 4 17-3 10-86
23 8 25-7, 9-84
23 12 9-6I 8-82
23 15 290) 7-79
23 18 23-71 6-77
23 20 53-8,1 5-74
m s j s
2 21-83 0.384
2 12-42 0-400
2 2 61 0416
I 52-43 0-431
I 41-90 1.1-446
I 31-03 0459
1 19-86 0-471
' 8-41 0-483
o 56-7o;o.493
44 '75 (0-502
3260 0510
20-27 0-517
0523
° 4-8310-528
° 17-35 |o-532
i8
Nautical A Imanac.
JUNE, 1880.
AT MEAN NOON.
I
43
o
Apparent
Right
Sr-'fAscension
Oil
Tues.
Wed.
Thur.
Frid.
Sat.
Sun.
h m s
4 39 III
4 43 7-09
i4 47 13-45
I
14 51 20- 19
14 55 27-28
(4 59 3470
THE SUN'S
Equation of
Time,
to be
added to
subt. from
Mean
Time.
Apparent
Declination.
Semi-
diameter.
Sidereal
rime.
/ //
N.22 9 0-4
22 16 43-6
22 24 3-5
22 30 59-9
22 37 32-7'
22 43 41-7
15 48-1 :
15 47-9 ;
15 47-8
15 47-7 ^j
15 475 :;
15 474 .i
m s !
2 21-82
2 12-40 j
2 2-59 !
I 52-41
I 41-88
I 31-02
h m s
4 41 2293
4 45 19-49
4 49 iG-05
4 53 12-60
4 57 9-16
5 I 572
Mon.
Tues.
Wed.
7 ,
8 j
9 '
5 3 42-43
5 7 50-44
5 II 58-71
Thur.
Frid.
Sat.
10
II
12
5 16 721
5 20 15-91
5 24 24-80
Sun.
Mon.
Tues.
13
14
15 '■
5 28 33-84
5 32 43-01
5 36 52-29
22 49 26-8, 15 47-3 ;;
22 54 47-91 15 47-2 Ij
22 59 44-8' 15 47-1
I 19-85
I 8-40
o 56-69
~3 4 17-5
23 8 25-8
23 12 9-7
15 470
15 46-9
15 468
23 15 290
23 18 23-7
23 20 53-9
15 467
15 467
IS 46-6
o
o
o
44-75
3260
2027
4-83
17-55
5 5 2-28
5 8 58-83
5 12 55-39
5 16 51-95
5 20 48-51
5 24 45-07
5 28 41-62
5 32 38-18
5 36 34 74
APPARENT PLACES OF STARS. 1880.
AT UPPER TRANSIT .\T GREENWICH.
Month
and
« Andromedje
Day.
R.A.
Dec.N.
Jan. I
II
21
31
h m
2
s
11-88
11-75 J3
11-62 '-'
11-51 ",
,j
28251
1
55^ 9i
54-4
51-8 'H
y Pegasi.
(AlgenibJ
h
O
s
4-24
4-13
402
393
8-6
77
6-8
5-7
'9-^' 11^5 fJ;26-3o ^^162-7
19-50
19-39
19-30
ii-«''' 3i
^81-6 -I
93 6^5 '"
^4-51 ^^i59-8 II
25-37
124-51
'2373
12
The Celestial Globe.
••
19
To revert to the subject of sidereal time: Sincrthe
sidereal clock stands at ^ero or 24!!. at the instant the ist
point of Aries is on themeridian, and as the clock keeps
time with the stars in ^apparent diurnal revolution round
the earth, it follows tlmt when any particular star is on the
meridian its right ascension is the sidereal time of the in-
stant. Thus, if the stars R. A. were 6h. the clock should
show thattime at the instant of the stars transit, and its
error may be ascertained by mountim^ a telescope .0 as to
move only in the plane of the meridian, and noting the
instant of transit. If we want to find the mean time of a
stars transit we have only to convert the star's R A
into the corresponding mean time of the instant, in 'the
manner to be presently explained. Conversely, a star's
transit gives us the sidereal time of the instant, and hence
the true mean time.
The celestial globe is of great use in studying astronomy.
It IS a model of the great sphere supposed to be viewed
from outside. The positions of the stars on it are the
points where straight lines, joining them with the earth
would intersect it. The equator and ecliptic-the lattei'
being the sun s annual path through the stars-are marked
on It, as also the sun's place in the ecliptic for every five
Jays. The axis on which it turns is that of the poles
1 he meval ring passing through the latter represents the
meridian and the flat horizontal ring the plane of he
rational horizon. ^ y ^i me
t.-o?"'fl'^' '^''^"''' °^'^" ^^^°^^ '^ t° «how the posi-
tZ. T T'\^' ^"^' '"''""' ^^'^^^ ^•^S^'-d to the spec-
tatoi. To do this we raise the pole by means of the
ToTrr- '''' ""7'^^" ^^ ^^ *« ^'- ^* - altitude
br2 tt °"i ''"'^ '" '^'' ^"^^^"^^ °f *he place, and
bung the sun's place in the ecliptic for the day to the
meridian The half of the globe above the horizon wl
now roughly represent the position of the visible hemis-
phere at noon. To find the position of the sphere at any
20
The Great Sphere.
other hour it is only necessary to turn the Tlobe through
ITL stirs at I r ""''!'" f "' °"* ^'^^ ^'^'^^^ P-'tions
ot tne stars at 8 p. m. we should have to revolve the dobe
westwards through an angle of x.o^ Conversely we can
find the name of any constellation or star by noting
IccSgTy.'" ''' ''' -^ '- '-- -^ s.U..X:Zl
easTtr'*.'^''^' P'f '"^ *^^°"^^ the zenith and the
angles to the mend.an, ,s called the "prime vertical "
Pig' 3.
In Figure 3 the small circle at the centre represents the
^nelk- '1 ')' ''''' ^'"'^ *^^ ^^-t ^Ph-- Strictly
to the Tal r: ''""'' '^ ^ "^^^ P«'"t ^" comparison
to the latter, and the pomts on the great sphere would ap-
f
ough
from
tions
flobe
2 can
3ting
jlobe
the
■ight
Explanation of Terms.
f
21
iionzon, and p pi the earth's polar axis meeting fh^
grea. sphere i„ .he points P l. z if/he .e"' N
he nad,r or po,n. on .he sphere diame.ricaliy opposi.e
"^ The plane of the paper represen.s .he plane of tl e
po.n.s of he horizon, . j ,s .he equa.or, E r O the
eq „,oc,.a r the firs. poin. of Aries, and P ,' P . .L
.n .,al dechnat.on circle passing through i., fro™ wh ch
'- Dorfiln f "" °" '"' "'"''"''''' ■""'dian. Z S B
Ind ri r ?•'"■'? '"■'=''=• P''^^'"'! "'""gh the .enith
and S and mee.mg the horizon a. B. Z B is of coursi
.'re',. s„h Tf '■'P'''''"' "'^ ^PP""^"' '""'ion of the
grea. sphere wi.h respec. to .he earth,
Do.^n','?"'' ^'f'r?' ""' ""«'■= '' ° '• '^ "-^ '^"'"de of .he
po n. A, and A O j=Z O Q, which is .he zeni.h distant
Mepij, /t;?et^.°1'^
It should be noticed .ha. .he whole of .he hemisphei
above the plane H B R is visible .0 the observer a A
(nor m'I ",^«"™" "f '•>' ^'ar S is zero, its declina;ion
(north) S hour angle S P Z, its al.itude S B, zeni.h
d,s.a„ce S / and azin^uth S Z R. The star S- hLs R. A
r «, decimation (south) Q s! hour angle nil, altitude
S' R, zen.th distance, S. Z, and azimuth ze o. The
dereal „me of .he instant is r P Q, or the arc r Q a
triangle P Z S ,s called the "astronomical trianrie " It
should be noted that in all calculations it north declina
..on ,s reckonedpositive,southdecli„atio„must becounted
negative, and wee z;m«. "uuiea
CHAPTER III.
Uses of practical astronomy to the surveyor.
Instruments employed in the field. Methods
of using them. Taking altitudes. Problems
RELATING to time.
The principal uses of practical astronomv to the sur-
veyor are that it enables him to ascertain his latitude,
longitude, local mean time, and the azimuth of any
given hne; the latter of course ^'iving him the true north and
south line and the variation of the compass. In fact the
only check he has on his work as regards direction when
running a long straight line across country is by determin-
ing its true azimuth from time to time, allowing (as will
be explained hereafter) for the convergence of meridians.
The mstruments usually employed are the transit theo-
dolite, sextant or reflecting circle with artificial horizon,
solar compass, portable transit telescope, and zenith tele-
scope. To these must be added a watch or chronometer
keeping mean time, a sidereal time chronometer (this is
not, however, absolutely essential), the Nautical Almanac
for the year, and a set of mathematical tables. With the
sextant or" reflecting circle we can measure altitudes and
work out all problems depending on them alone, and also
lunar distances. The transit theodolite may be used for
altitudes, and also gives azimuths. The solar compass
IS a contrivance for finding, mechanically, the latitude.
1
Instrumenta.
i
J
^
meridian line, and sun's hour anj^le. The zenith tele^
scope gives the latitude with great exactness, and is par-
ticularly suited to the work of laying down a parallel of
at.tude The transit telescope enables us to determine
the mean and sidereal time, latitude, and longitude. The
transit theodolite answers the same purpose, but is not
so delicate an instrument. It is, however, of almost uni-
versal application, and nearly every problem of practical
field astronomy may be worked out by its means alone if
the observer has a fairly good ordinary watch. The sex
tant has been called a portable observatory; but in the
writer's opinion the term is more applicable to the last
named instrument. The sextant is not so easy to manage
and only measures angles up to about Ii6,° so that s8" is
practically the greatest attitude that can be taken wJth it
when the artificial horizm has to be used. The latter
as generally made, is disturbed by the least wind, and
then gives a blurred reflection. maHng the observation
nearly worthless. There is little use in having the arc
graduated to read to within a few seconds if the contact
of the images cannot be made with certainty to within a
minute or two.
All observations taken with the transit theodolite should
If the nature of the case admits of it, be repeated in re-
versed positions of the telescope and horizontal plate and
he naean of the readings taken, as we thereby get rid of
r'of Thi '1"'"''^''^"i '"'^•^' '^"^^ "*''- '"^--"tll
errors hus, for an altitude, the plate having been
evened, the vertical arc set at .ero, and the bubble of Ihe
telescope level brought to the middle by the twin screws
the verticahty of the axis is tested by turning the upp
plate in azimuth i8o°, and seeing if the bubble is still in
the centre. If it is not it is corrected, half by the lower
plate sere., half by the twin screws, and the operation
repeated til the bubble remains in the centre in ever"
position. Ihe altitude is then taken, the telescope
H
A Ititudes.
turned over, the upper plate turned round, and the alti-
tude again read. In each case both verniers should be
read.
The first step after taking an alticude with either
sextant or theodolite is to correct it for index error, if
there is any. The following lists give the corrections to
be applied in each case to an altitude of the sun's upper
or lower limb to obtain that of his centre :
THEODOLITE.
Index error.
Refraction.
Parallax.
Semi-diameter.
SEXTANT.
Altitude above
water horizon.
Index Error.
Dip of Horizon.
Refraction.
Parallax.
Semi-diameter.
Double Altitude with
artificial horizon.
Index Error.
Divison by 2.
Refraction.
Parallax.
Semi-diameter.
The semi-diameter has to be added if the lev er limb
IS observed, and vice versa. When taking an altitude for
time with the artificial horizon the easiest way to get the
correct instant of contact is to bring the two images into
such a position that they overlap a little while recedinjr
from each other. Ai^the instant they just touch the
observer calls "stop," the assistant notes the exact watch
time, and the vernier is then read. This plan necessitates
observing the lower limb in the forenoon and the upper
in the afternoon. The dip depends on the height of the
instrument above the water, and, like the refraction and
parallax, is to be found in the mathematical tables.
In the case of a meridian altitude for latitude the sun
or star, after rising to its greatest height, appears for a
short time to move horizontally. When this is the case
the altitude may be read off.
Fixed stars require, of course, no correction for parallax
or semi-diameter. As the refraction tables require a
correction for temperature and atmospheric pressure the
height of the thermometer and barometer should be noted
t
.
Equation of Time.
r
25
If an altitude has to be taken with the sextant and
H.t.fic.al horizon, and the sun is too high in the heavens
for^t^he ,nstrun.ent, a su.table star n^ust be observed Tn!
In surveying operations the latitude is generally known
aF^roxjn.ately. This gives the approximate altitVdeTo"
a rner,d.an observation; for the altitude of the intersec-
K>n of the mend.an and equator being 90' minus the
lautude. we have only to add to or subtract from thi
tTtud"^ '^'''' declination, and we have t^e al-
\ T f . , , . y . f i ^ AVSE OF THE EQUATION OF TIME.
In Figure 4 P is the pole, E C
a portion of the ecliptic, and E Q
a portion of the equr.tor; each
being equal to go°. C and Q are
on the same meridian, and P Q is
also a quadrant. Now, let S be the
sun, and suppose it to move at a
uniform rate from E to C. Let ^■
S' be an imaginary sun (called the "mean" sun') moving
m the equator at the same rate as the real sun. Now let
the two suns start together from E, and after a cer ain
interval let their position be as shown in the figure
Since they move at the same rate. E S will be equal to
t ir> , but as a consequence the meridians P S and PS'
S p c?;°'"''^f ' ^; ^^"^"^ ^°t ^h^-d of S. The angle
time aZ.' '^" '"° "^'"^'""^ ^^ *h« equation of
IZn f ' •. ° '""' """"'^ ^"'^^ simultaneously at C
and Q It IS evident that, though S^ gains on S at first, it
will, after a certain point, cease to gain and lose insteld
Since the equation of time-in other words the differ-
ence between apparent and mean solar time-is con-
the'mtn ."'"^' '' "' """* *° '"^ ^^°- ^^e Almanac
the mean time corresponding to apparent time at any
'/^
rn:^ c ^'-K^t^ £^c<.f„^-c_^
'1-
<X (. ^i
'^ ' * Cj.a^
./
*^ Connexion of Sidereal
particular instant and longitude, we must allow for the
change in the equation that has taken place since noon
at Greenwich. For instance ; suppose we had to find the
mean time corresponding to three hours p.m. apparent
time on the 22nd April, 1882 at a place in longitude 6h.
west. By the N. A. the equation of time at apparent noon
that day at Greenwich was im. 34s. 43, to be subtracted
Irom apparent time and increasing, the variation per hour
o.s.496. At 3 P.M. at the place it would be 9 p.m at
Greenwich. 9x05.496=45.464. The corrected equation
of time is im. 383.89, and the true mean time 2h. s8m
ais.ii p.m.
GIVEN THE SIDEREAL TIME AT A CERTAIN INSTANT TO
FIND THE MEAN TIME.
Here we have given the right ascension of the declina-
tion circle of the great sphei-e that is on the meridian at
the instant, or— which is the same thing— the time that a
sidereal clock would show. Now the Nautical Almanac
gives the sidereal time of mean noon at Greenwich which
has to be corrected for longitude. These two data give
us the interval in sidereal time that has elapsed since
mean noon, and this, converted into mean time units
will be the mean time. '
Ex. Find the mean time corresponding to 14 hours
sidereal time at Kingston on the 28th April, 1882.
We find from the N. A.
Sidereal time of mean noon at Greenwich ,h ,,.„
Correction for longitude reenwicn 2h, 25m. 258-33
50-26
Sidereal time of mean noon at Kineston Tii e
Sidereal time of the instant. *^'"^''°" /h. 26m. 155-59
14". om. OS
Difference, or interval of sidereal time that has elapsed '
smce mean noon ^ ,
Which, converted into mean iime:\s::::::::::::::i^: ^^;^: ^^^^^^
The conversion of sidereal into mean time units and
vtce versa, is obtained from tables at the end of the Nauti-
cal Almanac.
f
and Mean Time. ,-
If the sidereal time of mean noon is greater than the
sidereal time given we shall obtain the interval before
mean noon. Thus, if on the same date as above we
wanted to find the mean time corresponding to sidereal
time one hour we should proceed as follows :
Sidereal time of mean noon ,k <
" oftheinstant .'i 2h. 26m. .jseg
I o o
Sidereal interval before mean noon ] ^ ~r
Which in mean time units is ~ 7 7
Subtracting this from i2h ^° '' 5^
la o o
We have mean time .o h. 33m. 58s 44 . m.
It is sometimes convenient to add 24 hours to the given
sidereal time to make the subtraction possible. Thus if
the sidereal time were ih.. and the sidereal time of mean
noon 23h.. we should have the interval elapsed since mean
Tan tL^^' " ' '°""' '-''-' ^' ''' 59-. 40S.3 P.M.,
TO FIND THE MEAN TIME AT WHICH A GIVEN STAR WILL
BE ON THE MERIDIAN.
fJ*;!! ^A^^^^" application of the preceding problem.
For the Almanac gives us the star's right ascension, which
IS the same thing as the sidereal time of its culmination,
and we have merely to find the mean time corresponding
GIVEN THE LOCAL MEAN TIME AT ANY INSTANT TO FIND
THE SIDEREAL TIME.
Here we must convert the interval in mean time that
has elapsed smce the preceding noon into sidereal units
and add to ,t the sidereal time of mean noon
Aoril' ^Z^ *he sidereal time at 9 a.m.. on the 29th of
April, 1882, at Kingston, Canada.
Here we have, as before:
Add 21 hours of mean time in sidereal units, or.' „ J^' 'f ^
" 3 20' Q8
Sidereal time "~~
ajn. 29m. 425-67
r
28
Sidereal Time.
If this process makes the result more than 24 hours
that number must, of course, be subtracted from it. Thus,
if we got 25 hours the sidereal clock would show ih. If
the sidereal time of mean noon is greater than the inter-
val in sidereal units we add 24 hours to the latter to make
the subtraction possible.
The correction on account of longitude for the sidereal
time of mean noon is constant for any particular place or
meridian
The subject of sidereal time may be thus illustrated :
In Fig. 5 let the small circle
represent the earth, and the
large circle the equator of the
great sphere viewed from the
north, the plane of the paper
being the plane of the equator.
Let P be the pole, A a point on
the earth's surface, and P A
the meridian of A. y is the
first point of Aries, S and S^ Fig. 5.
two stars, and s and s* the points where their declina-
tion circles meet the equator. Now the arc y s^ (or the
angle y P s*) is the right ascension of Sj, and the arc
Y s^ s that of S. Now suppose the earth (and therefore
the meridian P A m) to remain fixed, while the outer
circle and stars revolve around it in the direction of the
arrow; and at the instant that it is mean noon on a cer-
tain day at A let the position of the great sphere be as
shown in the figure. The arc ;* s^ m will be the sidereal
time of mean noon for that day at A. The star S will be
on the meridian at an interval of sidereal time after mean
noon corresponding to s m, while the star S^ has passed
the meridian by an interval corresponding to m Sy, and
by reducing these intervals to their equivalents in mean
time we shall have the mean times of their transits. For
r
Hour A ngle of a Star.
29
\
H'
instance, suppose we had to find at what time the pole
star would be at its upper transit on a day when the
sidereal time of mean noon
was 2ih. 30m., the right as-
cension of the star being taken
as ih. 15m. Now the state
of things at noon would be as
shown in Fig. 6. The star
would have passed the meri-
dian by an interval of 2ih.
jom. — ih. 15m., or 2oh. 15m.
(sidereal) and would there- pig^ c
fore be on the meridian at ah. 45m. sidereal, or sh. 44m.
23s. mean time after noon.
Sidereal time is usually found by calculating the hour
aVigle of a star from its observed altitude. This, added to
the star's right ascension if the hour angle is west, or
subtracted from it if east, gives the sidereal time. From
this the mean time can be obtained, if required. The
watch time at which the altitude is observed must, of
course, be noted.
TO FIND THE HOUR ANGLE OF A GIVEN STAR AT A GIVEN
TIME AT A GIVEN MERIDIAN.
Here we must find the local sidereal time of the given
instant and take the star's right ascension from the Al-
manac. The difference between these two quantities
will be the srar's hour angle, which will be east if the
star's R. A. is greater than the sidereal time, and west if
the contrary is the case.
TO FIND THE MEAN TIME BY EQUAL ALTITUDES OF A
FIXED STAR.
Fixed stars are employed for this purpose in preference
to the sun or planets because of the change in declina-
tion of the latter. A star should be chosen which de-
scribes a sufficiently high arc in the sky. Two or three
hours before its culmination its altitude is taken with the
30
Mean Time by eqnal Altitudes.
f
f.
sextant or theodolite, the exact watch time noted, and
the instrument left clamped at that altitude. Some hours
later, when the star has nearly come down to the same
altitude, the observer looks out for it (keeping the instru-
ment still clamped) till it enters the field of view of the
telescope, and waits till it has exactly the same altitude as
before, when he again notes the watch time. The mean
of the times of equal altitude will give the watch time of
the star's culmination, which should be the same as the
mean time (previously calculated), corresponding to the
star's right ascension, the latter being the sidereal time
of the culmination. If they are not the same the differ-
ence will be the watch error.
If the theodolite is used for this observation the verti-
cal arc only^s^kept clamped. When the star has nearly
come dowfi^o the ntiVnTd altitude the horizontal arc is
clamped and its slow motion screw used.
TO FIND THE LOCAL MEAN TIME BY AN OBSERVED ALTI-
TUDE OF A HEAVENLY BODY.
For this problem we must know the 'latitude of the
place, and, if the sun is the object observed, we must also
know the mean time approximately in order to correct its
declination.
The altitude should be taken when the heavenly body
is rapidly rising or falling— that is, as a rule, when it is
about three hours from the meridian, and the nearer to
the prime vertical the better.
If we take P as the pole, Z the zenith, and S the hea-
venly body (Fig. 7), PZS will be
a spherical triangle in which PZ
is the complement of the latitude,
PS the polar distance of the ob-
ject observed, and ZS the comple-
ment of the altitude. The three
sides being given we can find the
three angles from the usual for-
mulae. In the present instance we Fig 7.
/I
f
Timely Altitude of Sun.
31
/I
want P. which is the h";;;;:;;;^,^;^,-;^— ;
A convenient formula is
Sina~= !HLi?rPS)^ (s-PZ)
2 sin PS sin PZ~
wheics is ?^±PS+ZS
S\stt:^:n"°Vh'^ ",^^'^^^"* ^* ^^^ ^-^-^
to the altitud ; ndihe an^P hr""l""^ ^" ^^^"^^
is divided by i, Thl /"^ ^'"" ^^'"'^^^ ^^^
body. In tllZ o^t sTit "wil^b^ T' ^"^^^ °^ *^^
time, and by adding Z IT ^ *^^ apparent solar
of time we sh jl "f Z'l ^'^^S the corrected equation
the observation ' "" '"''" ^^'"^ °^^^^ ^"^tant of
If the body is a fixed star or nlanet th.r. t
known right ascension we subtract hrhn *^'"'/T '*'
east, or add it if it is west Th u ^"^'' '^ ^* '^
of the instant from l^Jh , J ^'"'^ *^' ^'^^^^^' ^^"^e
if required TheTn^ ' ""'"" *''"^ ^^" ^e inferred
•'astronomicaUrlangle '' " ^"^"^'^ ^^' '^ ^^^ ^ ^^e
The Ian of the a tS " .^'^ *'"^ ^' ^^^^ "°'^^-
tude to corresDonH .1 " *^'" '^^^" ^^ ^ ^i^gle alti-
sit theodol Ts "std two' rr/'*'^ ^^■'"^^' ^^^'<^tran-
versed pos ions of the M '^°"'' '^^ ^^'^^^ '" ^e-
- to co^rrectTst^t*^ r^^^^^ P^- -
- obtained by observing both an east aif" "'^^"^"^^
takmg the mean of the results .! ^''''* '^^' ^"^
refraction and nf ,\: ? ' ^ ^"^'"^ °^ observation
great measure ' '"'""^"^' ^^^" '^^ ^^^ "d of in a
EXAMPLE OF WORKING OUT A SKXtav^ ,,
Equation of t/^' I 1 r?""'^' ^'■- '''"■ ''=■ '■«•
e»t time. I„drV?oV't^-L^° be subtracted from appar-
^T
I
P
33 Example.
Double altitude 64° 4' o"
Index error c or.
, 2)63 58 30
/^ /
■ <-^- f^ A^^-e^ 31 59 15
a^> ^ <i/..^^pemi-diameter 15 57
■'^ Kefraction and parallax ... i 07
True altitude of sun's centre 32 13 49
90
n ,• ,. '^° ° ° '^^ 57 46 ii-ZS
Declination 10 40 o 79 20 o :=PS
45 46 20= PZ
79 20 o ,
2)182 52 31
90 O O
Latitude... 44 13 40 g^ 26 15=5
■ 79 20 o
45 46 20
91 26 15 12 6 T5=s_ps
Logsin(s-^l)i%.||44-7oo- ^' '' ''='~''^
" (s-PS)= 9.3215800
Log cosec PZ =10.1447400
cosec PS =10.0075700
2)39-3283600
i9-664i8oo=log sin 27" 19' +10
2
54 58=P
c .• r ■ ~3h. 39m. 52s.
Lquation of time = — o 34
True mean time =3 3^ 18
Watch time 3 ^y je
Watch slow im. 3s.
^^^ g h Star A Ititude. i^
EXAMPLE o^coK^^:^^^^^rr^^^~^^^
SINGLE ALTITUDE OF A STAR.
the'^aldtudetf ^"'l "' ' ^^'^^ '" ^h. 30m. west longitude
To find the watch error—
'^^' ?J^\"^^* ascension was 4h. 29m. 43
Add hour angle J 30 17
^^^®^^^'t™e of the instant = b"~Jg ^
24
""•:
Subtract the sidereal time of ^°^^ ^'
mean noon, corrected for
longitude
'^■' o p
Sidereal .nterval since mean ~~
noon o
•:: i. 59 12
And the watch was 2m i6s fast ^^"'" ^^' '" '"^^" *''"^-
C-— -°- ."Sir.':; :ii:'a^
Sin 8-^= 2°_l£iJn_(£ii_«) .
2 cos yi sin PS 'W^ere a is the altitude, X the
latitude, PS the polar distance, and 5=^'' + ^?
TO FIND THE TIME BV A MERIDIAN TRANSIT OF A
HEAVENLY BODY.
urvey. rhe theodolite is set up on the line,
34 Time by Meridian Transits.
the telescope directed on a distant point or mark on it,
and the horizontal plate clamped. The telescope will
now, if moved in altitude, keep in the plane of the meri-
dian, provided the instrument is in adjustment ; and the
instant of transit of any object across the vertical wire or
intersection being noted, we can deduce the true time.
As the altitude of an object at transit is equal to the alti-
tude of the intersection of the meridian and equator
plus or minus the declination of the object, we have the
equation
Altitude = 90° — latitude ± declination.
and can set the telescop^teforehand at the required
altitude. If the latter is more than 50° a diagonal eye
piece is necessary with most; instruments. In the case of
the sun we may either take the mean of the observed
instants of transit of the east and west limbs, or take the
transit of one limb and add or subtract the time required
for his semi-diameter to pass the meridian (which we
obtain from the Almanac). We now have the watch time
of transit of the sun's centre, which takes place at appar-
ent noon, and have only to find the true mean time of ap-
parent noon by adding to or subtracting from the latter
the equation of time (corrected for longitude), when the
difference will give us the watch error.
Example— At Kingston, on the 2nd of May, 1882, the
transit of the sun's west limb was observed at iih. 55m.
A. M. What was the watch error ?
Here we have
Watch time of transit of limb iih. 55m. os.
Time of the semi-diameter passing the
meridian ini. 6s.
Watch time of transit of sun's centre iih. 56m. 6s.
The equation of time, corrected for longitude, was
3m. 11.5s, to be subtracted from apparent time.
I
Htm
OS.
f
_, '^^Mertdtan Transits.
Apparent time of transit"^unTcentre '"17h' oi^"
Equation of time om. os.
T, 3m. 11.5s.
1 rue mean time of transit ~ [
Watch time of transit "h-56m.48.5s.
"h. 56m. 6.0S.
Watch slow ~-
In the case ofa Star itc: riLkf" ' . 42.5s.
time of .he ins.an „ f " /nfi ' a'dT" '\*' '"""''
(rasrt «( Gnenwich, correcting i, f , f P°"'"°° "'
directed in the explanX':?, the Ld"'""'^ '" '"^ ^'^
TheplanofthrowingtheSof* *^^ ""u'^" ""'""'■
Object „a. is obJeJonaVlll/ittV^rridr-^" ""
other. "P" P'™'= ''^'"S higl'er than the
mo^n^r^trarpt^eSr^^ire^t"-"^^^
pole are the worst. ^°'^ "^^^^^t the
l\
6s.
CHAPTER IV.
I I
METHODS OF FINDING THE LATITUDE, LONGITUDE, AND
MERIDIAN.
TO FIND THE LATITUDE BY THE MERIDIAN ALTITUDE OF
THE SUN OR A STAR.
The altitude may be taken with the theodolite or sex-
tant. The approximate direction of the meridian should
be known beforehand. If the theodolite is used the in-
strument is levelled, and the telescope directed on the
object a little before it attains its greatest height. The
horizontal wire is then made to touch the object (the
lower limb if the sun is observed) and is kept in contact
with it as it rises by means of the slow motion screw of
the vertical arc, the telescope being moved laterally as
required. When the object has attained its greatest
altitude it will remain for a short time in contact with the
wire, when the vertical arc is read off. The telescope is
then at once turned over, the upper plate reversed, and the
altitude again read. The mean of the two readings will
be the apparent altitude of the object.
When using the sextant and artificial horizon we bring
the two images into contact and keep them so by the slow
motion screw till they cease to separate, when the vernier
is read off.
The usual corrections having been applied, and the true
meridian altitude of the sun's centre or star thus obtained,
the latitude is found as follows :
Latitude by a Meridian Altitude.
57
nn!;? ''T' V'u *^^ °^J''* ^'''"'■^^^ culminate at~ the"op.
posite side of the zenith to the visible pole.
in Fig. 8 let A be the observer's
position, p k q pi e -e, section
of the earth passing through A,
and the poles {p pi) and therefore
«n the plane of the meridian.
Let O be the earth's centre and
\eteOq be perpendicular top pi;
then e and q will will be the inter-
sections of the equator with the meridian. W H A R
ouchmg the earth's surface at A and also in the pfant of
nd win f H "" ^r"' ■'" ^" ^^^ P^-^ °^ the hor on
and will he due north and south. Let S be the object
observed and let its declination be north. Draw A P
parallel to ^ /.^ and A Q parallel to e q. A P wiU be the
d.rect.on of the pole of the great sphe're. and A Q tha of
OAtT 7 °'-''' "^"''^" ^"^ ^^-"-t'^^- Join
O A and produce it to Z the zenith. A Z is at riht
angles to H R and P A Q is also a right angle S A k i
he me ed altitude of the object and S a'q its declfnl!
A O . 7 \n'''"i' °'^ '^ '''' ^^^ A ^ - the angle
the altitude of the visibl^poTe. Hell w! hav"';^''^' '^
Latitude=ZAQ=9o"-QAR=9o°-(S A R-S A Q)
=90 --altitude + declination.
If the object had south declination, as SS we should
have (since S« A R is its altitude and S^ A Q its d cHna
non) Lat.tude=ZAQ=9o«-Q A R=go-(s'rR "
Q A b»)=9o— altitude-dedination.
fi/u^'ln^lh ""' '''•'"''' '' '' ^'"^^^ ^^^t to draw a
hgure. In the one given / is the north and p^ the south
pole. ^ 1.U
38
Latitude.
li
i
c,
s\.?, s
and case. If the object culminates between the zenith
and the visible pole, as at S in figure g, its altitude will be
S A H, and we shall have :
Latitude =P A H = S A H— S A P
= S A H— (90°— S A Q)
=altitude + declination — 90°.
If the object is a star
which never sets, but de-
scribes a diurnal circle round
the pole, it will cross the
meridian twice in the 2.;
hours, and we may take its
altitude at what is known as
its lower transit, as at S?,
Fi^. g. Here we have :
Latitude = PAH = S^'\H
+ S* A P=altitude + 90° — Fig 9-
declination. Therefore, in the case of such a star we
have:
Latitude= star's altitude ± star's polar distance ; the
positive sign being taken if the star is observed below the
pole, and vice versa.
Case 2 can only apply to the sun when A is within the
tropics. In many books on astronomy the formuloe of
case I are made to apply to the sun in every situation,
whereas they manifestly fail when he culminates between
the zenith and the visible pole.
In the Nautical Almanac is given a very simple method
of finding the latitude from an altitude of the pole star
taken at any point of its diurnal circle round the pole.
The time of the observation has to be noted and the cor-
responding sidereal time calculated.
LONGITUDE.
Longitude cannot, like latitude, be measured absolutely,
4^
i
/ -n
r\^ft,..
«i»
\
Longitude.
as it has no natural zero or origin, and we have'tTTss"^
an .n.t.al mendian arbitrarily, the English adopting thai
o Greenwch But the difference of longitude of two
pi es ..n always be found. The simplest n.ethod of
doing this .s by comparing the local time at the two
places for the same instant. This is done by signa of
some kind or other, such as flashing the sun's'ray's from
station to station, or by the electric telegraph.
Since the earth revolves through 360 degrees of hour
angle ,n 24 hours it will pass through 15 degrees in i hour
That ,s when it is one o'clock in the afternoon at a certain
ea t of It. h fteen minutes and seconds of longitude in arc
tt w Th '' 'v °"^ "'""^^ ^"^ ^^^-^ °^''-^ -P -
Th r- Vl^ ^P?'"' '° ''^''^"' ^^ ^^" ^« to mean time
rhat.s If the sidereal clock showed i honr at one placTu
wrtir\"""'^'!"°^'^^- (T^'^-^W--^,.L^
. IS wor h thinking out, for it b often a puSetobeX^
Therefore, if at a preconcerted signal the observer atTwo
stations note the exact local time, either mean or s der II
he difference of the two will give the difference of long .'
tude Ordinary watches may be used if their exact rate
and their error at any given instant are known. The
the irsf T""' " '^ telegraphing star transits. If
he eastern observer signals at the instant that a certain
star IS on the meridian, and the western observer no "
he time of the signal by his sidereal chronometer and
afterwards takes the time of the star's transi at hfs
own station, the interval of time between the two ralit
allowing for the clock's rate, will evidently give the differ
ence o longitude. If the eastern observer note the fme
statoVThT''"'''^"^^^"" ^'^"^^^^ ^he transilat hi
station, the same result will be obtained, and by taking
the mean any time lost in the transmission of the signals
W.11 be corrected ; for it is evident that in the hrsf cat
the time lost will make the difference of longitude too
/ '<.<i ».
40
Longitude.
small, and in the second case it will make it too large.
By means of certain contrivances it is possible to register
the instant of a transit to a small fraction of a second,
and if a number of observations are taken the mean of
the results will be very near the truth.
Since an error of one second in the time will throw the
longitude out by about 360 yards in latitude 45° it is evi-
dent that for surveying purposes great care must be taken
to insure accuracy. When the local times are compared
by flashing signals a large number of observations should
be made and the results compared.
The subject of longitude will be more fully gone into
hereafter. It may be mentioned here that sailors obtain
their longitude by finding the ship's local mean time by
an altitude of the sun when the latter is about three hours
from the meridian, and comparing it with a chronometer
keeping Greenwich mean time, the latter being noted at
the instant the altitude is taken. It seems almost need-
less to remark that when using chronometers tiie correc-
tion for rate must always be applied.
METHODS OF FINDING TIIE MERIDIAN.
TO FIND THE AZIMUTH OF A HEAVENLY BODY FROM ITS
OBSERVED ALTITUDE.
This is a very similar problem to that of finding the
hour angle from an altitude; the only difference being
that, instead of finding the angle P of the triangle P Z S,
we have to find the angle Z. We have, as before, the
three sides of the triangle given, and may therefore use
the formula
Sin» ~ =
^ - Sin (s—Z S) Si n (s— Z P)
2
Sin Z S Sin Z P
Another formula that may be employed is
a
ti
V
t£
sc
la
ti<
ra
ed
of
tai
vei
• i-:.^^si »: m '' m> ' -ss im
Azimuth by an Altitude.
41
Cos» 1^ Cos 5 Cos (s-S P) Sec; Sec a
where a is the altitude of the object. S Pits polar dis-
tance, X the latitude, and s= "+'* + ^±
ve>^^ ^^^^^^.^.^^.r^ort^ce to sur-
^"^h as alSl^^e' "" ^*-^^'"^"th instrument,
of any line Z A (Fie Vo : ^'^^^^^^""^'"'cal bearing
directing the tel scope on th"e L°""' "J''^ "^^ ^^
horizontal plate reading and Then" ""'' *'''"^*'^
urmng it on the heavenly body and
S^r'd-'"'"'; ^"' the'horilTal
plate reading. It is better to repeat
the observation in reversed positions
of the instrument and take the mean
eneorf°'"^^^^ ^^^ediffer-
llne and 5h T'''"'^^ '""^^"^^ °" the
hne and the heavenly body gives the
angle A Z S, and the triangle P7^ u !^ "'
the angle P Z S, whence vSh a "?"" '^'"^"^ ^'^««
bearing, and therefore Z P fh^".^''^ ^ -^ P the required
In taking an alt-Xnth of .k ''''°" °^ *^^ "^^"'^ian.
altitudeonf, we mu"" ^^'s fbUct' th "^ '^'^ ' ^'"^^^
to get the altitude of the centre rfn' f"^'"^^^-^*^-
vertical and horizontal wire the sun! " '" ^
tangential to both. To ^et the Z%^^^^ '' '""^^
semi-diameterwemustmnf. 1 u ^""^^ correction for
Jatter by the seTanT .f T ? ^ '^" ^^'"""^^ ^^^^e of the
tions ar^ got ri<CoCv?nftr'^' ^°^' ''^'^ --'
rants of the crosVLre" Th^t^ °^P°"^^ ^"^^■
edge of the sun tangential ^o on^ JJ '" ^' *° ^^^P °»«
of aslow-motion scfew ^ the o,h ^', ^'"'^ ^^ "^^^"^
tangential. Thus, weight let th ' '' '"°"^^ ^^^°
vertical wire a ittle Zd J .^ '"" °^"''^P the
"ttle and keep ,t tangential to the
42
Azimuth by an Altitude.
horizontal wire by the vertical slow-motion screw till it
just touches the former. If
the wires of the theodolite are
arranged as in Fig. ii we take
the observation as follows.
Suppose the time to be fore-
noon and the apparent motion
of the sun in the direction of
the arrow. For the first ob-
servation get the sun tangen-
tial to the wires a b and e f
in the uppper position. This ^'fi'' "•
is done by making it overlap a 6 a little, and using
the vertical arc slow-motion screw to keep the lower
edge tangential to the horizontal wire e/ until the sun also
touches a b, when the verniers are read off. The instru-
ment is then reversed and the sun made tangential to the
two wires in the lower position, this time using the
horizontal plate slow-motion screw to keep the edge tan-
gential to a b. The mean of the two altitudes is taken
and also that of the two horizontal readings. The time
must also be noted so as to correct the declination. The
reading of the horizontal plate when the telescope is
turned on the referring mark (A) is taken both before and
after the sun observations.
Ex. At Kingston in latitude 44° 13' 40" o" the 3rd of
March, 1882, at 2h. 3°"!. p.m., or ^h. 36m. Greenwich
mean time, two altitudes were taken of the sun with a
transit theodolite in reversed positions for the purpose of
testing the accuracy of a north and south line, the hori-
zontal arc being first clamped at zero, and the telescope
directed northwards along the line.
READINGS ON SUN.
Altitude, Azimuth.
ist observation— 31° 8' 220"' o'
2nd •' 30 16 220 16
Mean 30 42 220 8
BttdPVWiRIlKfWWPMi'MK
^-fym
i
Azimuth by an Altitude.
43
To Correct the Altitude. To Correct the Declination.
30 42 o' Decimation at Mean) ^o
Refraction- 136 Noon at Greenwich ^ 44' 37" S
Parallax+- 8 Correction fory^hrs.
Truealtitudeao 40 32 at-57". 5 per hour} 7 3i S
True declination 6 37 6 S
90
Formula used ;
Sun's N. P. D.=96 ^y 6
Cos.2-^=cos. s. cos. (s— S P) sec. yl sec. a
**= 30° 40' 32'
^= 44 13 40
S P= 96 S7 6
2)171
31
18
s= 85
96
45 39
37 6
s-SP=.io
51
27
log COS s= 8.8687314
ogcos(s— SP)= 9.9921540
ogseca =10.0654637
log sec A =10.1447380
2) 39.0710881
, 19-5355440
=io+logcos69°56'
2
^=139° 52
360 o
Sun's Azimuth= 220* 8'
The direction of the line was therefore true.
The formula —
PZS
Cos2-- =cos s cos (s-PS) sec X sec a
where PS is the sun's (or star's) polar distance, a its alti-
tude, ; the latitude, and s=l^^, is thus derived.
We have, in the triangle PZS, if ^'- P'^+PS+ZS
2
Cos8?^z= ?HLil!i2jLz:PS)
^ 2 sin PZ sin ZS
Sin (s'-PS)=cos {9o-(.s'-«i^S)|
■ (I
44
Meridian by Equal Altitudes of a Star.
=cos
180-PZ-ZS + PS
2
X+a+FS
=cos =cos s
2
COS (s-PS)= cos9°-^^+9°-^S-PS
2
— ^^= i«« PZ+PS+ZSl . ,
= cos jgo [ = sins
Therefore cos»^^= cos s cos (s-PS)
2 cos X cos a
Similarly it may be shown that
sin*-^-= ^QS ^ s in (s—a)
2 ~~ cos ^ sin PS
TO FIND THE MERIDIAN BY EQUAL ALTITUDES OF A STAR.
Select a star which describes a good large arc in the
sky, and having levelled the theodolite direct the tele-
scope on it about two hours before it attains its greatest
height. Clamp both arcs, and by means of the slow mo-
tion screws get the star exactly at the intersection of the
wires. Having taken the reading of the horizontal arc,
leave the vertical one clamped, loosen the upper horizon-
tal plate, and look out for the star when it has nearly
come down again to the same altitude. When it enters
the field of view follow it with the telescope, using the
horizontal slow-motion screw, but still keeping the verti-
cal arc clamped, till it is exactly at the intersection of the
wires. Now read the horizontal plate : the mean of the
two readings will give the direction of the meridian. Set
the plate at that reading and send out an assistant with a
lantern. Get the latter exactly at the intersection of the
wires, and drive in pickets at the lantern and theodolite
station.
This method is rather a tedious one, but it mav be
shortened by observmg the starwheri nearer the meridian.
%*
Meridian by Transit of Pole Star.
be
45
TO FIND ^^HRinUN B AX OBS^^^::^^;^, THE POLE
: e^ STAR AT ITS MERIDIAN TRANSIT.
Ascertain the watch error bv anv of +ho ^-
i ' pole f 1° V'. rf "*?"' ^'"'^^ ^''°- - ''^'ow 'he
» pole. If the theodohte telescope is directed on it at thi,
.nstan. we shall evidently have the meridian Le prUd
ed the .nstrument ,s in good adjustment. But it s betler
n order to eliminate instrumental errors, to proceed ^
follows : at some definite time before the star wiU be o"
the mend,an-say. minutes-direct the telescope on ^
and take the reading of the horizontal plate. Reveri
Tnthk ., "•= '=5™ ""^™l of time after the transit-
in th.s , <^..,^ce 4 mmutes after the first observation-and
TO FIND THE MERIDIAN BY THE GREATEST ELONGATION
OF A CIRCUMPOLAR STAR. "'^CATION
nni^^!f '' ^ ""^'^ ^''"'"^*' "^'*^°'^- S*^" ^hich. like the
pole star, are very near the pole, owing to their slow
mo ion appear to n.ove vertically for some ht tZ
when at their greatest eastern or western elongation. We
will suppose the pole star to be the one observed. Aboul
s^x hours before or after the transit (the time of which
Z:u h' ^r.T^' calculated) the theodolite is carefully
levelled, its telescope directed on the star, and the hor^
zontal plate read. The operation is th;n repeated n
reversed positions of the instrument and the mean of the
two readings taken. If we have previously taken the
horizontal plate reading when the telescope was turned
onsome well defined distant object as a referring wo k
we can now obtain the azimuth of the latter as follows •
t* II BfijiJl^ U
' '' ■' I 'w-W | (i'^ ^ i iiM Mi »m i wi..i
t I
46
Meridian by Greatest Elongation.
Fig. 12.
In Fig. "" J let the plane of the paper
represent t le plane of the horizon, and
let Z be the observer's position, A the
referring mark, P the pole, and S the star
at its greatest eastern elongation . P Z S
will be a spherical triangle, right-angled
at S, and we shall have :
„. „ ^ Sin P S
Sin P Z S=^. — 5-^
Sin P Z
or, if 8 is the star's declination and X the
latitude of the place :
SinPZS=^i4
Cos /
since P S is the complement of the declination, and
P Z the complement of the latitude. The latitude need
not be very accurately known.
Now, having from this equation previously found the
angle P Z S, and having obtained A Z S from the plate
readings, we get at once the angle A Z P, which is the re-
quired azimuth.
If we have to use a star some distance f-om the pole
we must calculate the time of its greatest elongation by
solving the equation.
Cos Z P S=cot 3 tan X,
which gives us the star's hour angle, and hence the time
of the observation.
The altitude is given by the equation.
sin^
Sin. altitude:
sin d
If it is inconvenient to observe the pole star at its
greatest elongation we can use the following formula
which is approximately true in the case of a star very near
the pole.
tan A> • -7 Tj c
;;: 7-=sin Z P S
tan A
where A' is the star's azimuth, Z P S its hour angle
at the time of observation, and A its azimuth at greatest
elongation.
■>_
lu^ui^fm-miui. tn— r-in--Ti-iiTi-— r — r ntK tir'nninr-iin— ii-ni-i>wwTlr:itiiim i i t
J^''^'^^"«« h Hifrh and Low Stars.
47
TO FIND THE MERIDIAN BY OBSERVATIONS OF HIGH AND
LOW STARS.
This is a very useful method, as it is independent of the
pole star, and can therefore be employed in the southern
hcnusphere where that star is not visible.
Choose two stars differing but little in right ascension
oneof wh.ch culminates near the .enith and the othe;
near the south horizon (or the north horizon if in the
o hern hemisphere.) Level the theodolite very care
of the Vr ^"'' "^'^^ ^"^P^ °"^ ^^^ *^^ collimation Hne
of he telescope W.11 coincide with the meridian at the
^enith, however far U may be from it at the hori.or • and
the field T"*"^^ "'^'" -'^ ^^"^^^ -" -°^« the cent're of
he field of view at nearly the same time as if the ele
horizon Having^L^;,^^^^^^^^^^^ ^^^
stars will cross the meridian observe the tranlit of fZ
upper star, noting the watch time. Th s wm give the
watch error approximately, and we shall now knTw the
transit. By keeping the telescope turned on that starti
that mstant arrives we shall get it very nearly n th"
plane of the meridian ; and by repeating the process with
another pair of high and low stars we sh^all have the di^c
tion of the meridian with great exactness.
For this method we require a transit theodolite fitted
with a diagonal eye piece. The nearer the upper stars
are to the zenith the better. ^^ ^
The Canadian Government Manual of Survey recom
mends for azimuth the formula : ^
tan. P Z s=^^LPA55?jii!nAP^S
i-tanPStan/lcosZPS
as applied to observations of the pole star- but it r^
quires special tables in order to work it out
i WW,' ejl H P H I ^ ^i» B^.i ! . 'i i , - i"^ flB|BWWp W WWHBWllimwP
48
Meridian by Pole Star.
The following is the proof of this formula :
We have the fundamental formulce —
1
f sin fl sin C = sin c sin A
I ^ cos c — cos a cos b
cos C= ; ; — r
sm a sin b
COS. a — cos b cos c
cos A:
(I)
(2)
(3)
sm b sm c
From (3), cos a cos 6=cos^ b cos c-(-sin b sin c cos 6 cos A
^, cos c — cos a cos b
From (i & 2), cot C = -; -. — j-—. — -^ —
riuiii v* «. -a/, sm a sin 6 sm C
cos c — cos' b cos c— sin 6 cos b sin c cos A
•.tan C =
sin b sin c sin A
sin b sin c sin A
sin' b cos c — sin b cos 6 sin c cos A
cosec 6 tan c sin A
iii
I — cot b tan c cos A
In the triangle P Z S let P S=c, Z S= a, aud P -Tlrrfc
Z=CandP=A
Then—
cosecPZtanPSsin^PS
tan l'^S-^_^^^p2tanPScosZPS
tan P S sec ^ sin Z P S
■ I— tan P S tan ;i cos Z P S
t
t]
w
w
eq
ha
sh
tai
OS A
OS A
CHAPTER V.
SUM DIALS.
thrown on anv olanp «nrf=„ ''«<=earm, its shadow
of the sun alwaCn. ll """' '^" S'ven hour angle
be the sun's d™Hn 1" ' t^T T '^'" ''"" "■'^'""
Poin, in .h. line Slve asVhetT"'.'"^ "^"'"'^
will always lie in .he same straitlt H„« f" ™™'' ""
a"gk. On this principd al f„ ! ? '"'' «""" '"'"
The position of th'e sh?dow 1:: h^^.s'^r'™^'^-
at the instanf anri fi, r • "" ^ '^our angle
•tae; so tha in^"!',"::^^;';:"*-;- ""e .>^.„„, ,„L
>.ave to applv the equatL ofTme ""' """" "»' "'
.he°,ltercas?r"LH'*".'"'r°"'^' " -'->• ^
thrown on a ho^'o'a, Ite ttheT^ " '= ""^''- «
wall. P ^^ ' '" *^^ Matter on a vertical
equator the d'al wonM evidertl ""^ "' ""^ P°''^- A' the
having a hori.orta 'edg 'S'™^'!,' "' % -"'-' P'a.e
shadow lines would be P^a fd fo he ,L ar/tt ' ^'^
'ances apart for equal intervals of 'ti^ Cu d'"; ^^
mm
50
Sun Dials.
I
increase according to the sun's distance from the meridian,
and would become indefinitely great when he was on the
horizon.
At the poles the stile would be a fine vertical rod, from
the base of which 24 straight lines, radiating at mtervala
of 15 degrees, would indicate the hours. The line on
which the shadow was thrown at the time corresponding
to Greenwich mean noon might be assumed as the zero
or 24-hour line. At other places the stile must be set so
that its angle of elevation above the horizontal plane is
the same as the latitude of the place.
HORIZONTAL DIALS.
A horizontal dial generally consists of a triangular
metal stile fixed on a horizontal plate on the top of a
pillar. Fig. 13 is n elevation and Fig.
14 a plan. The angle of elevation of
the stile is made equal to the latitude
of the place, and if the variation of the
compass is known, the latter may be
used to get the dial with its stile in the
plane of the meridian. The hour lines
on the plate are marked out thus : let
Fig. 13. A B (Fig 14) be the base of the stile.
Fig.
14.
and A its south end. Draw A C so th^t n x~r~ 7~
the ,ati.„de and at any point^C^irSt C B^r
"ne E B F pTpta'Lfa,".; Isn'MtT^rfT'
D », D .., D b. D 4., &c., n,ee,i„g E B F in . j'!""
and n,akh,g ,ha angles B D ., b'd a'. I S ^.^ a, b ,."
ac, each equal to 1=; detrrees Fr^r^ a j y ^ u ,
lines .H.on,2 a. , „.,^, r%.::rj t^^e'^t
i'nes : A . for 9 A. m., A t for lo a. m., A a for „ am
and so on The proof of the correctnes of this con't'ruc"
tion ,s easily seen by imagining the trianele A R r , I,
turned round A B till i. is perp^endicil r tttha pLe" of
he paper or dial plate, and the triangle c D c'T K
turned up on . .. till it abuts on B C when D w n °
cde wuh C, and A C will be parallel lortepoIaT^isTd"
perpendicular to the plane of D cc'.
When the divisions on the line E B F run off the niaf .
we contmue them thus: In A C (the , p m 1 n.l , \
any point », and through it draw a H e para ieHo^ S
9 AM. l,„e) meeting A 6., A .. &., fn A >' & and
make », equal to .A o ,• to op^, &c.. and ttiugh'; ,"
onlhfothrsTdt"""" "™= '"'"= -orninghour lines
VERTICAL DIALS.
These have the advantage that they may ^e made nf n
very large size and placed in conspic L ^osl dot
There are vanous ways of constructing them As^Zle
.^ trnro'f '^ ^°,f^ \''' ''-'' havi^gaLndh^t
It, m front of a wall with a southerly aspect. (Fig. 15.)
1
i
52
Vertical Sun Dials.
Fig. 15.
The disk should be roughly per-
pendicular to the sun's rays at
noon about the equinoxes. The
The bright spot in the middle of
the shadow of the disk on the
wall indicates the hour. Tb'
hoi- lines are found thus: At
the time the sun is on the meri-
dian mark the position of the
bright spot on the v/all. Let A
be the hole in the disk and B
the spot. Measure A B. Through
B draw B C vertical, and draw a line B D so that B D is
equal to A B, and the angle C B D to the sun's polar
distance minus the co-latitude. Make the angle B D C
equal to the supplement of the sun's polar distance. It fol-
lows from this construction that if the triangle BCD were
turned round B C till it touched A the points D and A
would coincide, and C D (and therefore the imaginary
line C A) v^ould be parallel to the polar axis. Now take
a watch, set to noon at the time of the sun's transit, and
mark the positions of the spot on the wall at the success-
ive hours. Straight lines joining these points with C will be
the hour lines.
Of course a large triangular stile CAB might be
substituted for the disk ; o- we might use a rod C A fixed
in the plane of the meridian, and having the angle A C B
(which it makes with the verticaf equal^ to the co-latitude.
CHAPTER VI.
THE REFRACTING TELESCOPE
other. The former 1h ' '"'' " ^^= Pi"'^ =" ">e
objec. a. i.s foZi:; ^31" ^.rjs' ™^^^ °' '"»
tographic camera and th. ^- ''"' °' " P^o-
"u. ifihe .eSpe -^ :ar:er.: s"h""fH'''^r '^"^^
natural position a 00™^™!"™ of fo 7 °''''" '" "^
by means of which tt^Tu^T^ "' '""""^ '= 'mployed,
This has. however tl!e H? . ""^^^ " ^«='" '"^"'^d-
much ligi, and L '1 '^"f."^"'^^'' "f <^utlin^ off too
land objfct;. '' "'"' '" """" '^'«-°Pes and for
"ne^tivZ-'in^LHrh tte" '^ ''-'7 '■''"<''•• «-' '»^
t»o lense; of the et pLe "^h" '"T^" '"'""" "'-
"^ed in telescopes desi/ned for Th " "" "'"" ^=™""->'
objects without makTnf „/ ■""" examination of
"positive," in whkh h! Z T"'"' ^"""-"J"' 'be
is outside the eye pLee rr rf'''^'™'="=^^
'---Wean?-teto;rrra;:i:rs
54
The Telescope.
plane at the common focus of the object glass and eye
piece. The position of the focus of the former depends
on the distance of the object — that of the latter on the
eye of the observer. The one is the same for every indi-
vidual. The other has to be adjusted to suit the observer
— short-sighted people having to push the eye piece in,
while those who have long sight require a longer focus.
The larger the object glass is the more rays from the
object are collected on the image, and the brighter it is.
The greater the magnifying power of the eye pic. e the
more apparent are any defects of definition in the image.
The magnifying power of the telescope is measured by
the fraction
focal length of object glass.
Thus, if this
focal length of eye piece,
fraction were 4, the linear dimensions of the object seen
through the telescope would be four times what they
would be when vjewed with the naked eye. Therefore^, ^,
for a given eye piece, the longer the telescope is.^he"
smaller will be the field of view, or portion of the earth
or sky visible. 1 he angular diameter of the field is, in
fact, the angle subtended by the diameter of the eye
piece at the centre of the object glass.
In large telescopes the field of view is so small that it
is necessary to use a "finder," which is simply a small
telescope attached to it so that the axes of the two shall
be parallel.
A diagonal eye piece is one in which there is a mirror or
prism between its two lenses by which the rays of light
are turned at right angles and emerge from the side instead
of the end of the eye piece. It is used for observing
objects when the altitude is so great that it would be
uncomfortable or impossible to look up through the tele-
scope tube.
Lenses have to be corrected for chromatic aberration and
spherical aberration. Take the case of an object glass con-
-7^
P
it
ar
na
gh
m]
pr:
The Telescope. '
have different foci tL u °[ '^^'^"Si^ihty would
other. Such a lens is calle/ ..■r;^hr:Sic!" """" '""
By "spherical aberration ' ;.. mpai-l th. Hi.
rays caused by the central pc-, ^of let „ ^ T" "
surfaces having a different focu. om its ou J Ir '' "'I
St c^bitTor'^^ ^^ "■' -CeTtrr -r
pro;tiy"r:tedr:ifri'st °ut^^ ^'-= ^- ■!-
Spherical aberration is detprt^ri k,
portion of the lens wi h a ctc'la^dTrT*'^""*^^^
focusing it on an obie.r .V . ^ °^ P^P^'" ^"^
6 1 uu dn ODject, afterwards removmo- *i,^ ^ i
and covering the oiitPrr^orf -^u . "^ ''-'"oving the disk
.he focus o.'ZTZZT'ZT'''''''' "'-- -»-
If one part of the object pla<?Q f, .c: o ^•«-
power from another pa a b ^ht star w.'u Tr' "'"f"^
.nadiation, or m„ft at one side '^'^'" "'"' ""
THE MICROMETER
anjuiLTtrcr itis^r'rr- "--r^--"
nary theodohte, placed In ,e common fo^rofl'" "j"
glass and eye piece of a telescopror of the ' "d"'
microscope ■• which will be described pittlvTb''
Pnncple of the micrometer is simply thisfrpte tha
I
'.Wf
MMM-»k^..
m I
56
The Micrometer.
a point— such as the intersection of the cross wires-can
be moved across the field of view by means of a screw.
Let a be the angular diameter of the field, and let n be the
number of complete turns of the screw required to move
the wires through this space ; it is evident that one turn
of the screw will move them through an angle ^
The head of the screw should of course have an index to
mark the commencement and end of each turn. If the
head is made large enough to enable its rim to be divided
mto m equal parts we shall have the means of measuring
^" ^"^'^ l^nr ^y fuming the head through one division.
Thus, by making the thread of the screw sufficiently fine,
and Its head large enough, we have the means of measur-
ing small angles to an ^ktreme degree of accuracy, pro-
vided we know the angular value of one turn of the screw.
This may be ascertained by finding how many turns it
takes to move the wire across the image of an object of
known dimensions at a known distance. A levelling rod
will answer the purpose. The length of rod moved over
divided by the distance, gives, of course, the chord of the
angle subtended. There is usually a scale in the field of
view, the divisions of which correspond to the turns of
the screw.
Fig. 16.
There are several different forms of micrometer. The
accompanying figure {16) represents the one known as the
filar (or thread) micrometer. Two parallel wires m m,
The Micrometer.
57
^* slides .v,H„ a: r: .e'dSrr tH- ^"-^ ^^™^
the wire w w, and is mnv J k Tu ^ ^'"^"'^ '^ ^^"'^s
which has a graduated r 1 t^r^tL^'- ^'^ '^^' °^
'"oved by the screw B. In the ovZ ""''/ '' ^ ^"^ ^'^
2onta] wire at r.VJ^f , ' opening of « is ;; hori-
"av. ,0 ;Ls r fh :„TLVd° l'^ °'!:"^- ^"PPo- -«
->tal wire. If „o,v the screw A " .r„e7tnr ' '"'"•
one star, and B till n n cut« th? .V , '" " '^'"^
•ween the two is measured J ,^ "■'. """ *^'''"« be.
and fractional div" .^n^ofa t'„™ ItTakes'^t'-- "' ^
to » n. This is not theexact m.,f i / ""«^ "" ™ "P
"„, it serves to illnstr^e the prtadple ''""''"" ''""'"^^'
THE READING MICROSCOPE.
for reading the facH ai part^ o^f hT^' °' "^™'"
graduated circles ofl-,™.-. divisions of the
is fi«d, the ciS VZ'IZZTTL t^ """°="°p^
■nstrument and moving with it Th ''°P' °' '^^
one screw and moveabk f ame J*? ™"°'"^'" has only
of cross-wires in the common foculahTr^' T'^
and eye piece of the microscone Tl? '^"^ «'"'
used in exactly the same w!*^ 1^'^ cross-wires are
'elescope. only that thToh^ "• """'^ "^ " "-eodolite
arc, on which fh micrt eoolm Tf '^ ""^ S^-<i„ated
To make the matter cTearTe ^ffl' t° feTT ' "' '°''"'''-
measurement of a horizon,.] "' ^. "" '^^'^ of the
the arc of which is Sa'd'f '' " '"^' """"o'"'
<hat oneturn ofth^ij ort,rsc°rew isT'"', """'"'^^
"■nute, and that its head ^^1 j 7 equivalent to one
have thus the me^ns of meas" fe. "tt '° T"^- ^'
seconds. The circle with it a "fh d ,7* ° '° ""«''
vo.ved, and the cross-wires of thT^t^rdTtrLl-
li
:ax.
58
The Reading Microscope.
with one of the objects. On viewing the arc through the
microscope (which it must be remembered is a fixture)
the wire intersection of the latter must be made to coin-
cide with one of the divisions of the arc by means of the
micrometer screw, and the reading of the index of the
latter noted. Suppose the arc reading to be 10° 20', and
that of the screw head 15". Now move the circle and
bring the telescope to bear on the other object. The
cross-wires of the microscope will probably fall some-
where between two divisions of the arc, say between 50*
30' and 50* 40'. Turn the screw till the cross-wire is on
the 50* 30' division, and suppose that it takes between
three and four turns, and that the index marks 25'. The
micrometer wire will have been moved 3' 10", and the true
reading of the second object will be 50* 33 10". The
angle measured is therefore 40° 13' 10".
Two or more reading microscopes are placed at equal
distances round the circle, and the readings of all taken.
Errors due to eccentricity are thus got rid of, and those
due to faulty graduation and observation much diminished.
THE SPIRIT LEVEL.
The spirit level is used, not only to bring certain lines
of an instrument as nearly as possible into a horizontal
position, but also to measure the deviation of these lines
from the horizontal. For this purpose the glass tube is
graduated, usually from its middle towards both ends,
and the reading of the ends of the air bubble noted. The
length of the bubi .0 depends upon the temperature, and
the latter should therefore be also noted.
To obtain the value of one division of the level — that is,
of the vertical angle through which the level must be
moved in order that the ends of the bubble may be dis-
placed one division — a simple plan is to rest the level
upon some support (such as the horizontal plate of a
theodolite) that can be moved vertically and which is con-
^■■- i
)
The Spirit Level.
59
n«,ed„ith a telescope. The plate is levelled, the tele-
scope directed on a vertical measuring rod set up a- a
known distance and readings taken of the ends of the
.he r^d^Tht ' h'^ "'"""'"" °f ""= ■'""-■"^l -i- «Sh
the rod. The whole arrangement is then moved vertically
by means of the foot screws till the ends of the bubb e Save
^TofL'Tr """"" °"ivisions-say xo. T read-
ing of the telescope wire on the rod is now noted The
^fesThrch" rd f^n' "^''"^^' '"''^' '^y "' "ttaJce
frtht'dSd°;'t;r:th:Tar "°r v '=-''
CM uy iu gives the value cf one division.
In the case of the level of the transit telescope at the
Royal Military College ,t was found that at a dTs an e tf
383 feet a vertical movement that displaced the bubble
20 divisions altered the reading on a levelling saffo.i
t!f . r l"'"'""^ '^'='^™^' S've 0.0000313 as the
tangent of the subtended angle for one divLon, wh ch
made the value of the latter 6-.45.
J''l''VT °^ ^ ""'''"^ '«"^'' if "' <■«'. or the surface
other hand if the legs are of equal length, but the surface
of the level A and the o.Tef B. yhriev;i'eror tr^re
2f^:rel-:t-iL-\s?f-di-
greatest m one position. B will be greatest bv the same
amount on reversal. ' ^^^
especially in the case of the more delicate levels which
eas ly get out of adjustment. The amount of slopelf the
surface tested by the level is obtained thus: Take Jhe
"iiMlfr'-f
t^ntmtt?mm»i»^uaami»i^:'
I
i '!
60
The Spirit Level.
case of the pivots of a transit telescope. Placing the level
upon them take the readings of the bubble ends, and call
the reading next the west pivot W and the other E.
Then reversing the level take the readings over again and
call them W^ and E^ Thenumberof divisions by which
the bubble is displaced by the difference of level of the
pivots is given by the formula :
W + W^— (E+E^^
4
To find the actual slope of the pivots we must multiply
this quantity by the value of a division of the level.
The level error is obtained separately by simply chang-
ing the signs of W^ and E^ in the above formula, when
we have:
, , W— Wi— E + EJ W-E— (Wi~En
Level error = = i-I- tL. '
4 4
Of course if W+W ^ = E + E 1 there is no slope, and, in
practice, when the level is out of adjustment, we may get
the points of support horizontal by raising one of them
till this is the case: For instance; if W were «o and E
10, W^ would have to be 10 and E^ 20.
If the level is in adjustment we must have
W— E=Wi— El
In this case we have only to take W and E and the
slope is obtained from the formula
W E
X value of one division
Example — Take W=25, E=io, Wi=ri5, E»=20.
Value of one division =6".
Here we have '^ + ^g-io- jp^ jo^
4 4
Multiplying this by 6" we have 15" as the slope, the v. est
pivot being the highest.
The level error is — ^^^^ io-f-20 ^ ^n__ „
J
^ ^g Chronometer. i
two positions TherfL ,. °' '■^^'""S^ '■•' the
i...^ a. the L endirr;:; J ' --„i^>''''"-f
readings will give the slope. ^^^^ ""^ *^«
tail^7o?d»i«t^l:",j2t^ "^'' ^--" "' --
wh];Hr^t?„Vh":Lott';;iiS'''' =-^-' <->
by means of the ad- -tin, '^'^''■"•'°>"'- Then,
both its ends read ^tZ ' ""°'= '"= ^'■'^'>'' «"
THE CHRONOMETER
ourf^'nTa^^ranSSrh r ""'"' ^'^^ ""•"
changes of ,emperatu™have. he?"'r° "'^''™'^''^ "■"
the time of its osc llation ?» P"'"'"''"'''" "P°»
structed to Iceep either "l^^'''""™'^'"^ ""^X be con-
used on board BriSh shil "' ''"f^'' '™e. Those
wich mean time The Jea'-rnf"''' 't^''°" <^'-»-
that it should keep a regjlar rate th",' "^T"™"^"- '^
0"ly gain or lose a certf.n 12 ', "' ""' " ^'""'W
•his can be depe^ ed on ^ cfn T" "™- "
the true time at any instant h, T ^^^ ascertain
numberofdavsandhoaJst.lr''''?''"^''''' '*'= ^°' «he
error of the chronom r ^L la" d^ "^'"^ *"""•""
by comparison with other cZnteterroXt """"^
mical observation Th« r, "'"eters or by an a rono-
-re perfect ir.he c^ronrir ris^^, "'= "'P' '"«
venient to have a small than^a^^rge rltrtfalTr^-
eight daT'T^errr"^"^ ""^"^ '° -" ^""er tw^ or
seventh day. IttiZlTrr"^'"'' ""^ '>«- e-ary
"P at the rlgula: in e7:ra ' fie tr'™.'" '? " "'
-ed part of the spring c^m^f t'to'X, Tnd' ir^
I'm
t
ll
J
I
i
i 1
62
The Elecho Chronograph.
larity of rate may result. If a chronometer has run di-^vn
it requires a quick rotatory movement to start it aftt' ,'t
has been wound.
Transporting — On board sbip chronoireters are allowed
to swing freely in their gimba;-. so that t^.y may keep 3
horizontal position ; but en land they should be fastened
with a clamp. Pocket chrononieters should 'ilways be
kept in the same posifion, and if carried in the pocket in
the day shou'd be huag up at night.
Chronon;eiers h.ave usually a different rate when
travelling from WiuU: they keep when stationary. The
travelling rate r, ^y be found by comparing observr'tions
for time taken ai the same place before and after a
journey, or fro in observations at two places of which the
difference of longitude is known.
For mean time observations an ordinary watch may be
used by comparing it with the chronometer, provided ihe
rate of the watch is known.
Chronometers are generally made to beat half seconds.
THE ELECTRO CHRONOGRAPH.
Under this head may be included all contrivances for
registering small intervals of time by visible marks pro-
duced by an electro magnet, and thus recording to a
precise fraction of a second the actual instant of an
occurrance. By this means an observer at a station A
can record at a distant station B the exact instant at
which a given star passes his meridian, and thus the
difference of longitude of the two stations may be ascer-
tained.
REFLECTING INSTRUMENTS.
THE SEXTANT.
A person accustomed to work with the pocket si ^ nt
will have little diffit in using the larger kint ■ the
latter, with its adjai,.»i;ents, is so fully describe :^ most
J
- ' i
IL
J
The Sextant.
63
OntnT .:"^"^>';"^;hat little need be said ^ho~:^^^r^,
of.erc. ,,,,,t::r:2
or with a plate of glass floating on the mercurv Th;'
oof. when one haif of a set of observa.io,. has been ,aken
the roof should be reversed end for end R„r ,t I
sun. double altitude the da. glasl ^the ^ pt^,^
e:c^tj;;:t::h^.f^--
noted when the circles just touch. As this requir s h"
he images should be receding from each other he lui
tude of the lower limb must be taken in the for noon and
of the upper limb in the afternoon. For a lunar distance
of the sun direct the telescope on the moon and use one
or more of the hinged dark glasses for the sun The
A common fault of the sextant is that the optical power
of the telescope :s too small. There is httle use in beTng
ab e to read the graduation to .0 seconds if the eye cTn
not be sure of the contact of the images within 30^
THE SIMPLE REFLECTING CIRCLE
This is Simply a sextant with its arc graduated for the
Ic rt"""T"°''""'^'^"^^^'^^^h^-dexarmp o-
at h enT Th'' ""'''''1 T'"^^ ^"^ carrying a vernier
a each end. The mean of the two verniers can be taken
g t dd T'tT ^"' ^"^ '-'- ''- ^° eccentricity^!;::
got nd of. This arrangement also tends to diminish the
errors of graduation and observation
of i^r reflecting circles have three verniers at intervals
.jM
64
The Repeating Reflecting Circle.
THE REPEATING REFLECTING CIRCLE.
In the repeating reflecting circle the liori^on glass
(m Fip, 17), instead
of being immovable,
is attached to an
arm which revolves
about the centre of
the instrument and
which also carries
the telescope (0 and
a vernier (v). The
index glass (Af) is
^'^- ^7- carried on another
revolving arm, which also has a vernier vK The arc is
graduated from 0° to 720° in the direction of the hands of
a clock. To use the instrument the index arm is clamped
and its reading taken. The telescope is then directed on
the right hand object (6), the circle revolved till the
images coincide, and the telescope arm clamped. The
index arm is then undamped, the telescope directed on
the left hand object (a), and the index moved forward till
the images again coincide, when its vernier is read. The
difference between the two readings of the index vernier
IS twice the angle between the objects. This repeating
process may be carried on for any even number of times.
The first and last readings only are taken, and their
difference, divided by the number of J*p**i4i©»8, gives the
angle. If the angle is changing, as in the case of an alti-
tude, the result will be the mean of the angles observed,
and the time of each observation having been noted the
mean of the times is taken.
This instrument will not measure a greater angle than
the sextant. Its advantages over the latter are that there
15 no index error, and errors of reading, graduation, and
eccentricity are all nearly eliminated by taking a sutKcient
number of cross-observations.
f,
<■ ' /.i^.
i/<-.
,'
■
1
bi
J
about 4,' """f """""'lescope arm at an angle of
^ei.h. :/it;:,-,:':e<,';r„^t; LiT'^r ^^*»
altitude, of Ob;:" ^^L'lhelrUh. """ '"' '"""^ ^°""«
In ,t, ™-^ '""■SMATIC REFLECTING CIRCLE
I fixed prismatic reflector (/)
which halfcovers the object
klass. The index mirror
(w) IS carried on an arm
which revolves round the
centre of the circle and has
a vernier (v v^) at both
ends. This instrument will
I measure anj-les of any di-
'mension, and has also the
following; advantages: (i)
Eccentricity is completely
I eliminated by using both
Fig. i8. verniers. (2) Thp rpfl«,<f^^
■■nages are brighter than in the case o other Xt
:rr:r:raiSrjt'''^™-"---^
wa^^ri^^rniajit':;^"-^^^^^^^^
The pnmalic sextant differs from the circle in havin,
n;i^::r£:^;:t:!t:^nfs:^y"-
them, „■& Chauvenefs Astronomy ) °' "='"^
'mm»v^KiiMmi
CHAPTER VII.
TffE PORTABLE TRANSIT INSTRUMENT.
The transit instrument is a telescope with two trunnions
resting on Y-shaped supports so thnt its line of collima-
tion may move in a vertical \ lane, and is used for the
purpose of taking the times of transit of heavenl> bodies
across that plane — generally either the meridian or the
prime vertical. In the former case it enables us to find
the true local time, either mean or sidereal, and also
serves to determine the longitude by means of transits of
moon-culminating stars. Ir. th.. latter case it gives us a
very accurate method of ascertai'^ ing the latitude by
transits of stars ac ss the rrime v tical.
In the focus of the telescope are one or two horizontal
wires, and an odd number of equi-distant vertical ones—
generally five- jf wh.Ji .ne central rhould be in the opti-
cal axis of the instrument, and at righ: anghs to the a'is
of the trunnions or pivots; and if, in -' ,.uon, this axis is
truly horizontal, the line of colli' \tion vill move in a ver-
tical plane. The telescope is ovi 'ed with a ver cal
graduated circle, with a level a cht which serves as a
finder to set it at any required avigle of elevation. It v>hs
also a diagonal eye piece for transits of objects of consid-
erable elevation, and a very delicate striding level for
getting the pivots perfectly horizontal. At night the light
b
g
c
fl
P
w
ti
re
e I
f "I
The Transit TcUscopc.
. 67
of a lantern is thrown into the interior ,r. ,11 • T
w,res by „,eans of an opening v^,h"len° """""= "■'
»..d to u.: r„;iVoTanVht"^:3i::f::-.rt -"-■
server calling ortoo" r,h T- ''"■°"°'"^'"-' ">= "b-
In the case rf the un 1 'T T^'" '=^='> «'-•
noted is when the %":» 7.^ k' "'""^'^ '"^ ■"''^■"
Hther in connng :;t% tt^^T^tZ "^ "'!,■
for Its semi-diameter to na« ,1, _■ "* required
w^ds added or sub^a ^ed, ."' ""=?'';'." ''> ?"-
The first adjustment to be attende/^;= ,l\ r "
n^ation. This may be effected bTttinVth . r"'"
on some well-defined distant objecf or ^ ^^^ ""'"
- - at its greatest elongat on 'xh; tel." ' "f^TP^^^^
verscH in its supports, e'nd Z Jt^^TiiT ''"' "■
bis.crs the object, the collimation is aH n^h V^f^^
one side of it must be moved tova dth-,/;^ '^"^ *°
val by the limation screws The iL. ^' '"*"'■
moved laterally by means 7iLl '"^^'•""^^"t is then
one of the V 'sup^ ^ilUhT^ire^^ J^^^^^^^
when the telescope is again reversed and th. °^J''*'
peated till the collimation is perfect ^'°'"'' '"■
The horizontality of the axi? nf f h^ •
by the striding level and fooTslw " 'If th" ^'"T'^
generally an error of it. „, T '. ^^ ""« '««■ has
change (Ling to^Ltelt'ionsT tl^p til"" "1" '<-
flexure, &c.) it will be fnnn^ '^'"Pe'^ature, accidental
pivots by getting the!n in^ such a ^^1^? 'T' ''^
will have equal but opposite read^ntf *^" '^"^^
tions. Thus, if in one ^^si ion !hf ^ . '" J"'""'"'^^ P°^'-
reads xo, while the west end t .Th „ " /^'^ ^f ^^«
e .St end should read 12 and the west xo ''''"' *^'
'^.
r»m
68
The Transit Telescope.
9 1
1
If one of the pivots has a larger diameter than the
other it is evident th;. when their upper surface is level
their axis will not be so. This will entail a constant
error which will be investigated presently.
Thi verticality of the central wire must be tested by
levelling the pivots and noticing whether the wire re-
mains upon the same point throughout its whole length
when the telescope is slowly moved in altitude.
If the collimation is out of adjustment, but the level-
ling correct, the line of collimation will sweep out a
cone. If the collimation is correct but the levelling in-
accurate, it will describe a great circle, but not a ver-
tical one. If both are right it will move in a vertical
plane. We have now to make this plane coincide with
some given one— say that of the meridian. The north
and south line may have been already approximately ob-
tained by means of a theodolite, and we can now find it
exactly by one of the following methods.
(i) By transits of two stars differing little in right as-
cension, one as near the pole, the other as far from it as
possible. Let a be the right ascension, 8 the declination
and t the observed clock time of transit of the star near
the pole; «S ^S and t^ the same quantities for the other
star, d the azimuth of the instrument — in other words,
the error or deviation to be determined — and tp the lati-
tude. Then d is found from the formula,
, f , , . ,,, „ I cos ^cos ^'
d= Ha> — «)-
-it^-t)
) cos f sin (S — d^)
The rate of the clock must be known, but not its error;
the interval t^ — t must be corrected for error of rate; and, if
a mean time watch is used, converted into sidereal time.
d being in horary units must be multiplied by 15 to ob-
tain the error in arc.
If the declination of the southern star is sruth it will.
J
The Transit Telescope. g_
ton L? :r'v ''""""';' P^" °f '"'^ f°™"'> »' King.
on (Lat. 44 ,4, .^ o.oj;, fo, .^^ .^^^^ ^^.^ ^^ ^^^^^ S=
.he'd'e'*r^?eni;;r '7 '"-'" " too ^.V^ ^
safes. .0 draw aCelThtcfr "'""^ " ''^^'^^-^
the instrument is known. ^''*^"" ^"^^"^
To prove the formula:
J cos tp snT?^^
cos ^p sin (^~5')
^}^zJ^!;' ^^ P°'^' ^ the zenith.
^ ^B the plane m which the teJe-
scope moves, and A Z A- the tn.e meri-
dian. Wr ' " -
aza».
have to find the angle
Let S and S^ be the two stars at
transit, . the unknown clock error, t
the clock time when the star S was on
i
'*g- 19
M
I
70
The Trantit Telescope.
the meridian. The true time of the star being at S will
be t-\-e.
Let a be the R. A. of the star. Then a was the time
when the star was on PB; .*. Z P S=^+^— a.
Let f be the observer's latitude, d the star's declina-
tion, d the deviation of the instrument.
From the triangle P Z S we have;
Sin S Z sin S Z B=Sin P S sin S P Z (i)
And Z S=P S— P Z very nearly=y»-<5
.-. sin {<p — 8) sin ^=sin {t-\-e~-a) cos 8
or (sin tp cos 8 — cos <p sin 8) d={t-\-e — a) cos 8
or (sin <p — cos f tan 8) d=t + e — a (2)
If a' , 51 be corresponding quantities for star S* we have
(sin ^— cos <p tan c^) d=t^ +e — a» (3)
In equation (3) t^ includes the correction for the clock
rate between the observations.
Subtracting (2) from (3) we have
d cos <p (tan J— tan 8') = t^—t—{a'—a) (4)
Now tan 5-tan d^ = s'" (^'-^')
md-
-«)
cos O COS o
cos 8. cos 8'
1 1 — i— (a— « , . : --_
i 'J cos f. sm (5—5'
It is evident from equation (4) that for a given value of
d the quantity t^—Ha^—a) is larger as tan 5— tan 5' is
larger. In other words, one of the stars should be as near
the pole and the other as far south as possible.
Equation (1) may be put in the form
Sin Z P S =
i sin Z S
cos 8
As the angle Z P S is the error, in time, of transit,
caused by the azimuthal deviation d, this equation gives us
the means of correcting a transit where it has not been
convenient to correct the meridian mark.
TO FIND THE ERROR DUE TO INEQUALITY OF PIVOTS IN
THE TRANSIT TELESCOPE.
In Fig. 20 let A C, B D, be the diameters of two un-
equal pivots, E F their axis. If the side A B on which
the feet of the level rest is horizontal, the lower side
r
Inequality of Pivots.
CD will b. inclined a. a cer.;;;;;;^;;:::;:;;;;;^^
will be inrlinpH .,♦ 1 «
will be inclined at an angle
/'/. 4»
If, now, the instrument be
reversed in the Y's. A B will
evidently be inclined at an
angle 2 a, which will be given ^^
by the level readings in the usuaHST
rherefore, the inclination of the axk in tK. « * • •
will be one auar^Pr +h . ^ne axis in the first position
Mih-tary College tLfi% >nstrnment at the Royal
thickest m? i ""^^^ P'''°* ^^'^^ f°"nd to be the
uiicKest, moving the bnhhlp a a- • ■ •■" uc lue
Tllerefore. when th;!m„ %* ""°"' °" '"=>'"5»1-
their lower surfaf. "^ "", ° "' "'' P'™*^ ™= '^vel
theiraxTforo IVr,'"^ "=" ''"^ '- ^visions, and
corrected fo eh ' value of on T- '"'' "'"'f'"^' '^ '"'
Pi"0t highest bvthr^l ^ ""'"' '•-«•■• ^'h »"=«
error wiU b fl t^t""™!.""" ""'"" •="'• '"e total
pivot is highet^and': ■ :t:r "■' '"°-- '^''^" '"' «"^-
-0 APP.V THE LEVEL COaRECT.ON TO AN OBSEEV„,ON
The l,v ■ ™ "■= ™-*"=" ^'•'^SCOPE. "
.■el!;:„:ai,t£:'t;fr^^^^^^^^^^^
Now^'iJ'--'^_''^'"SR
Sin P
Sltl.
Sin R
cos^decjination
R
and P =JLgi j^- altitud e
COS. declinatron
Jiltitude
P
li
S'
r
\
I
(
72
The Level Correction.
West 35
West 55
The correction for the transit in time will be-— -
15
Example— At Kingston, Canada, the transit of Arcturus
was observed, the level readings being :
ist position East 45 —
2nd " East 25—
To find the correction in time.
Here we have
Latitude 44* 14' N
Star's declination 19 47 N
Star's altitude at transit 65 33
Level correction = 35±55_:^5zl25 ^^^ ^ ^.^.^j^^^^^
west end being highest, and the pivot correction altered
this to 6 divisions. The value of one division of the level
was 6".45, therefore the angle R was 38".7 east of the
meridian, and the transit took place too soon.
P = H X 38". 7 = A^y". 2, and the correction was
2.48s. to be added to the observed time of transit.
When the instrument is in perfect adjustment the error
of the watch or chronometer can be at once obtained by
means of meridian transits, as described at page 34.
FINDING THE LATITUDE BY TRANSITS OF STARS ACROSS
THE PRIME VERTICAL.
If S is a star on the prime vertical, P the pole, Z
the zenith, and W Z E the prime
vertical, S P Z is a right-angled
triangle ; and if we know the
angle S P Z and the side P S we
can find the side P Z by the
equation Fig. 22.
Cos S P Z=tan P Z cot P S
I
Prime Vertical Transits.
or. If a IS the star's right ascension, /the s^^^^^7^~^
Its crossing the prime vertical d ilT/ v ^ °^
i^ the latitude of the place ' ^^^^^'n^tion, and
re.u Cos(«_0=cot^tan5
•It the star is in thp nncit,/^,, c /
the equation becoLr '" *"' °' "*' »="<''-'»
Cos (<— a)=cot ». tan <J
vMiS':hr::n:it:itte''f;r^ ^"^ ^"">'''^' p-
observation may be m»T ^1 ™'" ''" ''°°™- The
instrument oTwUh TtntZ ™h'V^ """^"^ "»="
other delicate ,«»!<,:' latrude ?' '" '''^' " '"
Sin. altitude =^L
sin ^
(Since cos P S = cos P Z cos Z S )
exc^rth^^flls'edecl-r^^rb t*"^ ''""' -"-'
latitude of the place Th I ^ " ""'" ^"'^ *'
zenith are to be preferred h "" "'"""""^ "=" ">=
observed time of ^;S^:^//;C-:-;'-^ in the
telescope clamped'at n^' L^es t? b"'''°'=' ^"? *^
horizontal arc If „„ , ' ""^ ""^ans of the
transit telescope L nre Ld" d'"'. ^''^ "" """^"^
(from the approximate l,,rH,u°'""' '' '° '='''<="'="'=
which culn^'nyersTver d ; : 3t„:r^f :l '^'"^'^ =" ='"
cross the prime vertical, a.,d d ec'Tl \° '""'"'"
that instant. It will now be „ea Iv if^r °" " "
position. The emr m i ""any m the required
-ethesiderea,reoTL::i.t;:X^:^--
-r*^-
J
I
1
74
Prime Vertical Transits.
east and the west verticals. The mean of the two will
be the time of the transit over the meridian of the instru-
ment, and should be equal to the right' ascension of the
star. If the two results are not equal their difference
shows the angle which the plane of the instrument makes
with the true prime vertical.
In working these observations we may use either a
sidereal or a mean time chronometer, in the latter case
making the usual reductions, and always allowing for the
rate. If two transit telescopes are available, one of
them may be set up in the plane of the meridian for the
purpose of ascertaining the exact chronometer or watch
error by star transits. A large transit theodolite serves
. instead of two transit instruments, and in this case an
ordinary good mean time watch will suffice, the mean time
of the observations being reduced to sidereal time. If
both the east and the west transits tre observed the dif-
ference of time in sidereal units is double the hour angle
P, and the latter may therefore be obtained without any
reference to the actual watch error, provided the rate is
known. It should also be noted that if we reverse the
telescope on its supports any error of collimation or ine-
quality of pivots will produce exactly contrary effects on
the determination of the latitude. Two stars may be
observed with the telescope in reversed positions on the
same day, or the same star on two successive days, and
the mean of the two resulting latitudes taken.
It will be found advisable to calculate beforehand the
altitudes and times of transit (either mean or sidereal, as
the case may be) of a number of suitable stars.
If the plane of the telescope is not in the prime vertical
the calculated latitude^ will be too great. Suppose the
deviation to be to the ea^st of north and that the tele-
Prime Vertical Transits
cope describes a vertical circle
passing through fb^E > ZWi.
Then V Z\ which bisects
S S', will be the calculated
co-latitude. The correction
for the deviation may be
computed thus. The star's
R. A., minus the mean of the p- -
^me^of transit corrected for clock error, wiH be thetngle
JlL'e : ' "^ *'' "^ht-angled triangle Z P I,
tan PZ cos ZPZ' = tan PZ' = tan FS cos SPZ'
or tan<P= -^^^osZPZ'
cos"SPZi
te^to If •'''^f' '' ,"■' '"""= '°' ^" ='"^' -<i " i= bet-
er to obtain ,ls value from a star which culminate,
several degrees south of the zenith, since the same error
m the observations wi 1 have less effort „„„„ .1, I
lated azimuth. "P°" ""^ <==^'="-
exactlv'the'T'' '"'™™ '.'"= ^''■''='" -'■■« "e "Ot all
exactly the same a correction has to be applied. For
details on this point vide Chauvenet.
In the field the instrument is generallv mounted on a
ate the effects of vibration produced by the observer's
movements the ends of the legs may be made to res, ;„
no ches ,n flat blocks of wood placed at the bottom of
.wile .""d ''""''"' ^''"'"'''^'■'"" '-•>- deep a.°d
nvo feet in diameter, n,:, ,„,, t^, |^„ P f^
fZw T"'T^""""'-' *^""=>- b^'-^'^" 'he Lake of
the Weeds and the Kocty Ilountains.
The meridian mark snould, if possible, be at least half
a mile distant. ... black or white vertical stripeTa ntet
on a stone serves tor the day time. A, night a'buIlTe;t
76
The Personal Equation.
lantern may bo used, the glass being covered by a piece of
tin with a vertical slit cut in it. Or, as the lantern is
liable to be blown out by the wind, it may be enclosed in
a wooden box with a vertical slit.
The larger transit theodolites may be used as transit
instruments, and have the advantage over them that when
the meridian line has been ascertained the prime vertical
can be at once set off.
THE PERSONAL EQUATION.
It often happens that two persons, equally well trained
in taking observations, will differ by a considerable and
nearly constant quantity in estimating the precise instant
of an event, such as the transit of a star across a wire.
This difference is called their personal equaiion, and an
allowance should always be made for it when observations
made by two individuals have to be combined. In the
case of the transit instrument this equation may be de-
termined as follows: Let one observer note the passage
of a star over the first three wires and the other observer
note the transits over the remaining wires. If the two
observers' estimation of the instant of transit differ, it is
evident that (provided the wires are equidistant) the
difference will appear on comparing the intervals of time.
For instance, if A notes the transits across the first three
wires at los., 20s., and 30s., and B notes the remaining
two at 39S.5 and 49S.5, it is plain that A would consider
the star to be on any wire half a second later than B
would, and their personal equation is therefore os.5. By
repeating the same process on other stars, and taking the
mean of the result, a more accurate estimate is obtained.
The personal equation has been found liable to vary with
the state of health of the individual.
The difference in the estimated instant of a transit is
only a particular case of the personal equation.
CHAPTER VIII.
THE ZENITH TELESCOPE.
The zenith telescope is a contrivance for the exact d.
termination of the latitude by measurinc^wf/h7h
minuteness the differences or^tr^-T ' ^'^^*^'*
tancesoftwostars one of f ' ^ "^ "" ^'"^'^ ^^^^
the zenith distance of the equator, we have,
and adding, 2^= J+i^IjI7Z/
thett;:::7fi::L^;^Lrra„T;''^r -. ^-^
ing thei.. actual values/ Mo Lt, if /anT::! '""T
:nr;eir---7;i--— ^
r>f*u t .■ ^ '"^° account the difference
of the refractions at the two altitudes.
fnrln '"^[[""^^'^t is practically a telescope about 45 inches
foca length attached to a vertical a^s round whi^h
revolves, having been first clamped at a certain ang e o
li'
1
^«*te:*„j,*v,«L
'■ i
.1 i
78
The Zenith Telescope.
elevation. The latitude must be known approximately,
and a pair of stars selected which are of so nearly the
same meridian zenith distance at that latitude that they
will both pass within the field of view of the telescope
without our having to alter its angle of elevation. As a
rule, z and z must not differ by more than 50' at the most.
If the axis is truly vertical and the telescope remains at
the same vertical angle at the observation of both stars,
then it is plain that the difference of z and z may be read
by a micrometer in the eye piece.
It is usual to observe only stars which pass within 25
degrees of the zenith. The telescope has a long diagonal
eye piece with a micrometer in its focus, and the micro-
meter wire is at right angles to the meridian, ^^here is a
very delicate level attached to the telescope, and a vertical
arc which serves as a finder. By reading this level at
each observation we can detect and allow for any change
in the angle of elevation of the telescope.
The above is the merest outline of the principle
of the instrument, and reference must be made to
other works for the details of its construction. The
method of using it is this: The latitude being already
approximately known, a pair of stars is found from a star
catalogue, both of which will pass within the field of view
without altering the elevation, and which have nearly the
same right ascension. The reason for this is that their
transit may take place within so short an interval of time
that the state of the instrument may remain unchanged ;
but a sufficient interval must be allowed for reading the
micrometer and level and reversing in azimuth ; say, not
less than one minute or more than twenty. The meridian
line must have been previously ascertained by transits of
known stars, or otherwise, and the chronometer time
calculated at which each of the stars will culminate. The
telescope having been brought into the meridian, ready
for the star which culminates first, and set for the mean
The Zentt:
Celescope
79
wire at the calculated instant ofTt.anl H ™"°'"^'f
manner If'aftj^h , ' ''"=°"'' ^'^^ '" ">= ==>"<=
must 1; r LI ed b/r: ""• "° '^^^' '' ■""=•■ °"'. "
teleLope'^'' ^'^ ^""-""^^"^ ^^ --"^^^ '^^ the transit
This method of finding the latitude is known as Tal
cotts having been invented by Captain Takott oHhe'
U. S Engineers. Its defects are that it is often difficuk
to obta.n a sufficient number of suitable pairs o? stlVs o
which the dechnations are accuratelv known As a -X
we have to use the smaller stars, whose FJ;cef are not
very well known, and must therefore observe alarge num
her of pairs to eliminate errors.
TO FIND THE CORRECTED LATITUDE
ern^tr 'f *t ""'T''" ""''^^"^ ^'^ ^^^^ ^^ ^he south-
ern star, m, the same for any point in the field assumed as
he micrometer zero, and .„ the apparent zenith distance
represented by m„ when the level reaSingis.ero. Suppose
also, that the micrometer readings increase as the zen th
distances decrease. Then, if the level reading were ze 1
the star's apparent zenith distance would be
8o
The Zenith Telescope.
Let / be the equivalent in arc of the level readinK, posi-
tiwi when the reading of the north end of the level is the
greater. Lot r be the refraction. Then the true zenith
distance of the southern star, or z, is:
The quantity r„ +w„ is constant so long as the relation
of the level and telescope is not changed. We have, there-
fore, for the northern star,
Hence
3—z'—m'—m-\-l'-\-l-\^r—r'
and the equation for the latitude previously given will
become :
X=^ (^+5') + i (m'-m) + i {l'tl) + ^ {r-r')
TO FIND Ti'E CORRECTION FOR LEVEL.
Calling the readings of the north and south ends of the
bubble n ana s, and the inchnations at the observations of
the north and so-th stars, expressed in divisions of the
level, L' and L, we shall have
V-
n—s
L=
n-
2 2
and if D is the value of a division of the level in seconds
of arc, we have
l'=U D /=L D
and the correction for the level will be
i (/'+0=i (L'+L) D=''^-Jl±A D
4
TO FIND THE VALUE OF A DIVISION OF THE LEVEL.
Turn the telescope on a well-defined distant mark.
Set the level to an extreme reading L, bisect it by the
micrometer wire, and let the micrometer reading be M
Now move the telescope and level together by the tangent
\
The^enith Telescope . g
screw till the bubble mves s^^A\^^' T" " —
treme, bisect the mark LJn k 1 "^•^'- the other ex-
'nicrometer reading bM?THf ^ ""^ "' ''' *^^
the level in turns of th! n!'" "^^"^ "^ ^ ^'^^^''^n of
lurns ot the micrometer will be
^nd if R is the value in st^t of arc of . , r
of the micrometer th*. vai» r. r , •*,^^ "^ '^ volution
arc will be ' "'"' ^ °^ *^« ^^^^1 ^n seconds of
TO FIND Till
D=Rrf
^emth distance by the formulce ' ^"^''" ^"'^
cos ^-=cot <J tan >}
cos r=cosec «J sin /I
Whence, knowin/' the star'Q P a „ ^ .l .
error, .e find the chronome i ta. rf^h ^'■-"O'""-
Ration. Set the telescope for he^nhh ir"''''/'""-
it upon the star ,n „r ,„ • '""/'=""" distance », direct
«. elcngatio; 7. . rcrr^Ith':^ "'^ ""'"f «-'•
note the time if bisec^.^ "^'"""eter wire;
.■n.s. A3 the :Lr » ^es ; ^Irrea^ -' -^' -ad-
often as possible while it i"" LLTh "!'= ?™«= as
tU\:;:.t:.t^\"--™"-'.wroTbist^^^^
1 >
3>
s>
o , --- ""'"-o ui uisec-
readin,s, „ "the micromete'r Te'^Lrf,! '"•'■"°'""'='
greatest elongation (,). and T 7 ^f <he -nstant of
an..ardistances;the;the,at;rtre'?;ntSir^^^^^^^^^
sinz,=sin(^— <j)cos^
smf3=sin (f— /!,)cos5
™"roLtr'''rd1f":Le'V;erha:'r''^ ^T'"""™ °' '"«
w. since (.-., , irjii-r :rr.::
X
IMAGE EVALUATION
TEST TARGET (MT-3)
//
1.0
I.I
1.25
Hi p 28
tii
us
u
I—.
2.2
2.0
V
'Z
^^v
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Photographic
Sciences
Corporation
23 Wk:.>i MAIN STREET
WEBSTER, N.Y. 14580
(716) 872-4503
m
\
f
^\
^v
V
?
,^
^
(/.
82
The Zenith Telescope.
screw to move the thread through the angular distance ij
{m — ;»i) R=i\
Also (m — >Ha) R=«3
Therefore, subtracting
(Wa — Wi) R = /i — jj
or R:r=
Wj
-;m,
To correct for any change in the level reading, let l^
and /a be the level readings corresponding to m^ and ;«,;
then (/g — /j) D is the change required. The angular value
of D is unknown ; but, since D = (^/K, the correction to be
applied to (/j — i^) is {/g — /j) dR; and
(Wa — Wi) R=ti — ij, ± (/^ — /,) dR
or R=^
U-H
A value of R is thus obtained for each of the observa-
tions, and the mean of the results taken. This mean has
then to be corrected for refraction, thus : From the tables
find the change in refraction for i' at the i-enith distance
z. Let this change be dr ; then R dr will be the correc-
tion to be subtracted from R.
REDUCTION TO THE MERIDIAN.
If a star has not been observed exactly on the meri-
dian it may be taken when off it, and the observation re-
duced. The following is one method of doing this. Keep-
ing the instrument clamped in the meridian, the star is
observed at a certain distance from the middle vertical
thread and the time noted. This will give its hour angle,
and if we denote this by t (in seconds of time) the reduc-
tion is obtained by the formula
^ (15 ty sin x"sin 2d
This is to be added to the observed zenith distance of a
southern star, or subtracted from that of a northern one,
and, in either case, half of it is to be added to the latitude.
The Zenith Telescope.
REFRACTION.
83
When the .en'th distances are small the refraction
vanes as the tangent of the zenith distance.
Let r=a tan z
r'=ci tan z'
Then r~r'=a (tan s— tan z)
__ sm (2 — z')
cos z cos s'
=(^-^y') " sin i'
^^ ''^ cos* 3' "^^'■^y
a may be taken as 57".;, and the difference of th^
micrometer readings used for (z~z') ^
THE PORTABLE TRANSIT INSTRUMENT AS A 7ENITH
TELESCOPE.
adL'^'n T'^'l" uT' ''^'^^^"P^ ^'-^ - micromete^
added to It, and the level of the finder rim]. ;.- ^
sufficiently delicate, it may be used a^ a L " f te :scZe'
reversing the mstrument in its Ys between the oS^:
Th^'^!7'^^' ?^''^'"^"^^ give the mean places of the stars
The «/^«..;., places are those which have to be used and
must therefore be determined. ' ^
wliiilliHlili II
CHAPTER IX.
ADDITIONAL METHODS OF FINDING THE LATITUDE.
TO FIND THE LATITUDE BY A SINGLE ALTITUDE TAKEN
AT A KNOWN TIME.
Here we have in the triangle P Z S the hour
angle P, the side Z S (90°— the objects alti-
tude), and P S the polar distance. From these
data we have to find P Z. From S draw S M
perpendicular to P Z produced. Let cJ be the
declination, (p the latitude, and a the altitude.
In the triangle P M S we have :
cos P=tan P M cot P S=tan P M tan d
M Z=P M— P Z=P M + y>~9o°
Also 'A^- .
cos PM : cosZ M::cos P S : cosZS
or cos P M : sin (P M+^)::sin <5 : sin a
Therefore
sin (P M+^)=
sin ojcos P M
sin d
(2)
Equation (i) gives P M and (2) gives P M + ^
In this method, if the star is observed when far from the
meridian a small error in the hour angle produces a large
error in the computed value of the latitude. The altitude
should therefore be taken when the object is near the meri-
dian.
TDE.
"AKEN
tl the
large
itude
neri-
Lalitude by Altitude of Pole Star.
85
TO FIND THE LATITUDE BV OBSERVATIONS OF ThT^
STAR OUT OF THE MERIDIAN.
Up be the polar distance of the pole star
m circular measure p'^ h a very small
quantity.
Let P be the pole, Z the zenith, and S
the star at an hour anp^le h or SPZ. Draw
^ N at right angles to P Z and take ZU
equal to Z S. Let P N be denoted by .r.
MNby3..SPby;^, the star's altitude by
a, and the latitude by /". Then
P Z=Z M + M P=Z S + P N-N M '''' ''
or go~l=go—a+x~y
.'. l=a — x+y
We have to find .v and y.
(1) From the right-angled triangle S P N we have
cos S P N=tan P N cot P S
.'. tan .v-=tan p cos h
or, approximately, .-v =p cos A
(2) Denoting S N by , we have from the same triangle
Sin S N=sin S P sin S P N
or sin 7=sin p sin h
.'. approximately, q=p sin h.
(3) In the right-angled triangle S N Z we have
cos Z S=cos S N cos Z N
.'. sin fl=cos q sin (a+y)
or sin (a+y)= ^^'^^
or approximately
sin a+|^tcosa=-
:
COS^
sm a
=sin a (i-l-^ qi)
y cos a=^ q9 sin a
orj'=J<7» tan a
=i>» sin* /{tan a
MMi
!
i
f
86
Circum-Meridian A ItiUides.
Hence, in circular measnre
l=-a—p cos h-^^ p^ sin'* h tan a
or in sexag-esimal measure
l=a~p cos h + ^ />» sin i " sin^ h tan a
This is the method given in the explanations at the end
of the Nautical Almanac. To find the latitude we have
only to take an altitude of Polaris, note the time (which
will give us the sidereal time), and apply certain correc-
tions as directed in the Almanac.
FINDING THE LATITUDE BY CIRCUM-MERIDIAN ALTITUDES.
When the latitude has been found by a single meridian
altitude the result is only approximately true. It may,
however, be obtained with great exactness by taking a
number of altitudes of the sun or a star when within about
a quarter of an hour of the meridian on either side of it.
The altitudes may be taken with the sextant, reflecting
circle, or theodolite, and the observations should follow
ea-h other quickly, and at about equal intervals of time.
The watch error must be exactly known, and the time
of each altitude noted. The mean of the altitudes is
taken, but the hour angle for each must be obtained
separately. In the case of the sun this is done by cor-
recting the observed times for watch error and subtracting
them from the mean time of apparent noon. If a star is
used the mean time corresponding to its R. A. will, of
course, give the hour angles—
The formula is
Latitude=go* — a±d — x'
Where a is the mean of the altitudes, d the declination
of the object (negative if south), and x" a quantity equal to
• • ^»
2 sin' —
2
sin J. X cos. approx. lat. x cos. dec'n. x sec. alt. ; h being
the hour angle.
Circiim-Meyidian A ltitude%.
^_ 87
To prove the formula
2 sin» —
and .r"= ^ v ^°^ ^ ^°s ^
• //A
Sin I cos a
Let P be the pole, Z the zenith, and S the
sun or star near the meridian.
Let a be the star's altitude, h its hour anirle
and d its declination.
Let a + x be the star's meridian altitude.
Then a+x-=^d=Cio—l
We have now to determine the small quantity .v ^^
Now, sin PZ sin PS cos ZPS^cos ZS-cos v'z< ^^■
^^^^ or cos I cos d cos A=sin «-sin / sin d ' "'
.: cos I cos d (i-cos h) -= -sin a + cos (l-d)
= — sina-fsin(a + ;v)
.*. 2 cos / cos d sin* - -
2
Therefore, approximately
-2 sin -^ cos (a + ^)
2 smj —
X"= ?_ V ^°S ^ cos i
sin I cos a
d is, of course, negative if south
2 sin«~
The value of the expression __^'" 2 ..
sml"~ (^"own as the
"reduction to the meridian") is found for each hour
t7cire :.^:^ -' ''' -- -^" ^^e vai::^ tatn
the mean of ten altitudes of the sun's lower l!mb oh'
served w.th a powerful theodolite, was ,,■ 59 ao' Tht,"
:^-
-^■^■nn
88
Circum-Meridian A Ititudes.
\)
t
when corrected for refraction, parallax, and semi-diameter
gave 40° 14' 3i".55 as the true mean altitude of the sun's
centre. The sun's declination was 19° 53' 45".8 south
The mean of the values of the reduction for the observed
hour angles, as taken from the table, was i6".26, and the
calculated value of x was I7".36.
go o o
^•t'tude 40 14 31.5 5 -o
^ ,.■ . 49 45 28.45
Decimation... 19 53 45,80 ^_
29 51 42.65
1^36 '^
Latitude=29 51 25.29
Strictly speaking, a further correction ought to be made
for the change in the sun's declination during the obser-
vations.
In the case of a star we must add 0.0023715 to the log
oix" to correct the hour angle for the difference between
the sidereal and mean time intervals; for the star moves
faster than the sun, and therefore gives a larger hour
angle for the same time.
Additional accuracy is obtained by taking half the ob-
servations east of the meridian, and half west of it, the
intervalsof time between the successive observations being
made as nearly equal as possible. The hour angle
changes its sign after the meridian passage of the object.
I w
-diameter,
the sun's
".8 south,
i observed
3, and the
CHAPTER X.
> be made
le obser-
» the log.
between
ir moves
?er hour
the ob-
' it, the
ns being
r angle
object.
t^TERPOLATlON.
''^^^^ODS OF FINDim THE LONGITUDE.
and longitude of tre place 'fo7'' '°'" '^^ ^°^^^ ^'-e
given are for GreemvS"me. "''""' ^^"^^ *^^ ^^^-
accurate result. "^^ "^"^ *° obtain a very
su':s:s:::~:;::;^^v^thg.^^
^ -ain .ear, at a ^t:\Z^!: ^ :fJr:::^ Of
For Greenwich mean noon we find in the Almanac
Date c > J
'^ow, at apparent noon at the ohr^ :, -ii v
apparent time at Greenw.VI, VPf^ " "'" be 4 p.m.
^;-ree„..a.t.:r!;:;--:--;-rr
Th.s va„at,on is X3-.305, which n,„„ip„-ed by I X^
90
Intcipohition,
53".22 to bo subtracted from the dc-
dination of 2d January —
22' 57' lC".2
.^3".22
22 56 22,98=required dec'n.
IJ.2I
12) 1. 14
•095
13.21
TO one =^ V«rl»tlnn
^J •305 It a I'M
53-22
il V -3
INTERPOLATION BY SECOND DIFFERENCES.
The differences between the successive values of the
quantities fjiven in the Nautical Ahnanac as functions of
the time are called the first differences ; the differences be-
tween these successive differences are called second
differences; the differences of the second differences are
third differences, and so on. In simple interpolation we
assume the function to vary uniformly ; that is, that the
first difference is constant, and therefore that there is no
second difference. If this is not the case simple interpo-
lation will give an incorrect result, and we must resort to
interpolation by second differences, in which we take into
account the variation in the first difference, but assume
its variation to be constant and that there is no third
difference.
The formula employed is
f{a + k)=f{a) + Mi + \5k''-
where A is half the sum of two consecutive first differ-
ences and B is half their difference. It is thus derived:
We have by Taylor's Theorem
/ (.V + /;)=/ (.v) + Ml + hh 2 + &c . (A)
and if /t is small compared with .v ilie successive terms of
the series grow rapidly less.
VarUtlnn
at i r.M
(A)
Interpolation. m
Suppose a— I, 11, and </-f-i to be three successive ar^'u-
ments of a tal)le constructed from/(.v) in which it is as-
surDed that a is many times greater than i. Then, from
the table we knowy fa-i),/(a), and/(a + i), and there-
fore we knowthc(nfferences/(rt)-/0,— I), ,un] /0» + i)—
An), which we may (lesi<,Miate by A and A' respectively.
Knowin{( that third (hffercnces can be nc/-Iected we can
obtain tlie vahie of/ (a + k), where k is less than i, as
follows:
From (A) we have, if we nef,dect hij^her powers
/(a— i)=/(rt)_A + B (I)
/ («)=/ (a) (2)
/(« + i)=/(a)+A + B (3)
/{a-i-k)=/{a)+Ak+Bk^ (4)
Subtracting equation (i) from (2) we get
A=A— B
and subtracting (2) from (3) A'=A + B
.-. A = i (A'+A)
B = |(A'-A)
/. substituting in (4)
/ {a + k)=/{a) + ^ (A'+ A)/c + i (A'— A) k^
The signs of A and B must be carefully noted. If the
functions are decreasing the first differences are negative,
and if the first differences are decre- mg the second differ-
ences are negative.
The method can be better understood from an example
or two.
Ex. i.-Given Hie logs, of 365, 366, and 367 to 7 places
of decmials to determine log. 366.4.
Numbers.
365
366
3^7
Lor/.
IM Differ'ce.
SdDlffer'ce.
5622929
56348II "f2
5646661 "850
—32
9»
Interpolation.
Here k is ,<o, /I =11866, and li = ~ib.
5634811 ii«66+A
4746 A
3639557
3
47464
-»j^x(A)»=-3, nearly.
3639554
The tables give the log. as 3639555,
If the second difference had been neglected—/.^, if wc
had worked by simple interpolation, the result vvoulil have
been 5639551.
Ex. 2.-Given the log. cosines of 89° 32', 89' ^5, and
89 34. to find log. COS. 8g' :i^' 15".
M DifferenceJjml Difference.
Log. COS. 89 32=7.9108793 I
Log. COS. 89 33=7.8950854! -;57939 _ .
Log. COS. 89 34 = 7-8786953 —163901 5902
Here we have to subtract Uxhalf the sum of the ist
differences, and a^)«xhair the second difference or
40416 in all ; '
.'.log. cos. 890 33 '15" =7-89 10438.
TO FIND THE GREENWICH TIME CORRESPONDING TO A
GIVEN RIGHT ASCENSION OF THE MOON ON A GIVEN DAY.
Let T'=the Greenwich time corresponding to the given
right ascension a'
T=the Greenwich hi.ur preceding T' and correspond-
ing to the right ascension a
A a=the difference of R. A. in one minute at the
time T.
Then we shall have, approximately,
a —a
T— T =
r
A «
Interpolation. ^^
To r...Tcct for secnci differences we have now only to
fi.u the ,hf crenr.. „f R. A. for one minute at the nufhlle
nstant of the .nterval T-T. Call this a', and we shall
nave
-p, 'p « — a
T and T are in minutes.
INTKUPOLATION HV DIFFERKNCKS OF ANY OKDKK.
If it is required to find the intermediate values of a
unct.on w.th ^neater exactness than can be done by
M terpo at.on by second chfferences we can use any nun.
Der ol differences.
^f* I' J"^!',' '^+^ ^' &^" ''« the arguments.
^. P', F , &c., " the functions.
«. «, a" &c.,
" b, b' b" &c.,
&c.,
So that F>~F=a,
the 1st differences.
" and
&c.,
a' ~a=b, and so on.
Nmv, if FCO is the function corresponding to the argu-
nient I +n w we have
F(«'^F + «a+ "P^II^b+'K»-j) (n~2) ^^ , ,
u -r \-8cc. (a)
1.2
1.2.3
• If « be taken successively equal to 0,1.2, &c., we shall
obtam the functions F, F', F" &r nnH ;J a- .
, , i , r , (xc, and mtermediate
values are found by usmg fractional values of;/. To find
the proper value of n in each case let T-f^ denote the
va ue of the argument for which we wish to interpolate a
value of the function ; then
n w=t, and n= t ; that is, n is the value of / reduced
to a fraction of the interval w.
Ex.-Suppose the moon's R. A. had been given in the
Almanac for every 12th hour, as follows :
W
■ I
1 1
94
Mar. 5, oh
Muoit's R. A.
Interpolation .
i^TDiff^^ 2nd Dijf. 3rf niff, ^thDiff\5thDif
«»ai.i. on2in. 5Sni2Ss .30I „
" 5. 12h|22 27 15 .4i +28D1- 47S.O4
fM2h23 23 3 .39 27 37.89 32.18 6.53 + IS .74
" 7, oh 23 50 15 .03 27 12 .24 ^5 .05 ^ 53 j J ^g
" 7, I2h o 17 9 .8J 26 54 .20 ' i« oa 7
— OS.66
required the moon's R. A. for March 5, 6h.
Kere T=March 5, o^, t^C^., u>=x2\ n:=^\.=L. ^,,.
.f we denote the co-efficients of ., ,, ., &e. in^« Lv A
i5, L, &c., we have - '
„=+,8". 47..04, A^„ ,, F=^" 58" a8..39
b=~
4.79, C=B ^ =+^.^, c'c^-.-+
ip.74> D^C '
M— 3
.r= 6
— tItt, D^=<=—
4^62
o«.3o
o».o7
Og.02
Moon's R.A. on March 5, 6^ or F^^^ =22» 12-56.74
TO FIND THE LONGITUDE BY TRANSITS OF MOON-
CULMINATING STARS.
This is a simple and easy way of finding the lon^ritude
tTLT/''^ ''" '^ '"°^^"' ''^^^^ -^ ^ -y at
ZT2 ' r '" '''''' °^ °"^ ■'^^^^"^ '" an observed
^ans t may throw the longitude out as much as half a
mmute m t.me, or 7-i n^inutes in arc. It is, howeve a
^^es 11 T' '^""'"' *° ^ ^"'•^^^-' -- ^"'h
wa'c Of " ' ''■'"''* ^'^^^^^''^^ ^"d -" ordinary
prefted'f rS,:.-^^^^^^ ^^^^ ^— is to d
The instrument is set up in the plane of the meridian,
■'--r
I^^^ii^^deby Moon-Culminations. gg
knew either Greenwich or Ic'tim uHL'TT 1°
watch should be taken into' accoT^ 't fn lal J
transit are noted, and the interval of ti„,e between t
.s reduced from mean to sidereal time! ''™
In the Nautical Almanac are given, for everv dav of .(,„
year, the sidereal times of transit a C^TZthllt^
moon and of certain suitable. •,s c-ill,. '•"'"^'^ °f "'<=
in-r" ^, I . ''"^'J'e^ ^^s, callei. noon-culniinat-
■ng stars ; also the rate of change per hour fat f hT
of transit) of the moon's R. A As the m„
rapid^- through the stars from west t Z. TL7ZZ
tl.at,a sta.,o„ not en the meridian of Greenw.cl the
.nterva, between the two transits will be diffe,e„Tto ,h,
at Greenwich; and, the moon's rate of mo.irpeHou
being known, a simple proportion will ,if the stat o nl
longi^,dr"?f M '""°" """ ''"•'■"*''• ""'I 'hence the
longitude If the station is far from the meridian of
Greenwich a correction will have to be made for "he
change i„ ,he rate of change of the moon's R A xt
rate of change at the time of transit is fonnH f? .f
Nautical Almanac by interpolation by ^ c™d iffrncer
and the mean o the rates of change at Greenwich and at"
he station is taken as the rate for the whole interval o
time between the transits. mierval of
An example will best illustrate the method :-
At Kingston, Canada, on the 24th Februarv .sx, .1.
transits of the star -. Tauri and of'the „ 00 's ti^ iimb
were observed at 6h. om. ys., and 6h. rm. gs r sp"
vely, mean time. Uifferenee, 46 seconds, „i 46 ,?,,
Sidereal units. 4"s.i^ in
1^
I 1
i
96
Longitude by Moon-Culminations.
Greenwich TransitslL^^"""-^^- ^9^. 163.62
Moon I... 4 7 57 .44
Difference in sidereal time=
Add interval at Kingston^
iim. 19S.18
46 .12
Total change of moon's R.A= 12m 53.3=7253.3
By interpolation by second differences the variation of
the moon's R.A. per hour at Kingston at the time of
transit was found to be 142s .23
At Greenwich it was 1^.2 .68
J
2)284 -91
Mean rate of variation 142.455
1-2:^5 ^^h. = 5h.09i6
5h. 5m. 293.76 west longitude.
It should be noted that in this case the moon was west
of the star at transit at Greenwich and east of it at
Kingston, having passed it in the interval.
The following is a specimen of the part of the Nautical
Almanac relating to moon-culminating stars.
n
Mom-Culminating Stars.
!5s.3
tion of
ime of
[itude.
s west
it at
utical
o
00
00
U)
O
^
O
NH
H
h
<
:?;
h
V
O
o
H
■V^s„jo-joo
■u a
« o
apnjiuSBi^
2 N ix
O CO
■*oo
<> d
"> 't- fO 5h
to to
i
_oo'_
I-H l-H .^ ^
CXI >-
o c rt
"3
FINDING THE LONGITUDE BY^Ti^dISTANCES.
This method is an important one to the travelh'n^ n
r ::;r';; ^'^ -^^^^"^^ -^°^^ chJnTmit^r;
other Ifl^;- " '"^trument used is the sextant or some
other reflectmg one, and the observation is a very siZe
eve; an"erTl'°''^T^''"^ ^^^ angle, causes how!
ever, an en or m longitude of about a quarter of a degree
1 ■ I
98
Longitude by Lunar Distances.
The moon moves amongst the stars from west to east
at the rate of about 12° a day. Its angular distance from
the sun or certain stars may therefore be taken as an in-
dication of Greenwich mean time at any instant— the
moon being in fact made use of as a clock in the sky to
show Greenwich mean time at the instant of observation.
The local mean time being also supposed to be known,
we have the requisite data for determining the longitude
of a station.
In the Nautical Almanac are given for every 3d hour of
G.M.T. the angular distances of the apparent centre of the
moon from the sun, the larger planets, and certain stars,
as they would appear from the centi of the earth. When
a lunar distance has been observed it has to be reduced
to the centre of the earth by clearing it of the effects of
parallax and refraction, and the numbers in the Nautical
Almanac give the exact Greenwich mean time at which
the objects would have the same distance.
It is to be noted that, though the combined effect of
parallax and refraction increases the apparent altitude of
the sun or a star, in the case of the moon, owing to its near-
ness to the earth, the parallax is greater than the refrac-
tion, and the altitude is lessened.
Three observations are required— one of the lunar dis-
tance, one of the moon's altitude, and one of the other
object's altitude. The altitudes need not be observed
with the same care as the distance. The clock time of
the observations must also be noted. The sextant is the
instrument generally used. All the observations can be taken
by one observer, but it is better to have three or four. If
one of the objects is at a proper distance from the meri-
dian the local mean time can be inferred ."rom its altitude.
If it is too near the meridian the \vatch error must be
found by an altitude taken either before or after the lunar
observation.
I
Longitude by Lunar Distances.
99
written down in their proper order:
latob?;i>"^^*'.'.r'="- ^'^•.°^«'»'- -'t- Of moon'B lower limb. Dist.ofmoon'B f.rllmb
ard "
4th "
4)
Mean
ToUli.
If there is only one observeVit is best to take" the ob-
servations in the following order, noting the time by a
watch, ist. alt of sun, star or planet; ad, alt. of moon;
3c., any odd number of distances; 4th, alt. of moon; 5th
at of sun star, or planet. Take the mean of the dis!
ances and of their times. Then reduce the altitudes to
the mean of the times; ,.c., form the proportion-differ-
ence of times of altitudes : diff. of alts.::diff between
t.ine of ist alt. and mean of the times : a fourth number
vvaich IS to be added to or subtracted from ist alt ac
cc rdmg as it is increasing or diminishing. This will give
the altitudes reduced to the mean of tSe times, or Z-
responding to that mean.
The altitudes cf moon and star must be corrected as
added t"?h.'!"'™"''^ semi-diameter of the moon
added to the distance to give the distance of its centre.
1 iie lunar distance has then to be cleared of the effects
of parallax and refraction.
TO DETERMINE THE LUNAR DISTANCE CLEARED OF
PARALLAX AND REFRACTION.
Let Z be the observer's zenith, Zm
and /,v the vertical circles in which the
inocn and star are situated at .he instant
of observation. Let m and s be their
observed places, U and S their places
after correction for parallax and refrac-
tion : then Zm, Zs, and ms are found by
observation, andZ iM and ZS are obtained
by correcting the observations. The ob- Fi. 27
100
Longitude by Lunar Distan
ces.
ject of the calculation is to determine M S.
Now, as the angle Z is common to the triangles mZs
and M Z S, we can find Z from the triangle mZs in which
all the sides are known. Next, in triangle MZS there
are known M Z, Z S, and the included angle Z, from
which M S can be found. M S is the cleared lunar dis-
tance. The numerical work of this process is tedious.
The cleared distance having been obtained we proceed
in accordance with the rules given in the N.A.
The Greenwich mean time corresponding to the cleared
distance can be found either by a simple proportion or b}-
proportional logs.
It admits of proof that if D is the moon's semi-diameter
as seen from the centre of the earth (given in N.A.), D'
its semi-diameter as seen by a spectator in whose ;;enitli
it is, D" its semi-diameter as seen at a point where its alti-
tude is a, then
D" — D=(D' — D) sin a, very nearly.
For details of the methods of finding differences of
longitude by the transportation of chronometers, and by
the electric telegraph, vide Chauvenet or Loomis.
CHAPTER XI.
MISICELLANEOUS.
TO KIND THE AMPLITUDK AND HOUR ANGLH OF A
GIVEN HEAVENLV RODY WHEN ON THE HOK,;^ON.
Tho ampUtHdc is the nn-^Ie that the plane of tl,e
vertical c.rcle thron.h an ohject n,al<es Jith the pi
of the prime vertical. *
ane
Let N SEW he the north, south,
'■•'i.'^t, :in(l west points of the hori/on
respectively; P the pole ; and H the
heavenly body. Suppose H to be
hctween N and W. Join P H.
Here W H is the amplitude {a)
Fi, .S vn u '^ *'"'^"^'" H P N we have
, ^. '^^ ^he latitude (<p), H P the object's
poar distance (90 -Jj, and H N P a right angle. Also, if
Hence "'^"^ " ^ ^^'^°~'' ^"'^ ^ H^go'-a.
sin a= sec <p sin 3
cos ;:= — tan <p tan S^
From the second of these equations we can calculate
the tune at which the heavenly body rises and sets.
i
'i
102
Parallax.
TO FIND THE EQUATORIAL HORIZONTAL PARALLAX OF A
HEAVENLY BODY AT A GIVEN DISTANCE FROM THE
CENTRE OF THE EARTH.
Referring to the Hgure in the next article, if A is the
observer's position H' will be the apparent position of the
heavenly body, and if C be the centre of the earth the
equatorial horizontal parallax will be the angle H'. Desig-
nating A C by r, A H' or C H by d, and the parallax by />,
we have
sin p = — ,
a
TO FIND THE PARALLAX IN ALTITUDE, THE EARTH BEING
REGARDEI'^ AS A SPHERE.
In Fig. 2g A is the observ-
er's position, Z the zenith,
C H the rational horizon.
A H' the sensible horizon, and
S the heavenly body. Let p
be the horizontal parallax
(H'), p' the parallax in altitude
(S), h the altitude (S A H'),
and d the distance of the /,,„ .,,
heavenly body (S C). From the trian ,de S A C we have
!i" ^__ — _ s'" S _ A C
sin Z A S ~ sin S A'"C~ S^
= —7- = sin 6
a ^
or sin/)' = cos // sin p
The angles/) and p' being (except in the case of the
moon) very small, we may substitute them for their sines,
and the equation becomes
p'=p cos h
STAR CATALOGUES.
If we want to find the position of a star not included
amongst the small number (197) given in the Nautical
Almanac we must refer to a star catalogue. In these
J
J
Star Catalaf;ues.
103
catalogues the stars are arranj^^ed in the order of their
right ascensions, with the <hita for fin.ling their apparent
right ascensions and declinations at any given date These
co-ordinates are always changing, ist. by precession, nuta-
tion, and aberration, which cause onlv apparent changes of
position ; 2ndly, by the proper .notions of the stars them-
selves amongst each other. In the catalogues the stars
are referred to a mean equator and a mean equinox at
some assumed epoch. The place of a star so referred is
called Its mean place at that time; that of a star referred
to the true equator and true equinox its Irne place; and
that m which the star appears to the observer in motion
Its apparent place. The mean place at any time can be
found from that of the catalogue by applying the
precession and the proper motion for the time that has
elapsed since the epoch of the catalogue; the true place will
then be found by correcting the mean place for nutation •
and, lastly the apparent place is found by correcting the
true place for aberraticn.
The most noteworthy star catalogues are the British
Assoaation Catalogue (B. A. C.) containing 8,^77 stars,
the Greenwich catalogues, Lalande's, containing nearW
50,000, Struve's, Argelander's, &c., &c.
DIFFERENTIAL VARIATIONS OF CO-ORDINATES.
It is often necessary in practical astronomy to deter-
mine what effect given variations of the data will produce
m the quantities computed from them. If the variations
are very small the simpler differential equations may be
used. The most useful differential formula, as regards
spherical triangles, are deduced as follows :
We have the fundamental equations:
cos a = cos b cos c + sin b sin c cos A
sm a cos B = cos 6 sin c-sin b cos c cos A 1
sin a sin B = sin 6 sin A ( \
smacosC = sin6cosc-cos6sinccosAl'
sin a sin C=sm c sin A I
I
104
Differential Variations.
Differentiating: the first equatici of this ^aouirand
charijing signs, we have ^
sin a da=sm b cos c db + co. b sin . rfc-cos b sin . cos A .//.
^ —sin b cos c cos A ^/c-f sin /; sin c sin A c/ A
-(sin b cos c— cos6 sin c cos A) (/6 +
(cos b sin .-sin 6 cos c cos A) .^c + sin /. sin c sin A </ A
=sin a cos C rf6 + sin a cos B ./c-fsin 6 sin c sin A ./«
or rffl=cos C rf^ + cos B rfc + sin b sin c'!" ^,/ A
=cos C rf/; + cos B dc+m^ b sin C d A ^'" "
Similarly we obtain
~nn! ^ Z^+^^^-^o/ A ^c = sin c sin A d P, ' (2)
-cos B rffl-cos A db+dc^sin a sin B d C j
From these, by eliminating da, we obtain :
sm C db~cos a sin B rfc=sin b m'i r ^ \ , ■
-cousin CV.+s,;„ B ..JsTnlTos B ^;J::;L' H^i^
and by eliminating rf6 from these :
«'"«sinBrfc^cosWA+cos«./B + c^C (4)
If we eliminate d A from (3) we get
cos 6 sin C i6-cos c sin B ^c^sin c cos B d B
—sin 6 cos C ^ C
and, by dividing this equation by sin b sin C or its
equivalent sin c sin B, we have
cot 6 db-cot c dc^cot B rf B-cot Cde (5)
and put" '''"^''' '^^" '^" -^tronomical triangle P Z S.
A «-Z fl=9o°— ^
^ =9 c=go° — ^
Then the first equations of (2) and (3) give
which determme the errors i ^ and rf ^ in the values of ^
and t computed according to the formul«,
Differential Variatims,
K^S
cos (J sin /= ^ '^"^ . -sin ^ sin ; cos Zr (7)
(which are derived directly from the'^un l"" ^' ,
tions(i)), when '/-.n.i . I 'undaniental cqua-
In a similar manner we obtain
-sin c/ir~i°'''i'?*^i" ? =« ?'' '-cos Z,(^ I
(It seems almost superfluous to point out that in ..
formula. ^ is the latitude. J the starCZ ! '^
EQUAL ALTITUDES OF A FIXED STAR
.h 'U.r,i:„t2:;:."r. r'it°"ob' '"^ ^'-^"'■"^
apparent altitudes will no, , °''='-'"«'ons, equal
find tl>e cha g„ 1 ' "IT "''"" .''"^ '"""<'"• To
change A» in^hetl'tirude' trL^^titrd";^" '' "
the equation. "'^ **^ differentiate
Sin«^sin^sinoVcos^coso-cos^
Regardin^r <p and o^ as constant : whence
Cos«A«=-cos^cos<Jsin^i5 A^
wh.e.« is in seconds .fare, and ./in 3econds of
If the altitude at the west observation IS the greater by
|i
xo6
Effect of Errors.
£f.m the hour angle is increased bv a/, and the middle
time is Jucr-^ased by -—, wliich is thei "fore the correction
for the difference of altitudes. From the above equation
its value is
A « cos «
30 cos tp cos d sin t
If A is the azimuth of the object, we have
,,. . cos 8 sin t
Sm A =- ^
cos u
and the formula may be written
A a
30 cos <f sin k
which will be least when the denominator is greatest ;
that is, when A=(jo° or 270*, The star is therefore best
observed on or near the prime vertical. Low altitudes
are, however, best, owing to unccrtaintv in the refraction.
If the star's declinat'cn is nearly equal to the latitude
the interval between th3 observations will be short, which
is an advantage, as the instrument will jc less liable to
change.
EFFECT OF ERRORS IN THE DATA UPON THE TIME COM-
PUTED FROM AN ALTnUDH.
We have, from the first differential equation (8), multi-
plying At by 15 to reduce it to seconds of arc,
15 sin q cos d dt = ^^-)-cos Z t/^-f cos q d d
If the zenith distance above is erroneous we have
d<p=o, and d 8=0, and
sm q cos o cos ^ sir Z
from whirh it follows that a given erro;- in the al.itude will
have the 1. -^t effect upon the time when the object is on
the prim.- v. 'n : A!fo, that thest observations giv«;
the most ncc., te r* suits when the pla:e is on the
equator, aifd ib«.- ;east accurat? when at the poles.
Effect of Errors.
toy
have
By putting d ^=0, </ J=o, and sin q cos (J=cos ip sin Z,
we have
by which
that a
d<p
cos <p tan Z
latitude also pro-
dares the least effsct when the star is on the prime verti-
cal, or tlje observer on tlie equator. In the former case
tan Z is infinite; therefore, if the latitude is uncertain,
we can still j^et {jood results by observing,' stars near the
I>rime vertical.
U d(^ — o and d(p=o we have
*^ cos tan q
Hance an error iu the star's djclination produces the
l«;ast effect when the star is on the prime vertical (since
tan (7 is a maximum when sin Z=i), and that, of different
stars, those near the equator are the best to observe.
In high latitudes it will often be necessary, in order to
avoid low altitudes, to obsf.rve stars at a distance from
the prime vertical. In this case small errors in the data
v/ill affect the clock correction. But if the star is ob-
served on successive days on the same side of the meri-
dian at about the same azimuth, the clock's rate will be
accurately obtained, though its actual error will be un-
certain.
If the same star is observed both east and west of the
meridian, and at the same distance from it, constant
errors d<p, dd, and d^, will give the same value oi dt, but
with opposite signs. Hence one clock correction will be
too large, and the other too small, and by the same
amount, and their mean will be the true clock correction
at the time of the star's meridian transit.
EFFECT OF ERRORS OF ZENITH DISTANCE, DECLINATION,
AND TIME, UPON THE LATITUDE FOUND BY
CIRCUM-MERIDIAN ALTITUDES.
^ The formula for finding the meridian zenith distance
C' from a circum-meridian zenith distance Q is
i
I
|[*gr;*f
io8
The probable error.
c'-:-~A
m
where A =^^^l^^^^ ^
Differentiating
have, since dip=clC + dd
sin (J*'
and m
2 sin^
sin i'
and regarding A as constant,
we
the,, whole amount. The coefficient of dl ha, onnosite
signs for east and west hour anrfes ■ therefor^ IT
va.,o„s are taken of a number of I'lis of"nna ' a ti^nT"
east and west of the meridian, a Lah eons^t e,™ „'
n e"„lt' (o.- clock correction, win he eli,ni„:.:
takint th^^"' ,' '""" "= P"<:'i':ally attained by
taking the same number of observations at each side of
the mend,an, and at nearly equal intervals of time
elimina'ted h'" "" '''™'"^ '""""'<= ^''''^'' ''ff<=c's A is
fo^rb'yihtTsni'r"'"""""""'™ "'"■ "- '-■"■"=
THE PROBABLE ERROR.
, '""^' ^ "■= "■'" suppose a rifleman to have lired ,
Ia.^e number of bullets at a targe, at the sam rang and
target 'it f" "J !™'"^' '"^ '^'"' °" -amini^rthe
the fe of^h" 'f '^^"f ?f"- have struck Within
one shot iLng withif^ r^'tt'cete^treve-n-h','
other words, that it is an even chance whether or no. th
bullet w,ll stnke within that distance or not. It^d this
d,sja„ce may be taken as the probable error of any on,
car1?uT' 1 "" ""''" " '"""= "f independent but equally
careful measurements of a given quantity, such Is an
i
The probable error.
?tant, we
log
anjjle or a base line, they will all differ more or^^^T^e
closeness of the agreement depending on the instruments
employed and the care exercised; and the problem is to
decide what value is to be taken as the most likely to be
he correct one-in other words to have the smallest
piobable error.
If «n «2, 'H,8cc., are thedifferent measurements,;? their
number, and m their mean, then ;«=- -»-'^-2±_^^-__JLf^nJ
and it follows as an arithmetical consequence" that the
algebraical sum of the quantities {m~a\) (m-a ) Z
irr;! ''^ ^? ^^^-' ^° -- These^^iJ: titt't;'
called the "rcstduals." Another property o the mean I
&c. IS a miniimini. '' '
Now it admits of proof timt the mean is that value de-
nved from the various measurements, which is likely to
'her:,:?^ '"■"■• ^'-™-o rtheprohahleerroVof
~~~ir^~= - "" 0.674489 '
And the probable error of any one measurement is the
probable error of the mean multiplied by ^-
It must be borne in mind that by the probable error
m, aken as so much is meant that it is an even ch nee
that the value taken ,s within that much of the truth
without i.gard to sign. Thus, if / be the mean of a
nnmber of measurements of a base line, and i foot ts
probable error, it is an even chance that its real vLue
lies between /— i and /+ 1.
Instead of using the probable error of a result we often ^ ^
employ what is called its .ei.kt; a function which Ldi
cates he relative value to be assigned to the results as
regards precision. ^^
t^z
no
The probable error.
The formula for the weight is
'■' Probable &vvot=^^^M^
V weight
(2)
JnVu ^^^^^ ''^"'' ^"^^^'^'y ^= the square of the
of the residuals being a minimum in the case ofthe mean
thzs method zs often called the "method of least squares"'
As a simple example ofthe calculation ofthe probable
error we w.ll take a side of a triangle forming part of a
tnangulat.on carried out near Kingston in 1881-82. Four
length, and the results were:
I
2
3
4
1060.1 yards
1060. g
1060.6
1060.4
4)4242.0
Mean= 1060.5
Here the squares ofthe residuals, in tenths of yards, are
I.
2.
3.
4.
16
16
I
I
Total 34
And the probable error ofthe mean is
V 34
-.-X 0.674489x3-6 inches.
=:3.54^nches.
e of the
squares
e mean,
uares."
robable
rt of a
. Four
ain its
Part II.
in i
GEODESY.
'I '
s, are
CHAPTER I.
. K
THE FIGURE OF THE EARTH.
Geodesy is a word of Greek derivation, and signifies
division of the earth." Broadly speaking, it comprises
all surveying operations of such magnitude that the fig-
ure of the earth has to be taken into consideration.
The earth is an oblate spheroid— that is, the figure
forqied by the revolution of an ellipse round its
minor a:as-the polar axis being shorter than the
equatorial by about 26-88 miles. This has been
proved in two ways. Firstly, by pendulum experiments,
which shew that the force of gravity increases from the
equator towards the poles ; secondly, by actual measure-
! Kl
i\ 1
11
112
Figure of the earth.
ments of portions of meridianal arcs. A^it^considor-.
ion wm show that if the curvature of a mcridiana 1 '
elhpt.caUnd therefore decreasing towards the poles the
length on the earth's surface of a degree of latitud'mu
be greater in high than in low latitudes. That is Ta
and B are two points on a meridian near thf ec ua of
and C tTL t' ' "'•''" """"^^ ^" astronomical ladt^.d:.'
and C and D two ponits on a meridian in a high latitude
and also differing by the same amount, then ft,
tances A B and C D are n.easured on the ground 11
will be found to be less than C D. This has a tually
been done at various parts of the earth's surface-La'^
land, Peru France, Russia, (where an arc of ovear^vTs
adopted ,s to measure a base very accnratelv -ind
from It to connect by means of a chain of trian' Jl.t on
two distant stations which are as nearly as po Sbl o"
he same meridian. This being done we can calculate
the actual distance from one of the stations to the pJi I
where a perpendicular drawn to the meridian of th it s "
S. f "f r^her station meets the meridian. The
latitudes of the two stations are found by very care ul
astronomical observations, and their difference, ^akenTn
connection with the calculated distance on the meridian
gives the curvature of the arc, since the radius of curva
atitude in circular measure. There is, however, one
the la;> 7"V" ^^.*^™"-^--^ of this kind. In fi iding
the la tudes of stations we are in general dependent on
the direction of the plumb line; and should there, as
often happens, be a local abnormal deviation of the
later from the true perpendicular, the resulting latitude'
wi 1 be erroneous. This was proved many years agX
taking the latitudes of two stations on oppJte side^ o a
mountain in Perthshire, and measuring Jhe true hoi 'l!
tal distance between them, when it was found that the
^^^e of the Earth.
Betweeen A iind B... 2=;' co
•tJ and C 17"
while .he actual differences, as found by triangula.ion.
Between A and B 24" 2^7
B and C r4".ig
As a rule, the deviation seldom exceeds a f.yv seconds
except in the neighbourhood of great mount.i^ Z
as at the foot of the Himalayas. tvher^eT -"rmXas'
Where there is considerable deviation in level countries
It .s no doubt caused by neighbouring portions orthe
Z %TJ:7 '''-' ''''-'' - li.lfter than the avt!
sy..rnetrica,ly g.ouped round ita.::Wn ^n tL t^l^^
and ong..uae of each obtained by astron^micll obta
t ons. The actual distance and azimuth of the central
station rem each of the others being known by tr'an 1
latitude and lon"itude of th. 7 1 ^'^ ""^^^
Dared with ,v\ *''^ ''^"''"^^ station being com-
pared wuh the latuude and longitude as obtained by
ill
r-i
i 'i
1^
V
114
Figure of the Earth.
if
astronomical observations will give the deviation of the
plumb line.
If a and b are the semi-major and semi-minor axes of an
ellipse, the distance of the centre from either focus is
|/rt»Z_^3, and this quantity divided by a is called the
"eccentricity." This is generally written e. The quantity
a
-b.
is called the "compression" or "ellipticity," and is
denoted by c. The latest calculations make the com-
pression of the earth about ^^j, the ratio of the semi-
axes being believed to be 292 to 293. The true measure of
the compression is the difference of the semi-axes divided
by the mean radius of curvature of the spheroid. The
equator has also been found to be elliptical, its major
axis being about *oo yards longer than its rninor axis.
It should be noted that the expression e has different
meanings in different books. English writers occasion-
ally employ it for the compression or ellipticity, while in
American books it is used in the same sense as here,
namely, for the eccentricity. Even in different chapters of
the same work the letter e is often used both for the
compression and the eccentricity.
The accompanying |
figure represents a
section of the earth.
PP' is the polar axis, |
QE an equatorial di-
ameter, C the centre, I
F a focus of the
ellipse, A a point on
the surface, A T a
tangent at A, and!
Z A O perpendicular ^''s- 3°
to A T. Z' is the geocentric zenith, and Z' C E' is its
declination. The latter is called the geocentric or reduced lati-
Figure of the Earth.
"5
/«rf. of A. Z O' E' is thegeographkal ^r^^^tZ^^iiMA^^^
t •.°' ^^^'^ ^^"^d tJ^e reduction of the latitude.
It IS evident that the geocentric is always less than the
geographical latitude.
LetCE=:«. CP=6. Let .- be the compression and .
the eccentricrty.
a—b
a ,
CF
_b_
a
e=
C F2
CE3'
CF
CE ~P F
C E3 "■^~ (^£2
63
^i-^^i_(i_,).
That is, e2.
or, c= i/gcH^ (I)
TO FIND THE REDUCTION OF THE LATITUDE.
Taking the centre of the ellipse as the origin of axes
the equation of the ellipse will be
«? ^ 62 ^
Let ^ be the geographical latitude
9* " geocentric "
We have, tan a>=^— ^
dy
and from the triangle ACB, tan ^'=
X
or.
Differentiating the equation of the ellipse, we have
_y_^ b^dx
X a2 dy
tan 5^= -^tan ^—(1-^2) tan ,p
lo hnd the reduction, or^_^', we use the general
development m series of an equation of the form
tan A'=-/) tan y, which is
x—y=^q sin 2y-fj ^2 sin 4v4-&c.
1:
i
1^
ii6
Figure of the Earth.
in which (7=^-—
Applying this to the development of (2) we find, after
dividing by sin i" to reduce the terms of the series to
seconds, and putting x=^<p' , y=^(p.
9—9 —
- J „ sm 2 <p ^ „sm 4 W—&C. (z)
sm I 2 sm I ~ ^"^
in which q-
I — C2
The known value of e gives ^, i.nd thence <p—tp' for
any given value of f .
N.B.— 5^ is negative, and q^ is very small compared
with it ; therefore (f — if' is positive.
In some books on geodesy t'l ; ex;iri!S'iion "rcluction of
tbe latitule" is applieJ to the aiig!.; A' C E, whire A' is
the point in wliicli BA p-jhcid me-ts the circle de-
scribed with centre C and radius C E. Let this angle be f" .
tan (p' _ B A_ 6
a
Then
tan <p" BA'"
by the properties of the ellipse. And since tan '/
.2 t^" <P
we have
and
tan ^ a2 ^ ^
T — -»r-= 7— tan (p -~ -r-tan c>=- —
tan <p b2 ^ b ^ b
tan <p tan <p"
tan f" tan ^'
TO FIND THE RADIUS OF THE TERRESTRIAL SPHEROID FOR
A GIVEN LATITUDE.
Let p (or A C) be the radius for latitude ^,
We have, p — \/ ^9 ^ yi
To express x and y in terms of cp, we have, substituting
I — e* for-^ in the equation to the ellipse and its differ-
ential equation,
"^
(5)
tj on i>f
"^
Figure of the Earth.
u9
117
^» +
J'^
I— ^a
-a^
-~ =• (i~e') tcixi <p
X
whence, by elimination, we find
^^ a cos Y>
Vi—e^
^nd hence, p=a{^IZ3jl^}l
\ I — I
Vi~e^ sin' <p
(i — «') a sin ^
sin^ ^
= a i ^ZZ3J^^}^9 + e* sin» <p . ^
■e^ sin' <p j
(4)
TO FIND THE LENGTH OF THE GREAT NORMAL, A O, FOR
A GIVEN LATITUDE.
From the figure we have
Great normal=^.^^L?l
cos f
the"'normaI.'° ^''" "°™" "'" '" '""'''" "' =™P'^ ^=
g_cosj p sin i'' _
TO FIND THE RADIUS OF CURVATURE OF THE TERRESTRIAL
MERIDIAN FOR A GIVEN LATITUDE.
Denote this radius by R.
We have, from the Differential Calculus,
^{^ - (I)')*
i-1 f\
zx8
Figure of the Earth.
From the equation to the ellipse we have
dy _ bj X
dx~ a^ y
d^ _ _ _b*_
dx^ "
whence
R
a2y*
{a*y2-\-b^x^)i
a* h*
Observing that b^ = a^ (i—e^), we find, by substitut-
ing the values of a- and j/ in terms of ^ (page 11^.)
R =^ ci(i-e^) ^ jy^, , (6)
(i— c2 sin2 (p)\ o^-^y' ^
This last equation gives the length of a second of lati-
tude at a given latitude, since it is equal to R sin i"
The following formula is sometimes used for the radius
of curvature of the meridian,
R = — \ {a — b) cos 2 y
It also admits of proof that the normal at any point is
the radius of curvature of a section of the earth's surface
through the normal and at right angles to the meridian.
From equations (5) and (6) we see that the normal at
any point is always greater than the radius of curvature
of the meridian at that point.
If the earth were a sphere the shortest line on- its sur-
face between any two points A and B (otherwise called
the geodesic line) would be an arc of a great circle, and
the azimuth of A at B would differ from that of B at A
by 180 .^ But on the surface of a spheroid the geodesic
line is, except when both points are on the equator or on
the same meridian, a curve of double curvature. The two
azimuths, also, will not, except in certain cases, differ
from each other by exactly *8©^.,^ The reason of this is
that the vertical plane at A passing through B will not
-- !'.
Figure of the Earth.
"9
coincide with the vertical plane at B v^^^i^^^^~;^
These two planes will, of course, intersect'at A and b"
, /'^^'V"^^""^^^^'""^ ^vith the surface of the spheroid
pace I„ add, ...n to these two lines and the geodesic
l-ne there w.il also be what is known as the line !/ «Z
'-"/o the two points-that is the line on everVpoi '"f
wh.ch the hne of sight of the telescope of a theoVoh h
dotted "'J"^^•"^"^ -'^ ^->'y levelled would, wh
directed on one station, intersect the other on the tele
scope being turned over.
CHAPTER II.
OEODETICAL OPERATIONS.
The methods adopted in the old world for mapping
large tracts of country have been reversed in America.
Instead of starting from carefully measured bases, and
carrying out chains of triangulation connecting various
principal points in such a manner that the relative
positions of the latter with respect to each other may be
ascertained within a few inches, though several hundred
miles apart, the system pursued (if we except the U. S.
Coast Survey and some other triangulations) has been
to take certain meridians and parallels of latitude inter-
secting each other ; to trace and mark out these meridians
and parallels on the ground ; to divide the figures
enclosed by them into blocks or "checks ;" and to further
subdivide the latter into townships, sections, and quarter
sections. Although the method of triangulation is incom-
parably the most accurate, the American plan has the
advantage of rapidity and cheapness. As the latter is
very simple, and is fully explained in the Canadian Gov-
ernment Manual of Survey, it will not be further touched
upon here.
At the commencement of a triangulation a piece of
tolerably level ground having been selected, a base line,
Tnauffulatiofi.
. lai
cliam of tna,n.|,.s ,, s,,^,t«l. I„ ,i,e f„r,„e,. case th^
nancies are expanded as rapidly as possible ,11 ,he„ Ire
lame enourf, to cover the whole country with a netwo k
"f pntnary tr,a,„des. This is done by taking ang losfrl
d tan , f " "" "'°""""" '"""■ ■•""' calcnia ing thei
I stances by ,ng„,„„ctry. The instru.nont is then placed
enTt : in '":r"'r r'""': -■" -«'- -"- S
;.ei; Id';': rb :ere,''° ■r',:;,'':j"'-'--'^ »-
and e.te„de<l til, the whl^isV;;:, ':::r:d Irh^e
p™,ary tnangles the sides of which should be astargH:
Smaller, or secondary, triangles are formed within the
pr.mary ones to fix the position of important poins
wh.ch may serve as starting points for raverses &
da;;'rer""' " ""^°'-'™- f°™^<' withi„ theL™;
cit^uLtScl.'" T5;eir:^es '::etft 't '"""'-"^ '»
=>• iiitir siaes die often from ^o to 60 nr
The louges s.de ■„ the British triangulation was ,T
The stdes of the secondary triangles are from abou , to
20 mtles, and those of the tertiary triangles fiveor"Ls
The larger triangles should be as nearly equilateral as
.rcmnstances admit of. The reason for having te„ so
■s ha, w„h ,h,sform small errors in ,he measuremen"
of ,he,r angles wll have a minimum effec, on the cdcn
.e, lengths of the sides. Such triangles are a ed
"well-conditioned" ones.
The original base has to be reduced to the level of the
re^tTc fthri^'h :r f "-^^ '^^^^^^" ^'^ ^^^-^^ -^^r
verticals through its ends intersect the sea level must be
V-.** e
122
Tnangulation.
II
ascertained. The exact geographical position of one end,
and the azimuth of the other with respect to it, must of
course be known. The angles of all the principal triangles
must be measured with the greatest exactness that the
best instruments admit of, the lengths of the sides calcu-
lated by trigonometry, and their azimuths worked out.
The work (when carried on on a very large scale) is still
further complicated by the earth's surface being not a
sphere but a spheroid. The accuracy of the triangulation
istestedby what is called a "base of verification." That is,
a side of one of the small triangles is made to lie on
suitaole ground, where it can be actually measured. Its
length, as thus obtained, compared with that given by
calculation through the chain of triangles, shows what
reliance can be placed on the intermediate work.
As instances: The triangulation commenced at the
Lough Foyle base in the North of Ireland was carried
through a long chain of triangles to a base of verification
on Salisbury plain, and the actual measured length of the
latter was found to differ only 5 inches from the length as
calculated through several hundred miles of triangulation.
An original base was measured at Fire Island, near New
York, and afterwards connected with a base of verifica-
tion on Kent Island in Chespeake Bay. The actual
distance between them was 208 miles, and the distance
through the 32 intervening triangles 320. The difference
between the computed and measured lengths of the base
of verification was only 4 inches. In Algiers, two bases
about 10 kilometres long were connected by a chain of 88
triangles. Their calculated and measured distances
agreed within 16 inches.
If the country to be triangulated is very extensive—
as, for instance, in the case of India— instead of covering
it with a network of triangulation, it may^ntersected in
the first place by chains of triangles, either single or double.
Base Lines.
123
and bases measured at certain places/usually wher--^ these
chains meet. In India the chains run generally either
north and south or east and west, and form a great frame
or lattice work on which to found the further survey of
the country. A double chain of triangles forms, of
course a series of quadrilateral figures, in each of which
both the diagonals, as well as the sides, may be calculated.
The following is a brief account of the measurements
ot some celebrated base lines :
In 1736 a base line had to be measured in Lapland for
the purpose of finding the length of an arc of the meridian
by triangulation. A distance of about 9 miles was mea-
ured in mid winter on the frozen surface of the River
Tornea. By means of a standard toise brought from
France, a length of exactly 5 toises (about 32 feet) was
marked on the inside wall of a hut, and eight rods of pine
terminated with metal studs for contact, cut to this exact
length. It had been previously ascertained that changes
of temperature had no apparent effect on their length
The surveying party was divided into two, each taking
four rods, and two independent measurements of the base
were made, the results agreeing within four inches. The
time occupied was seven days. The rods were probably
placed end to end on the surface of the snow.
The same year a base 7.6 miles long was measured near
yuito in Peru, at an altitude of nearly 8000 feet The
work occupied 29 days. Rods 20 feet long, terminated
at each end by copper plates for contact, were used
The rods were laid horizontally, changes of level being
effected by a plummet suspended by a fine hair. The
rods were compared daily with a toise marked on an
iron bar which had been laid off from a standard toise
brought from Paris. This base was the commencement
of a chain of triangles for the measurement of a meri-
dianal arc. Three years later another base, 6.4 miles long
124
Base Lines.
was nieasured near the south end of this chain and onlv
occnp,ed ten days. The party was divided into'.wo c:™"
pan,es wh.ch measured tne line in opposite directions
melced'trtTr'™'' """"'■ "^ "'"'' ''"«^'" ™^ "-
He^th I , "^^f "'■"•"ent of a base on Hounslow
Heath, which was chosen from the great evenness and
openness of the f;ro„nd. Three deal rods, t.pped „' h
bell metal and .o feet long, were used at firs . But it was
™™i y'tf'th;-;""' r ^"1='^" ^^ ='"'"^- '" '^""■
Cth ^f t, fr^P'^"'' t''" Slass tubes of the same
iscfrti: d" » rP'""°" '°' temperature had been
ascertained, were substituted, the temperatures of the
ubes betng obtained by attached thermometers The
^eng h of the base when reduced to the sea lev and 6^
Faht. was 9,134! yards. This distance was subsequentfv
40 I nks half an mch square in section. A second siniH.r
chatn was used as a standard of comparison The chai,^
was laid in five deal coffers carried on trestles Id
kept stretched by a weight of a8 pounds t'c act Z
olT'' f "mf " ™= """''' '' =" ="''- ^^ •"
steel TJL r° "?^''^'«-^'"ems (glass tubes and
steel chains) agreed within two inches.
Two bases, each about ji miles long, were subse-
quently measured in France-one near Par s, the other at
Carcassonne in the south. Four rods were ised The '
were composed of two strips of metal in contact (patilum
and copper), forming a metallic thermometer carried o^
a stout beam of wood. Each rod was supported on two
.ron tripods fitted with levelling screws, a^d there was In
arrangement for measuring their inclination.
The Lough Foyle base was measured with Colbv's
compensation bars; an arrangement in which the ,,„e„,«
expansions and contractions of two parallel bars o differ
ant metals (brass and iron), ,0 feet long, are utili/ed o ktp
Base Lines.
bury Plain base was measured in the same way Colby's"
bars were subsequently used for ten bases inlndiabu
were not found togive very reliable results there '
sattn J7'°'?'"' ""^ ^°'^^'^ arrangement is the compen-
atmg apparatus used in the United States coast smvev ^
It consists of a bar of brass and a bar of iron a I ttl.!^
than SIX metres long and parallel to each ohe"' Th bar'.'
Sr";?^^"^'^^^"^"^^"^' but free to movl'^t th
other Their cross-sections are so arranged that llll
versejy as tHfespecific heats, allowance being made for
hei difference of conducting power. The bras bar is
tl r"; -^'--^f on rollers mounted in suspen !
to'itts ruJo: r brLvr °" ''''' -'-' '-^-'
i-
'iM,Cl tS-.
11
II
126
Base Lines.
fig. 31
The annexed figure shows the arrangement at the two
ends the left hand part being the compensation end. It
will be seen that the lever of compensation (/) is pivoted
on he lower bar (a), a knife edge on its inner side abutting
on the end of the iron bar (b.) This lever terminates at
Its upper end in a knife edge (^ in such a position that
whatever be the expansion or contraction of the bars it
always retains an invariable distance from their other
end. This knife edge presses against a collar in
the shdmg rod (d), moving in a frame (/) fixed to the
iron bar, and is kept back by the spiral spring (s). The
rod IS tipped with an agate plane (p) for contact. The
vernier {v) serves to read off the difference of lengths of
the bars as a check.
At the other end where the bars are united a sliding
rod terminates in a bluftt horizontal knife edge (g) its
inner edge abutting against a contact lever (A) pivoted at
(»)• This lever, when pressed by the sliding rod, comes
in contact with the short tail of the level (k), which is
mounted on trunnions and not balanced. For a certain
position of the sliding rod this bubble comes to the centre
and this position gives the true length of the measuring
bar. Another use of the level is to ensure a constant
pressure at the points of contact, p and g. To the lever
and level is attached the arm of a sector which gives the
inchnation of the bar.
J
^
J
Base Lines.
127
level sector and vernier, are read through glass doors
tTesVe"s T^'n'^' :"" ^"^ "-"«'<' on a ptof
rSgned'^rAl*!''""""^^'''""'-"'"^-^''^^
paSr'the'°;r'T".'""'' '°"«' '"=^^"'-='' -i"- "•- ap.
paratus, the greatest supposable error was commit.^
th 'onrtr^"'- ""^ '"^ *''^" -.::„thT^ an'
men. On another base, six and three quarter miles ?„n„
annr"''' '"" *^^ '^^^ "-an one ten"h oTan °ch'
and the greatest supposable error less than three-temhs'
GellTrttaes".'"" '"•'' '■' -"^^="""8 a base in
eorgia thre . t.mes, twice in winter and once in summer
at temperatures ranging from i8- to 10/ Faht Th^
discrepancies of the three measures with their respecliv!
-cans were, in n.illimetres,-8.ro,-o.3., and "sTi '
qul^trfec7ilr."''' "TT' "'^' "■= >PP-«- - n°t
.r^^^ure is'r„";oX •''"''"^ ™ "-""" "■'
a w^;^str:rt„^%rst:: ■" ^^r^s
surveys ,t may be sufficient to nn.asure the b».. .
F -V.C ui pjanx, which is made to adhpr#> ♦« fK^
ground by means of pointed spikes on its under L^ce!
oarnrd'^^^'' ^"" ^^^«°"^d' baked, boiled in drying oil
painted and varnished, may be u«?pd Th u f j"^ , '
be levelled or h=.v« fk T , ^"^^^ ^^°"^d either
ieveued or have their angle of inclination read. If the
li
u
<M>»i'— t'-C<i £{
t^^VU
i'Mii
ir
s
128
Base Lines.
i
Bf
I
ground is uneven they may be levelled on trestles with
sliding telescopic supports. The ends of the rods should
be capped with metal, either wedge-shaped or hemi-
spherical in form, and either placed in actual contact, or
the spaces between them measured by graduated glass
wedges. If the end of one rod has to be placed on a dif-
ferent level to that of the next a fine plumb line may be
u«=ed • or the rods may have fine lines marked at each
end of the unit of length, so that one rod may be made to
overlap the other with the two marks exactly correspond-
ing. This plan answers well on ice.
Before measuring an important base it is usual to make
a preliminary approximate measurement of the line and
also to get an accurate section of it by levelling. Suit-
able points are selected for dividing it into sections, and
these points are accurately adjusted into line by means of
a transit at one end. It may happen, however, that it is
impracticable to have all the segments in a straight line,
in which case the angles they make with each other niust,
of course,- be exactly measured. Any deviation also from
a true horizontal line must be recorded in order that the
base may be reduced to the sea level. The ends of the
base as well as of the sections, are generally marked by
microscopic dots on metallic plates let into massive
stones embedded in masonry, and are thus permanently re-
corded. The mark itself may be a minute cross on a piece
of brass, or a dot on the end of a platinum wire set
vertically in a piece of lead run into a hole in the stone.
If the rods used in measuring the base expand and con-
tract with changes of temperature the latter must be re-
corded at regular intervals of time, as the rods are at
their true length only when at a certain standard tempera-
ture.
If the base, or any portion of it, is not level, its inclina-
2s with
should
hemi-
:act, or
i glass
n a dif-
may be
it each
nade to
sspond-
to make
ine, and
. Suit-
ms, and
neans of
hat it is
jht line,
er must,
Iso from
that the
Is of the
irked by
massive
lentlyre-
)n a piece
wire set
he stone.
and con-
st be re-
is are at
tempera-
ts inclina-
Then b=.B cos d. ^^^ ^"^^« °^ inclination.
above . K As .ivenTnJnrsrhair '''''' °^ ^
B-b^B (i-cos 0)=^ B sin»i-
=i B 0i sin» I'
==0.00000004231 09 B_
If the base is intersected hv =, . •
cannot be conveniently measured .'"' °' ""'""^ ^^'^^'
as follows : ^ "measured across we may proceed
Let AB CD be I
the base, and BC
the interrupted!
portion (Fig. 32).
Let AB^a, Cd]
=b, and BC=.r.
Take an exterior]
station E and!
measure the
angles AEB (a)'
AEC (/9; and AED pi^^ 3,
(r). Then if ^ is such an angle that
tan» ^= -i_^_ j^ sin /9 sin (r~a
It may be proved that
2 2 CO? »
The base is, of course, « + 6+;,.
1
t
*-;
I
130
Base Lines.
Ifthe
nature of the ground necessitates an angle C be-
tween two portions of the base A C, C B, we can find
the direct distance A B thus : The angle C (which is very
obtuse) is measured with great care. Let i8o°— = ^,
A C=6, C B=«, and A B=c.
Then c'=a^+b^ + 2 ab cos
and (if G is not more than 10°)
COS 6=1 —, nearly.
= {a + by—abd^
(a b 0^ X
and c=(a + 6)|i-7;;3:x,2-^
t
= (fl + 6)]i-i
abe^
(« + 6)8 "^
ab 6^ sin* i'
&c.
a + b
2 irt+6)
= « + 6^<).ooooooo42 3 1 ■
6 being in minutes.
To reduce a measured base to the sea level we must
know the height of every portion of it in order to get its
mean height. Let /be the length of a rod, and/f its height;
I' its projection on the sea level, and r the radius of the
earth.
JL- J—
r ~r-\-h'
Then
or
r + h
I 1, nearly.
If w be the number of rods in the base and n l=h;
then the length of the base reduced to the sea level will be
L 1 1 — MJ — ~-^ being the mean height of all the
{ r n J 7t
rods. ,
Base Lines.
;le C be-
can find
.u
: IS very
—C=d,
we must
;o get its
its height;
us of the
J n /=L;
vel will be
of all the
131
The base thus reduced is a curve. To find the length
of th?l^." [ '.'■ ^^i'^^^^vided by .4 times the «.«;
ot the earth s radms.
MEASUREMENT OF BASES BY SOUND.
This is a rough method which has sometimes to be
adopted in hydrographic surveys of extensive shoals which
have no pomts above water. It should, if possible, only
be adopted in calm dry weather. The velocity of sound
in air is 1089.42 feet per second at 32" Faht. It is un-
effected by the wind, the barometer pressure, and the
hygrometic condition of the air. The observers are
posted at both ends of the base and are provided with
guns, watches, and thermometers. When the gun at one
end IS fired the observer at the other notes thelnterval in
seconds and fractions between the flash and the report
The guns are fired alternately from both ends at least
three times, a preparatory signal being given.
The value of the velocity of sound given above must be
quantty ^^^P-^^ure (f) by multiplying it by the
i/i+(f— 32"))Ko.o"^^^
Of course the distance is the corrected velocity multi-
plied by the mean of the observed intervals of time The
errors of observation are always considerable, but are no
greater for long distances than for short ones.
ASTRONOMICAL BASE LINES.
In cases where no suitable ground for a measured base
IS available two convenient stations may be selected as
the ends of an imaginary base line, and their latitude
and longitude, with the azimuth of one from the other
ascertamed by astronomical observations. We shall
then have the length and position of the base with more
or less accuracy, and a triangulation can be carried on
II
j •
'il
u
L
132
Bait Lines.
from it. The base chosen should be as lonpf as possible,
but not greater than one degree. None of the sides of
the triangles should be greater than the base. The azi-
muths of the sides being known, the positions cf the
observed points can be plotted by co-ordinates.
If the zenith telescope and portable transit telescope
are used the latitude can be determined within 10", the
longitude and azimuth within 30". With the sextant
these errors are at least doubled. Differences cf longitude
may be determined by flashing signals.
•^
T
■^
CHAPTER III.
TRlANaULATION.
Having diseased the measurement of base lines we
ave now to consider the triangulation. It is evTde^
that t e , tter may be commenced without waiting to om
plete the former. The first thing to be done isi select
the stations and to erect the necessary points to be ob
served, or ''signals" as they are called! L a hilly count
the mountam tops naturally offer the best stations, as being
conspicuous objects and affording the most distant vlws
In this case the s:.e of the triangles is only li.nited by the
distance at which the signals can be observed. Thul
n the Ordnance Survey of Ireland the average length of
the sides of the primary triangles was 60 miles! while ome
cTrrdTxS " "" i'" ^'^ triangulation' which wa
earned in 1879 across the Mediterranean between Spain
oDserved at a distance of 170 miles.
In a flat country lofty signals have to be erected not
only that they may be mutually visible, but in order'tha
the rays of light may not pass too close to the surface of
he earth, as they would be thereby .00 much affected b^
':!:rr,jr:ti^- ^^-^^^^^ ■•= consid.
limit advisable. If h', h"
H
are the heights of two
Hi!
134
Triangulation.
signals in feet and d their distance in miles;, then, on a
flat country cr over water, they will not, under ordinary
circumstances, be visible to each other if d is greater than
J ( W*'+ W* ) ^'I'c most difficult country of all in which
to carry out a triangulation is one that is flat and covered
with forest.
Formerly, conspicuous objects, such as the points of
church spires, were commonly used as signals; but of late
years this has not been done, because in all large triangles
it is necessary to measure all the three angles, and this
cannot well be done directly in the case of such objects.
The form of the signals varies much. Whatever kind be
used the centre of the theodolite must be placed exactly
under or over the centre of the station, and if a scaffold-
ing has to be employed the portion on which the instru-
ment is supported must be disconnected with that on
which the observers stand. One kind of signal is a ver-
tical pole with tripod supports, the pole being set up with
its summit exactly over the station. It may be sur-
mounted by two circular disks of iron at right angles to
each other. A piece of square boarding, painted white
with a vertical black stripe about four inches wide, can be
seen a long way off. Flags may be used, but are not al-
ways easy to see. A good form of signal is a hemisphere
of silvered copper with its axis vertical. This w Jl reflect
the rays of the sun in whatever position the latter may be,
but a correction for "phase" will be required, as the rays
will be reflected from different parts of the hemisphere
according to the time of day. The ordinary signal used
in the United States is a pole lo to 25 feet high, sur-
mounted by a flag, and steadied by braces. With respect
to its diameter, the rub is that for triangles with sides
not exceeding five miles it should not be more than five
inches. If more than five miles, five to eight inches.
Various other forms of special signals are used in the U. S.
Coast Survey. Amongst others may be mentioned a
Triangtilation.
135
on
pyramid of four poles, with its upper portion boarded over
and termmat.n^' in a point, <lirectly u.ulcr which the theo-
dolite IS placed. In En^dand do,d,lc scaftbldin,^s as
high as 80 feet were used, the inner scaffolding^ carryii,.
Uie instrument and the outer one the observers. In
Russia a triangulation had to be carried on over an arc
of more than 500 miles across a flat swampy country
covered with impenetrable forests, and scaffoldinf^s of as
much as 146 feet high had to be erected. On the prairies
of the Western States towers have had to be bu.It ; as also
has been done in India, where solid towers were used at
first, but were afterwards superseded by hollow ones, which
allowed the instruments to be centred vertically over the
sta ions. The centres of trigonometrical stations are gen-
erally indicated by a well-defined mark on the upper sur-
face of a block of stone buried at a sufficient distance be-
low the surface. In the Algerian triangulation the stations
were marked by flat-toppod ron. of masonry havin-^ a
vertical a.xial aperture c„ unicnting with the station
mark.
In sunlight, stations may be rendered visible at a great
distance by means of the heliostat, and at night the elec-
tric light is now much employed. . In the triangulation
across the Mediterranean already alluded to the signal
hgh s were produced by steam-engines of six-horse power
working magneto-electric machines. Tliese lights woio
placed m the fc. us of a reflector 20 inche. in diameter
consisting of a concavo-convex lens of glass with the
convex surface silvered. The curvatures of the surfaces
corrected the lens for spherical aberration, and it threw
out a cone of white light, having an amplitude of 24'
which was directed on the distant station bv a telescope'
A refracting lens, eight inches in ' diameter, was also
used, and threw the light one hundred and forty miles
There were two Spanish stations Hfty n^iles apart."
Mulhacen, 11,420 feet high, and Tetica. 6,820 feet. The
I 1
136
Triangulation.
m
two Algerian stations, 3,730 and 1,920 feet, were 66 miles
apart, and were each distant from Mulhacen about 170
miles. The labour of transporting the necessary machin-
ery, wood, water, &c., to such a height as Mulhacen was
very great. It was twenty days after everything had been
got ready before the first signal light was made out across
the sea. After that the observations were carried on un-
interruptedly. In France, night observations have been
carried on by means of a petroleum lamp placed in the
focus of a refracting lens of eight inches diameter.
MEASURING THE ANGLES.
Of late years the only instruments used for measuring
the angles of a triangulation have been theodolites of
various sizes ; the larger natures being really "alt-azimuth"
instruments. The more important and extended the sur-
vey the larger and more delicate are the instruments em-
ployed. In the great triangulation of India theodolites
of 18 and 36 inches diameter were used, the average
length of the triangle sides being about 30 miles. For the
Spanish-Algierian triangulation they had theodolites of
16 inches diameter read by four micrometers. In the
United States Coast Survey the large theodolites have
diameters of 24 and 30 inches. For the secondary and
tertiary triangles smaller instruments are used. The
method of taking the angles varies with the nature of the
instrument. The smaller ones have usually two verniers.
Those of about 8 inches diameter have three, while the
arcs of the larger ones are read by micrometers, of which
some have as many as five. In all cases errors due to
unequal graduation and false centreing are almost entire-
ly eliminated by the practice of reading all the verniers
or micrometers, and taking the same angles from differ-
ent parts of the arc. It is usual to measure all important
angles a large number of times. / r*,^' oi,^/./^-<''
/.
^<x:-:
l-UU./i.^
i^'*-* -»•:■««♦''■
•*■/-
(xxwi-tH.
i'
* X-*1->v_J_-,
56 miles
out 170
machin-
;en was
ad been
t across
! on un-
.'e been
in the
asuring
lites of
imuth"
he sur-
ts em-
•dolites
.verage
"or the
ites of
In the
3 have
ry and
The
of the
rniers,
le the
which
lue to
jntire-
irniers
differ-
Drtant
'l!<,fH-V.-
Triangulation.
1 , ^Z7
Of the smaller theodolites there are two ki^^^dT^il
ticukr reading, the telescCo can be dSed ,„ "' ''""
required I„ a ^eUe.atin/.heodomf le C:,X'^
fixed to the stand, and when the instrument Ts set 1 ' ,nr
.he purpose of measuring a horizontal angk it s ' ui.e ,
verniers. If an angle is read off on each and th« * i
Srhl""'"'^' °™' ^"'' •"= -rr;m a's ed.t
tx:zT '"^ --- ^^ '^^ saUairr Ch!
h-d.object.andclamped,!::fad"Vt r^t ;",r
Th ', f °'" '^ "6"'" =" °" 'h^ l^ft hand obtct
The lower plate ,s then clamped, the upper one set f^l
ve"rn er'ma °'\'""''"' °" ""^ right^Ldob e The
^a-;::-r^;e:^:^tr:r-ii--
a=Xti"-o--Lt<Teo^£^s
not usually give such good resul Is mighf be "m!
II
T
138
Tri angulation.
TO REDUCE A MEASURED ANGLE TO THE CENTRE OF A
STATION.
It may happen than an inaccessible object — such as
the summit of a church spire — has to be used as an angle
of an important triangle. It cannot, of course, be meas-
ured directly, but it may be found indirectly as follows :
Let ABC be the triangle and A the inaccessible point.
Take a contiguous point A' and measure the angles ABC,
BCA, BA'C, AA'B. Calculate or otherwise obtain the
distance AA' on plan.
CallBAC, A; BA'C, A';
ABA', a; and ACA', /9.
Now A + a=A' + /3.
Therefore A=A'+/3— a. .
Also, AB and AC are
known, and
(AB sin a=AA' sin AA'B
(AC sin /9=AA' sin AA'C Fig. 33.
or, since a and /9 are very small angles, if they are taken
in seconds,
JAB X o sin i" =AA' sin AA'B
(AC X /9sin i" =AA' sin AA'C
Therefore, A=:A' -"^4^^^+ ^^' ^^" ^^'C
AB sin i"
AC sin i'
CORRECTION FOR PHASE OF SIGNAL.
If the sun shines on a reflecting signal — such as a
polished cone, cylinder, or sphere — the point observed will,
in general, be on one side of the true signal, and a correc-
tion will have to be made in the measured angle. The
following is the rule in the case of a cylinder.
Let r be the radius of the base of the cylinder, Z the
horizontal angle at the point of observation between the
sun and the signal, and D the distance.
^
Triangidation.
139
L^
Then, the correction = ± ^ ^°^T
L
D sin i"
The proof is very simple.
In the case of a hemisphere the value of r will depend
on the sun's aUitude. If we call the latter A. r will
become r cos -|i , which must be substituted for r in the
above equation.
TO REDUCE AN INCLIMED ANGLE TO THE HORIZONTAL
PLANE.
It often h-pocns, as in
the case ' • gles meas-
ured wit:, u.e sextant or
- repeatinpr circle, that the
observed ang^le is inclined
to the horizontal, and a
reduction is necessary to
get the true horizontal
angle. In Fig. 34, let O
be the observer's position,
a and 6 the objects, andaO 6 Fig. 34.
the observed angle. If Z is the zenith, and vertical arcs
are drawn through a and b, meeting the horizon in A and
13, then A e B IS the angle required, a Z 6 is a spherical
triangle and by measuring the vertical angles Aa. Bb
we shall have its three sides, since ZA and Z B are each
90. Also, « Z 6.^A O B. If we call aft, A; za,.; and
Zb, z , we can obtain a Z b from the equation
sin ^-^= (si n js-z ) jm^-^-y^
2 ( sin s sin z ]
where s = ^l±£±£'
2
The arcs A«, B6 are generally small, and the differ-
C^
I'-
j;
140
Triangulation.
II i
ence of z and z therefore also small. The arcs may
therefore be s ostituted for the sines, and we have for the
correction (in seconds)
AOB— /i = |go°-l±£
V 2
f'tan^sini"-^^-^''" ■ ^
2--- ( a j cot ^ sin i"
This formula is applicable when z and / are within ^°
of go . "^
If one of the objects is on the horizon we shall have
AOB-/t=- - 2 {45 -^j 'cot h sin 1"
If, in additio: , the angle h is 90° the correction will be
ml.
THE SPHERICAL EXCESS.
The angles of a triangle measured by the theodolite
are those of a spherical triangle ; the reason being that at
each station the horizontal plate when levelled is tangen-
tial to the earth's surface at that point. We must
therefore expect to find that the three angles of a large
triangle, when added together, amount to more than 180-
and this is actually the case. The difference is called the
spherical excess." From spherical trigonometry we
know that Its amount is directly proportional to the area
of the triangle. In small triangles it is inappreciable.
An equilateral triangle of 13 miles a side would have an
excess of only one second. For one of 102 miles it would
be one minute.
Taking for granted that the spherical excesses of two
triangles are as their areas we can easily find the excess
for a triangle of area s-thus : A trirectangular triangle
has a surface of one-eighth that of the sphere, or -''-^', and
The excess, in seconds,
X v; r and s being,
its excess is go', or 324000"
will therefore be equal to ^ ^ 3^42°°
of course, in the same unit of measure
■
■
■
Triangulation.
141
Since s is very small compared~^^^ithT»~if^;^ 7"
obtained with sufficient accuracv J 7h ^ ^^
treating the triangle as it we^e Ypl e 0^'"^" '^
thus use either of the formula ^^ "^^^
« 6 sin C
or
s —
«2 sin B sin C
J. 2 sin (B+C)
according to the data given
2 y2 sin i"
of "irei^LTtrrct^nr xh" '-'- ''- ^^^^^-^^^^^^^
become ^^^ expression will then
a^ sin Cj(i-f^os^2 L) 7
2 y2 sin i''" ^ ^
radius, and L the mean latitude of the three stations
CORRECTING THE ANGLES OP A TRIANGLE.
In practice the sum of the three measured anries of a
orreSd' T' ""■'' " °"«'" '° ''=' ^'^ '^^y I' ' 'o b
me u1 wifhTuairelh!" ,"'"7"'^^ "^™ """
wuii equal care, the plan adopted s to -.dH t^
spHerica, . th^ .hp^X^.e? o,;:^. '.:%:? ^^
:ndtttwer"eTeZttE''^^Tr°""'t '? '^°-^""
e greater tnan b. Then we should subtract
from each angle -Z::£_
3
If some angles have been measured oftener, or with
greater care, than others, the amount of correction to be
z_
142
Triangulation.
L
applied to each v/ill be inversely as the weights attached
to the results of the measurements.
In the Spain-Algiers quadrilateral triangulation the
spherical excesses of th6 four triangles were
43".5o ; 6o".7 ; 7o".73 ; 54".i6
and the errors of the sums of the observed angles were
+o".i8 -o".54 +i'.84 +i".i2
CALCULATING THE SIDES OF THE TRIANGLES.
The next step is the calculation of the sides of the
triangles. Treating the latter as spherical this may be
done in three ways.
1. Using the ordinary formulas of spherical trigonome-
try. This is a very laborious method, and others which
are simpler give equally good results.
2. Delambre's method. This consists in taking ^he
chords of the sides, calculating the angles they make
with each other, and solving the plane triangle thus found.
To reduce an arc a to its chord we have
Chord=2 sin J a
or, if the arc be in terms of the radius.
Chord=a — -^^ -*
The angles made by the chords are obtained by a well-
known problem in spherical trigonometry.
3rd method, by Legendre's Theorem ; which is, that in
any spherical triangle, the sides of which are very small
compared to the radius of the sphere, if each of the
angles be diminished by one-third of the spherical excess,
the sines of these angles will be proportional to the
lengths of the opposite sides; and the triangle may
therefore be calculated as if it were a plane one.
All three methods were used in the French surveys. In
the British Ordnance survey the triangles were generally
■
well-
Triangulation.
cakulated by the second method and checke^^T^
Legendre's theorem gives very nearly accurate results
The following investigatiru shows under what circum-
stances small errors in the measurements of the anTe^f
^.:^^jt ''-' *^' "- '"» -='-/
Suppose that in a triangle r. b c we have the side b as a
measured base, and measure the angles A and C ; we have
a sin B=b sin A
If we suppose b to have been correctly measured we
-ay treat ,t as a constant; and under this supposhil
If we differentiate the above equation we shall get
, 6 cos A , .
a cos B , „
or, since —^
sm B
sin B
a
sin A
rffl=a cot A d A~a cot B i B
^ A and ^ B are here supposed to be positive and
represent small errors in the n.easurements'0 A a^d B
shill h '''r:r^ ^° ^^ ^^-^l and of the same sign we
shall have for the error of the side a,
da=a d A (cot A— cot B)
which becomes zero when A=B
sJs,?.rhaii''h!v:"=^"^^''=^'''='"'^' '>■" °' °pp-'-
du'^ ±ad k (cot A + cot B^
and smce '
sin (A + B)
cot A+cot B=ii5LiA + Bl
sm A sin B
f^osTA^=BP-JE5F(ATB)
',
144
Triangulation.
it follows that
da= ± ad A-
2 sin C
cos (A — B) -(• cos C
and da will be a minimum when A"=-B.
In either case we have the result that the best con-
ditioned triangle is the equilateral.
1
1
w i
■J
m
;st con-
CHAPTER IV.
DETERMINATION OF THE GEODETIC LATITUDES LONOI
TUDES, AND AZIMUTHS OF THE STATIONS OF A
TRIANOULATION, TAKING INTO ACCOUNT
THE ELLIPTICITY OF THE EARTH
Where the lengths of all the sides of a triangulation
have been computed it becomes necessary, in order to
plot the positions of the' stations on the chart, to obtain
their latitudes and longitudes.
The first step to be taken is to determine by means of
astronomical observations the true position of one of the
stations, and also the azimuth of one of the sides leading
from it. We can then, knowing the lengths of all the
sides of the triangles and the angles they make with each
other, deduce the azimuths of all the sides, and calculate
the latitudes and longitudes of the other stations.
Before geodetical operations had been carried to the
perfection they have now attained it was considered suf-
ficient to solve this problem by the ordinary formulae of
spherical trigonometry, taking as the radius of the earth
the radius at the mean latitude of the chain of triangles
/
L
146
Geodetic Latitudes, 6-c.
Thus in the triangle PAA'
(fig- 35) where P is the
pole of the earth, and A,
A', two stations, if the
latitude and longitude of
A were known, and also
the ler.jth and azimuth of
A A', we should havo tin-
two sides A P, A A', and
the included angle PAA',
and could use Napier's .^j,,^ ^^
analogies to determine the remaining parts of the triangle
and thus obtain the latitude and longitude of A', and the
azimuth of A at A'. But this method is deficient in
exactness, especially as regards the latitude, and the fol-
lowing has been adopted as giving better results.
Let A N be the normal at A, and suppose a sphere to
be described with centre N and radius N A meeting the
polar axis at^. Also let p A, p A' be meridians on this
sphere. We then calculate the geographicr.l position of
A , not by the ordinary formulas of spherical trigonometry
(since the side A A' is very small relatively)" but by the
series
I. a-^b—c cos A+J c2 cot b sin»*'A
+J c3 cos A sin2 A (J+cot2 b)+...
II. i8o°— B—A + c sin A cot b
4- ^ c2 sin A cos A (i + 2 cotj" b)
+ i c« sin A cos2 A cot 6 (3 + 4 cot^ b)
— J-c' sin A cot b (1 + 2 cot2 b)...
III. C=
'sif6'^"^+srrh'^"-^^°^Acot6
'^i^b ^^" ^ ^°^' ^ (I + 4 cot2 /;)_i _li sin A cot^ b...
sin
V
C^^
n .
■
y. iif<i
liaiigle,
ind the
ient in
the fol-
here to
ing the
an this
tion of
ometry
by the
J9 \\\
Jt^i...
Geodetic Latitudes, &c.
H7
Let L be the latitude of A
M be the longitude of A —A Pa
" ^^' " " A'-A'Pfl
Let AA'» K, and let Z and Z' be the angles it makes
with the meridians /A and M', respectively. Then, sub-
st.tutmg this notation in the sph- al triangL' ABC. and
expressmg by u the value of K in cernis of the radius, we
have
« — 90" — L 6 = go _ L
^ ^ Z B =^180°— Z'
'^ - " C = M' — M
which would be the values to introduce.intc. the series
I,n,m^; but m practice it is mn^^f^^oniont to count
the- a;mru.ths from o to j6o°, starting at the south and
go.ng round by the west, north, and east. TiH^k^ Z L
the azunuth of A' at A, and Z' the azimuth of A at A'
IhcMx4efe-in Fig. 35 V=i8o-Z, and ¥'=360 -Z' and
the series I, 111,111 will be changed respectively into
(a)
^ ~"L— w cos Z— ^ u2 sin i" sin^ Z tan L
-i «2 oS., ," • rj tan L
t u^ sin I sm 2 Z—
M'-M-f
sin Z
cos L
cos L
Z'—iSo' + Z— K sin Z tan L
+i M2 sin i" sin 2 Z (1 + 2 tan^ L)
the arc u being supposed to be in seconds.
-ii-*om«ti4fies-bappe«s-th*t-Ilie latitude L' is not quite
the true latitude of A'; for the latter is A' N' Q.' or the
angle made by the normal A' N' with N' Q' vvhile the
latitude given by equation (a) is the angle A' N Q The
correction of the latitude {</') is the angle N A' N'- for
A 'x\g-A 'N 'Q '=A 'RQ '-A 'N 'Q '=N 'A 'R
andsins^=-N-Nls'"-PJij^'
:' V . . N' A'
****f n n , inv G cti4itttiug the exact value of this angle it
should be noted that when the geodesic line K is more
I
i-t«
Geodetic Latitudes, Sc.
"^■5
than half a decree its amplitude in latitude on the sphere
— '■ y (/L — becomes a dilforent (]uantity — say aL — on
the ellipsoid, and that these two amplitudes of arc^s of
the same length being inversely proportional to their radii
of curvature N, R, we have
A L : rfL::N : R::i : - ' "fL
I — c' sm'' L
whence we have, very nearly
A L--i L (i+c" COS* L), and consequently
i}^^d L c* cos2 L
and therefore the corrected latitude L' is
(fl')L'— L— (»cosZ + 1j(2 sin I'sin^Ztaii L)(i+e» cos'L)
and we have in seconds,
™ K (I— g ' sin 2 L) \ _^ K
tf sin i" N~sin i"
The formulas (6) and (c) are not ordinarily used, for when
the latitude L is known on the spheroid it is used to de-
termine M' and Z' . But in this case we must introduce
L' into the values of these two unknown quantities. Now
we have the spherical triangle/) A A', giving
/»*' Tk*\ sin %i sin Z
sm (M — m)= :r-i —
cos L
and, since u is very small
^ cos L
Also, in the same triangle
cot \ (A + A')=tan \ (M'-M)^-^" ,*-.^J^J^2
cos J (L — L)
. A + A')
2 I ■
but 90° — and M
■tan \ go'
-M being
always very small angles, and A + A' being the same as
Z' — Z, we have
sini(L + L') ' ''
v---
(c') 2'=i8o° + Z— (M'— M)
cos \ (L— L'y
't-tv,
/i<>- f
6t.
^-r
■Vij Lt-,
/
f
-1l^
/
f
Geodetic Latitudes, S-c.
149
The imaginary sphere used in the above investigation
will, of course, coincide with the spheroid for the parallel
of latitude through the point A. Any plane passing
through the normal will cut the surface of the sphere in
the arc of a great circle, and the spheroid in a line, which,
for about three degrees, will be practically a geodesic
line.
The following is another way oi treating the sub-
ject. Instead of taking the n jruial at ne of the points
A A' as the radius of the imagi-iurv sph.re let us take the
normal at the point B, mid-way v^■ee.n them, as in I-i.r.
36, and for the sake
of simplicity let these
points be on the same
meridian. Let A N,
A' N' be the normals
at A A', produce them
to Z and Z' respec-
tively, and draw A c,
A' e parallel to the
major axisOE.
The astronomical lati"
tudesof the two points
are Z A e, Z' A' e. If
now we draw B C the
normal at B, C will fall between N and N'. The curve
given in the figure is the elliptical meridian. The circular
curve drawn with radius C B is not shown ; but it would
pass a little outside of A and A'. For practical purposes
we may suppose it to pass through those points. Join
C A, C A', and produce them to s and z respectively.
2Ae,zA'e' will be the latitudes of A and A' on the
imagmary sphere, one being less and the other greater
than the latitudes on the spheroid. The differences
ZAs!,Z.Az may be considered the same. Let each be
c
I ,
^-n^-u^^j^ ^
f^-V^
/ i-6
J
^srsSBESwa
i
150
Geodetic Latitudes, 6-c.
3
designated--. Let L and L' be the astronomical lati-
tudes of A and A', /. and /' their latitudes on the sphere,
and X the latitude of B. Then <J=L—L'— (/-/')
3 L + L' /+/'
^^~ — or-— , and
JzzL^^ ra dius of c urvature at B
L— L ^ normal at B "
AlsOj^-^-,-=-j__^,_ ^
therefore
I — e^
I — e^ sin2 X
I
1 + ^2 COS2 /.
L— L— (J= -
L— L'
1+^2 COS2 X
and ^=(L— L') j 1 i_ __)
( 1+^2 COS2 ; /
= (L-L') -^i^?!LL_
1+^2 COS2 yl
'^(^—L) ez COS2 ;, nearly.
The angle .J is therefore nearly the same as the correc
tion </> already investigated.
In what next follows K is the distance A A' in yards
of any two stations A, A', u the same distance in seconds
ot arc K the radms of curvature of the meridian, N the
normal (both in yards), e the eccentricity (=0.0817), and
a the equatorial radius.
Equation («') gives us the values of wand V, (V) ^ives
us M . and (c) gives Z'. If we neglect the denominato
of the fraction in (c) we have
2'=i8o- + Z— (M'— M)sini(L + L') '
or Z'^i%d' + z-
u sin Z . ,
-^^^,smnL + L')
The last term of this equation, which is the difference
t^::::^:''''' ^^^ ""^^"^' '- ^'^ —genceof.
■ ■ . cU,.-
^•f-'^i-
■tr:
/ :^a >
/ / i v . -
^yr ,.
r.^.^. I/H-Al' ••■.
■^v<.yu^
- ( M ' - /ii/ -:...vx
C : 7.. ., .-
J
I
srence
nee of r-^'"^" ^^"^
■ ■ ■ d
J
J
_^Geodetic^atitudes, &c. '■
If the triangnlation is limited^hT^^^i^i^Tr^T
convenient to express L' Af n ^^. '^ '"^y ^^ "^ore
lar co-ordinates re eld' to axes ha " " 'r^ °' "^^^"^"■
station A. the axis of/b h.rthe '?• '' °"^'" ''^'^^
axis of .the freodesic-'line iourr'^'" '' ^' ^"^ ^^^
to the meridian.The equations are ' P-P^-^icuIar
L'=
'^Rsinr-^-^iNsHTr"
M'=M± ^-
N sin I"
tan fL± — ^_ )
^ ^Rsini'V
X
cosL'
tan L'*
N sin i"
"=edi„ the next three probkms! ' '"" ''^ "»^.*«
THE LINE JOINING THEM DIRECTION OF
Here we have eiven T r ' »*
Lan<lL'weobtai„T ' ' ""'' *''• ^-"i *»">
W« have then to fl„d^/ and /'from the equations
^"~L and /'=-L'+~^
2
2
^icuIartothe^eHdiathLn^h^olh^tin-''^-''^-'-
Let y be the number of secondQ in f k
".eridian het.een L and CZlu^^Z^::"'
pointy ' from 1. ^ ' ^ ^^*^^ ^^''""th of the
.:: .1
^--o
<^'
I-IATH^
,/^ //';
_;,2i-^«='5— i-Stess;:^:'
1
152
Geodetic Latittides, &c.
Then we shall have
.V'-=-(M'— M)cos/'
j"— =/ — /' — \ sin i" x-i" tan /
x=~x N sin I"
^_jy" N sin I"
tan Z= —
M =
sin Z cos Z
K=-w* N sin I"
The signs of (L — L') and of (/ — /') must be carefully at-
tended to.
EXAMPLE.
'
\t
Given '
L=49« 4' 25"
L'=49 22 33
M' — M, or difference of longitude— 38' 47"=2327*
to find Z and K
Here L + L'-=98° 26' 58"
A ^ ^ (L+L)=49 13 29
L'— L-» o 18 8
i(L'— L)=o 9 4=544'
»
To find the value of —
2
log £2—7.81085
log \ (L— L)=2.73549
2 log cos HL + LV 0.62994
Jog — —0.17628
2 ^
Z— L-
=49° 4' 26".5 (5 being negative)
5
/'=L'+ ^-=4q°22'3i".5
1
irefully at-
=2327*
;ive)
Geodetic Latitudes, &c.
To find x"
Log (M'-M) =3.3668785
log cos //= 9.8136470
Jo&^:=3.i8o5255
X =1515"
To find the value of the 2nd term oiy"
log i sin 1 "=4.38454
2 log a;' =6.36105
log tan /=o.o6ig7
log 2nd term=o.8o756
2nd term»=o° o' 6"
/'-/ -=0 18 5
:v'=o 18 IlcsrIOgi"
To find the azimuth Z
Log a;"=3. 1805255
logy'=3.o378887
Jog" -V =0.1426368
^=125° 45' 21"
To find log N sin 1"
Log N (in yards)=6.8443224
log sin i"=4.6855749
Log N sin I "=1.5298973
To find log u"
Logj/';=3.o378887
iog cos ^=9.7666596
To find k'"^'''^^-'''"''
log «"=3.27I229I
logNsini"=i.5298973
4.8011264
J^= 63226 yards.
I
•if
\t
154
Geodetic Latitudes, &c.
'
To find the co-ordinates.
Value of X.
Log a'=3. 1805255
log N sin i"= 1.5298973.
log x=\'yio^?'i%
^=51336 yards
Value of y.
Logy;=3.o378887
log N sin i"=i. 5298973
log ^=-4.5677860
J' =36965 yards
TO COMPUTE THE DISTANCE BETWEEN TWO POINTS,
KNOWING THEIR LATITUDES AND THE A2IMUTH OF
ONE FROM THE OTHER.
Let L and L' be the latitudes, Z the azimuth, and let^ ■^'^
<, >->-
Then we sh.j.\\ have, as before
J- = g ^ (L— L ') col';
2
2
N=
,0
2
(i— e2 sin3 /l)J
/'=L'H-
«?
Assume
tan / ^
cos
,, . , „, sm / . -^
then, sm (<p — « )—- ^ — :-sm v*
'^ sm / ^
which gives «; and K-=-«" N sin 1"
The algebraic sign of cos Z will determine the sign of f,
and, consequently, whether u" is to be added to or sub-
tracted from <f.
Example —
L= 49» 4' 25" N
L'— 49 22 33 N
Z = 125 45 21
Here, as in the last example, we find d, and hence
=49 4 27
/'=49 22 32
it*
[78887
598973
177860
yards
OINTS,
PH OF
^^{.•iv
sub-
Geodetic Latitudes, &c.
155
To find the value of f.
log tan /=o.o6i9727
log cos 2=9.7666566
log tan f =0.2953161
To find <p — u.
log sin 9'=9-9503895
log sin /'-g.8802377
co-log sin /=o.i2i7320
log sin (^-«) =9-9523592
^~^=:-(63°39'3")
«=o° 31' 8"= 1868'
eifaS yard's! "'""'°" "^'""^ ™ '" "<= ""d K to be
Using the same nomenclature as before, let L be the
gtven latitude and ,n the difference of the longitudes:
Take L"=L' + ^
Assume tan ^=sin L tan Z
tan L''=taiLLsiii^-w)
2 . -i _^
2
^ W COS^/
sin Z
^.
'i-<^
««
K=?r N sin I"
The algebraic sign of tan Zwfll determine the sign of ,,
fshtd bTr""" ""^'"^^ " '^ •" "^ '"---^ - "-'n
Example-—
Let L= 49° 4' 25'/ N
2=125 45 21
w^=38' 47"=2327"
*%
156
Geodetic LatUudes, &c.
To find y>.
log sin L=9.8782652
log tan Z=o.i426358
log tan ^=0.0209020
f =— (46° 22' 42")
^— 38 47
To find L"
log tan L=o.o6i9663
log sin (f—;ji) =9.8643024
co-log sin 0-0.1463154
iog tan L' -0.0665841
- ig" 2 ^ 30
yj — w— 47* I' 29"
To find t?
L
49 4 25
49 2-z 30
L-L"=— 18' 5 "=1085"
L+L"=98 2655
log c» ^.7.81085
log(L— L")=3.0353i
2 log cos \ (L+L'0=9.62994
log ^=0.47610
8= -3"
L'_L"-^=49°22'33"
=49° 4' 26/'.5
/'=L'-
d
=49 22 31 .5
;■ !
i^
To find u" and K —
log ^=3.3668785
log cos /--g.8136471
co-log sin Z—0.0907036
log m"'=3.27I2292
log N sin i'/==-i.5298973
log K- , >ii265
K^^L.3^26 yards.
•
G<^odetic latitudes, &c.
On the North American boui^d^i^rVev "irTTs^Tr
following method was omploved in\lT2 '^^ ^^^
muths of two distant noinV I , ^ *^^ """^"^^ ^zi-
of which were known.' '' ^'^'^"'^^ ^"^ ^-^'^"^es
noJjLlrndPthUl/^^Tr'-r' °^ -^-^ B is the
sphere, we have in Ihe s^h T' '''"'^"^ *^^ ^^'"^h as a
sides PA, PBfanVtLTnrA pTef "^^J^^ ^^
find the angles A B Th; J^ ^'''^"' ''"^ have to
g'es A, B. This IS done by the usual for^I^,
tan A (A+B)-^
cos
AP-BP
cos
. AP-BP
sm — tif
tan \ (B-A)-z. ___^ v o . P
. AP+BP "^^^^a
sm ^— - 2
2
which give -^ilB and fc^
2 2
Then,A=^B_B-A
2 2
B=A±B B-A
>
«, /?, from the formul ' ' "' ''^'"'^^^ *^^ ^"^^^
sin a=
sin AP
sin /?=
sin BP
^75 -'- 4/--^
'' tln'/^o"-;' ^' ^^^ *^^ *-^ spheroidal azimuths,
igo —A )-=cos a tan (go'-A)
tan(B'-go")=cos^tan(B-9o°)
cut'^from'oXfntT ^'T t^" ^ ^°"^ ''^ ^^ ^« ^^ /
one point to another through forests. ^
L
I
p
158
Geodetic Latitudes, &c.
To find the accurate length of the arc on the surface of
the earth between two very distant points of known lati-
tude and longitude is a very difficult and not very useful
problem. It is, however, often advisable to calculate the
distances between stations that are within the limits of
triangulation, as a check upon the geodesical operations ;
and in the case of an extended line of coast, or in a wild
and difficult country where triangulation is impossible,
this problem is most useful for the purpose of laying down
upon paper a number of fixed points from which to carry
on a survey.
In the triangle PAB mentioned in the last article we
have, as before, the sides PA, PB, and the angle P, as
data. By solving the triangle we obtain the length of the
arc AB. If the azimuths can be observed at the two
stations the accuracy of the result will be greatly increas-
ed, and we can obtain the difference of longitude of the
two stations as follows : — It may be proved that the sphe-
rical excess in a spheroidal triangle is equal to that in a
spherical triangle whose vertices have the same astrono-
mical latitude and the same differences of longitude :
from whence results the rule
PA-PB
A+B
, P
tan — =
2
cos-
cos-
PA + PB
X cot
cos ^ diff. lat.
xcot
A+B
sin J sum of lat.
which gives P, or the difference of longitude.
As a rule, a small error in the latitudes is of no import-
ance unless the latitudes are small : but the azimuths
must be observed with the greatest accuracy. The angle
P being known we can get the length of the arc AB, and
must then convert it into distance on the earth's surface,
using the radius of curvature of the arc for the mean lati-
tude.
surface of
lown lati-
ery useful
culate the
! limits of
lerations ;
in a wild
npossible,
^ing down
ii to carry
irticle we
igle P, as
gth of the
: the two
y increas-
de of the
the sphe-
that in a
astrono-
)ngitude :
Devillc's Methods.
o import-
azimuths
rhe angle
AB, and
; surface,
aean lati-
,
to only a few inches in loo miles ^' ''™°"""
of solving certatan„M °'"';™"'^ ^'"P'" "^'hods
tables of Lrrhmsofl^."' '" "^'"""^ ""^ "'^''"^ °f
the distance ot oi„f I. J ^0?:;:: ^Tr 'ot°/ "T"
oT.ts.'.rt- i^ r T 4-™^^^^^^^
A and B wil, be. pra^iXls^^i^TSr^r"
.a"n7fre:s I'e^xi^i':-' rr='^- <'■^-
less than 90' by ha f thecnnv " '^ """^ '^ "'" ^^
constant the conve I "ce will f "" " "" *='^"« ^
the equator (where ttk„rh- ? "' "" ''"'^^ f"™
problems involv S^ Vo station^' ' fT^^ "", ""'^^^ '"
convergencensed ,! .. f<?.r:e:?reri:tt^^
by fo^riZ!" Tn'the'f'ir'"'™ '"'''"^ -"-^ -■■^«' ™'
thepLipie:„iy'::f:j:rciLtSd."-^'™^
1
z:.^*^ '4
ir'i
1 60
DeviHr\ hfdhods.
Prob. I. — To find the con-,. agcnceot meridians between
two points of given latitude.
Here we have only to find by a traverse table the de-
parture in chains and multiply it by the convergence for
one chain for the mean latitude.
Prob. 2. —To refer to the meridian of a point B an
azimuth reckoned from the meridian of another point A.
Calculate the convergence between the two points, and
add or suMract it from the given azimuth according as B
is east or west of A.
Prob. 3.— Given the latitude and longitude of a station
A, and the azimuth and distance of another station B, to
find the latitude ;ind longitude of the latter.
The distance and azimuth being giveM we can find the
departure and distance in latitude of B approximately by
the traverse table, and have the approximate mean lati-
tude. We next l..id the mean azimuth by multiplying
the departure by the convergence for one chain at the
nriean latitude, and applying the convergence thus ob-
ained to the azimuth of B at A, which gives the azimuth
of A at B, and hence the mean azimuth.
To get the correct In^-'tude of B we aultiply the dis-
tance by the cosine of t» mean azimutii and by the value
of one chain in seconds of latitude. This gives the differ-
erence of latitude of the t-. - tations in second?.
Similarly, the difference of longitude 01 ihe stations is
found by multiplying the distance by the p- . of the mean
azimuth and by the value of one ' hain 'n seconds of
longitude.
Prob. 4.— To correct a traverse i the ^un' s azimuth.
On a traverse survey of any extent the direction of the
lines must be corrected from time to time by astronomi-
cal observations, usually either of the sun or the pole star.
1
■
Deville's Methods.
,
If the traverse is commencod at a station A JiTiT""!
be equally d.stnbu.ed among the cour.es CSn.i" t
a westerly direction, A. station H, or the e„d of X /h
^ui:erti:ttxts:r-^^'"-^'-='^
rerUv rnn f I , ■ ^ ' *"® tfaverse had been cor-
267* Ji' 50'
49 5
268 055
267 59 10
M5
have been the same as the plate
readiP': but the latter was i' 45"
tooliuie. O ■ seventh of this,
or 15", is th( rection for each'
course, and we nave to add 15"
to the plate reading of the fir^st
course, 30" to that of the second,
and so on.
Unless the line is a north and co„fK
Ji" be continually chan At^plt" " 'poi^rTts
direction can he checlied at anv time 1,„ fil- "
niuth astronomically to ascertain fthis'isXf,-,"' "t"
to be after allowing for the convergence Thet ^^
.0 find the approximate difference of a.Lde fro n'fh"
ed. Th,s w,ll g,va the latituae of the station l^d
ptp"
\
162
Deville's Methods.
the mean latitude approximately. The latter being
known, the azimuth an ' distance give the convergence,
which being applied to the initial azimuth the true azi-
muth is obtained.
Prob. 6.— To lay out a given figure on the ground, cor-
recting the courses by astronomical observations.
Take as an instance a square ABCD, the side AB
being commenced at A with a given azimuth. The course
is to be corrected by observations at the other three corners.
The convergence between A and B being found in the
usual manner and applied to the original azimuth (in
addition to the angle rit the corner) gives us the azimuth
of BC. Similarly, the convergence between A and C will
give us the azimuth of CD ; and so on.
Prob. 7. — To lay out a parallel of latitude by chords of
a given length.
The angle of deflection between two chords is the con-
vergence of meridians for the length of a chord, and the
azimuths will be go° minus half the convergence and 270*
plus half the convergence. The convergence is found in
the usual way.
Prob. 8.— To lay out a parallel of latitude by offsets.
A parallel may be laid out by running a line perpendicu-
lar to a meridian and measuring offsets towards the near-
est pole. The length of an offset is its distance from the
meridian multiplied by the sine of half the convergence
for that distance; or (since the distance is in this case the
same as the departure) the square of the distance multi-
plied by the sine of half the convergence for one chain.
As this angle is small the logarithm of its sine is ob-
tained by adding the logarithm of the sine of half a second
to the logarithm of the convergence for one chain de-
parture. *
i
When the offsets are equidistant any one of them may
%^J
:er being
ivergence,
true azi-
und, cor-
s,
side AB
he course
e corners,
id in the
muth (in
; azimuth
nd C will
:hords of
the con-
and the
and 270*
found in
jffsets.
■pendicu-
the near-
from the
vergence
case the
ce niulti-
le chain,
e is ob-
a second
hain de-
lem may
DevilU's Methods.
163
be obtained by multiplying the first one by the squli^f
the nnmber of the offset. j h »i
It is almost superfluous to point out that in practice all
these problems are worked out by means of logarithms.
TO FIND THE AREA OF A PORTION OF THE SURFArP ni.
A SPHERE BOUNDED BV TWO PARAU ELS OF Lt,"
TUDES AND TWO MERIDIANS (SPHERICAL SOLUTION.)
Let AB and CD be the meridians and AC, BD the
parallels. Let if be the latitude of A, <p' of B and n the
ditterence of longitude of the meridians.
Now the area of
the whole portion
of the surface com-
prised between two
parallels is equal to
the area of the por-
tion of the circum-
scribing cylinder
'^'S- 37-
(the axis of which is the polar axis) contained between
the planes of the parallels produced to meet it. {Vide
second figure showing a section, in which a is the point A
and 6 the point B.)
Let r be the radius of the sphere and h the perpendicu-
lar distance between the planes.
Then the area of the spherical zone will be
27: rxh
~2Tr rxr (sin f '- -sin f)
=—2 7T r^ (sin (p'—.AXi <p)
:. the area of the portion between the two meridians
will be
^^^i , . . . ,
~ 180 ^^^'" ^ ~^'" f^
..-par-
i
164
Offsets to a Parallel.
TO FIND THE OFFSETS TO A PARALLEL OF LATITUDE.
Let PA, PBC, be meridians, AB
a portion of the parallel, AC a por-
tion of a great circle touching the
parallel at A.
It is required at a given latitude to
find the offset BC for a given dis-
tance AC.
Let X be the circular measure of AC
^o. do. BC
do.
Fig. 3«.
y
" I
do. PA
AC and BC are very small.
In triangle PCA we have cos PC=cos I cos x
=cos / (i—
) nearly,
Therefore cos /—cos PC=cos /~
2
■ l+PC . BC
or 2 sm— !— - sin — -
2 2
-cos l-
or 2 sin /-^^'^cos/, nearly,
therefore, ^=^ a?" cot /.
(or, if a; and j' are measured lengths, and R is the radius
of the earth, j>/==-^ cot/)
2 K
Next join AB by a great Circle arc. The angle BAC will
be half the convergence, and AB=AC, approximately.
Draw PD bisectmg P, and therefore at right angles to AB.
In the triangle APD we have D— 90*
converg ence
2
and rAD'='9o'
i
'
Offsetts to a Parallel.
Therefore, cos PAD=tan AD cotT
or sin^^^— I^"^^ _ a t
2 -^AD cot /, approximately
=J X cot / „
Therefore, j;=^,y2 cot /— .v sin ^5BX51?2£5
2
This is equally true if .r and ^ are measured lengths.
165
i
CHAPTER V.
METHODS OF DELINEATING A SPHERICAL SURFACE ON A
PLANE.
Since the surface of the globe is spherical, and as the
surface of a sphere cannot be rolled out flat, like that of
a cone, it is evident that maps of any large tract of coun-
try drawn on a flat sheet of paper cannot be made to ex-
actly represent the relative position of the various points.
It is necessary, therefore, to resort to some device in order
that the'points on the map may have as nearly as possible
the same relative position to each other as the corres-
ponding points on the earth's surface.
One method is to represent the points and lines of the
sphere according to the rules of perspective, or as they
would appear to the eye at some particular position with
reference to the sphere and the plane of projection.
Such a method is called z. projection. The principal pro-
jections of the sphere are the "orthographic," "stereo-
graphic," "central or gnomonic" and "globular."
A second method is to lay down the points on the map
according to some assumed mathematical law, the con-
dition to be fulfilled being that the parts of the spherical
surface to be represented, and their representations on the
map, shall be similar in their small elements. To this
^Wk-
Projections.
167
class belongs Mercator^s Projection, in which the meridians
are represented by equi-distant parallel straight line "d
the paralle s of latitude by parallel straight lines a d^t
ang es to the mendians, but of which the distances from
each other mcrease in going north or south from the
equator m such a proportion as always to give the t ue
hearings of places from one another.
The third method is to suppose a portion of the earth's
surface to be a portion of the surface of a cone whose
axis coincides with that of the earth, and whose ve ex
somewhere beyond the pole, while its surface cuts or
ouches the sphere at certain points. The conical sur
ace as then supposed to be developed as a plane. whTh
It of course admits of being. The only conical develop
as tfte ordinary polyconic."
in l\L°''"'Tf"' '''°'""°'' '^ ''"P'y ""= °"= employed
in plans and elevations. When used for the delineat on
of a spheneal surface the eye is supposed ,0 be a an "n
fin, e distance, so that the rays of ifgh. are para 1 1 the
P^ane of projection being perpendicular ,„ their direc ion
eitner that of the equator or ol a meridian. When a hemi
sphere ,s projected on its base in this manner the rela.™
positions of points near the centre are given w'h o r
able accuracy, but those near the circumference are com
dtdutd ^'T'^'- ^"l'^"^ °f "- P.ojection arTs":
deduced. Amongst others it is evident that in the case
of a hemisphere projected on its base ail circles pL,n.
through the pole of the hemisphere are p o ected »«
straight lines intersecting at the centre. CircTes hti I'g
heir plane, parallel to that of ,he base are projected I*"
equal circles. All other circles are projected as e fin
of wh,cl, the greater axis is equal to the' dia„,e r o T
r':;3:,":'or:b^u!.7
^Xi
!l
i68
Projections.
Stereographic Projedion.—ln this projection the eye is
supposed to be situated at the surface of the sphere, and
the plane of the projection is that of the great circle
which is every where 90 degrees from the position of the
eye. It derives its name from the fact that it results from
the intersection of two solids, the cone and the sphere.
Its principal properties are the following: i. The pro-
jection of any circle on the sphere which does not pass
through the eye is a circle; and circles whose planes
pass through the eye are straight lines. 2. The angle
made on the surface of the sphere by two circles which
cut each other and the angle made by their projections
are equal. 3. If C is the pole of the point of sight and c
its projection; then any point A is projected into a point a
such that c a is equal to
tan (arc CA
rx
..„ — I —
-S
where r is the radius of the sphere. From the second
property it follows that any very small portion of the
spherical surface and its projection are similar figures ; a
property of great importance in the construction of maps,
and one which is also shared by Mercator's projection.
The astronomical triangle FZS can evidently be easily
drawn on the stereographic projection. Z will be the
pole of the point of sight. The lengths of ZP and ZS
are straight lines found by the rule given above, and the
angle Z being known the points P and S are known.
The angles P and S being also known we can draw the
circular curve PS by a simple construction.
The orthographic and stereographic projections were
both employed by the ancient Greek astronomers for the
purpose of representing the celestial sphere, with its
circles, on a plane.
Gnomonic or Central Projection. — In this case the eye is
at the centre of the sphere, and the plane of projection is
Projections.
i6g
a-plane touching the sphere at any assumed point. The
projection of any point is the extremity of the tangent of
the arc intercepted between that point and the point of
contact. As the tangent increases very rapidly when the
arc IS more than 45°, and becomes infinite at go*, it is evi-
dent that this projection cannot be adopted for a whole
hemisphere.
Globular Projection.— This is a device to avoid the dis
tort.on which occurs in the above projections as we
approach the circumference of the hemisphere. In the
accompanying figure let A C B
be the hemisphere to be repre-
sented on the plane A B, E
the position of the eye, O the
centre of the sphere, and EDOC
perpendicular to the plane A B.
M and F are points on the
sphere, and their projections are
N and G. Now the representa-
tion would be perfect if A N :
N G : G O were as A M : M F :
F C. This cannot be obtained Fig. 39,
.^'^^^y^;.^"* ''' ^^'''1 ^« approximately so if the point E
IS so diown that G is t!.e middle point of A O and F the
middle point of A C. In this case, by joining F O and
drawing F L perpendicular to O C, it may easily be shown
that L D ,s equal to O L, which is O Fxcos 45" or
rxo.71 nearly.-M«.««^ G O is half the radius and F L
halt the inscribed square— tfeererfase
FL:GO::OC:OL
but F L: GO::LE ; O E
.-.LE : O E::OC : OL
consequently, LO : O ■• : . C L : O L, or O L^.=0 E C L
but O L^=.F L»=D .. L C, .-. O E. C L^D L. C L
or O E-»D L
that is, E D—O L
I
170
Projections.
The above projections are seldom used for delineating
the features of a single country or a small portion of the
earth's surface. For this purpose it is more convenient
to employ one of the methods of development.
Mercator's Projection is the method employed in the
construction of nautical charts. The meridians are repre-
sented by equi-distant parallel straight lines, and the
parallels of latitude by straight lines perpendicular to ^he
meridians. As we recede from the equator towards the
poles the distances between the parallels of latitude on the
map are made to increase at the same rate that the scale
of the distance between points east and west of each
other increases on the map, owing to the meridians being
drawn parallel instead of converging. If we take / as the
length of a degree of longitude at the equator (which
would be the same as a degree of latitude supposing the
earth a sphere), and /' that of a degree of longitude at
latitude ;, then /'-/ cos >^, or /'.•/:: i : sec ;. Now
/.' : / is the proportion in which the length of a given dis-
tance in longitude has been increased on the map by
making the meridians parallel, and is therefore the pro-
portion in which the distance between the parallels of
latitude must be increased. It is evident that the poles
can never be shown on this projection, as they would be
at an infinite distance from the equator.
If a ship steers a fixed course by the compass this
course is always a straight line on a Mercator's chart.
Great circles on the globe are projected as curves, except
in the case of meridians and the equator.
In this projection, though the scale increases as we
approach the poles, the map of a limited tract of country
gives places in their correct relative positions.
The Ordinary Polyconic Projection.—In conical develop-
ments of the sphere a polygon is supposed to be inscribed
in a meridian. By revolution about the polar axis the
vxmf-sss.^i^^ifi.fi^,^,
Projections.
171
polygon will describe a series of frustums of cones, li
the arc of the curve equals its chord the two surfaces wilF
be equal. In this manner the spherical surface may be
looked upon as formed by the intersection of an infinite
number of cones tangential to the surface along succes-
sive parallels of latitude. These conical surfaces may be
developed on a plane, and the properties of the resulting
chart will depend on the law of the development.
The Ordinary Polyconic is a projection much used in
the United States Coast Survey. It is peculiarly appli-
cable to the case where the chart embraces considerable
difference in latitude with only a moderate amplitude of
longitude, as it is independent of change of latitude.
Before describing it it must be noted that whatever
projection is used the spheroidal figure of the earth must
be taken into account, its surface being that which would
be formed by the revolution of a nearly circular ellipse
round the polar axis as a minor axis.
In the Ordinary Polyconic each parallel of latitude is
represented on a plane by the development of a cone
haying the parallel for its base, and its vertex at the
point where a tangent to a meridian at the parallel in-
tersects the earth's axis, the degrees on the parallel pre-
serving their true length. A straight line running north
and south represents the middle meridian on the chart,
and is made equal to its rectified arc according to scale.'
The conical elements are developed equally on each
side of this meridian, and are disposed in arcs of circles
described (in the case of the sphere) with radii equal to
the radius of the sphere multiplied by the cotangent of
the latitude. The centres of these arcs lie in the middle
meridian produced, each arc cutting it at its proper
latitude.
These elements evidently touch each other only at the
middle meridian, diverging as they leave it. The curva-
/
. -SCTssssTjiBaBSsM:,
I.l
172
Projections.
tureofthe parallels decreases as the distance from the
poles increases, till at the equator the parallel becomes a
straight line.
To trace the meridians we set off on the different
parallels (accordinfj to the usual law for the length of an
arc of longitude) the true points where each meridian
cuts them, and draw curves connecting those points.
To allow for the ellipticity of the earth we must use for
the radius of the developed parallel N cot /, where
N=
a
(i—e^ sin 3 /)J^
a being the equatorial radius, e the eccentricity, N the
normal terminating in the minor axis, and / the ar >!« it
makes with the major axis.
It is evident that
the slant height of the
cone— say y— is N cot /,
and that the radius
of the parallel on the
spheroid is N cos /.
The length of an arc
of M° of a parallel will
be »°-^o N cos /.
In practice, instead Fig. 40
of describing the arcs of the parallels with radii, it is more
IZrV"- T''''''' *'^" ^^°- ^^^'^ ^<i-tLs
r n 1 w '"^'^^'^'^'^^^^^ the meridians and parallels
can also be found in this way. Express . and ;., the ec
angular co-ordinates of a point, as function of the
(6)7 f.U 'r ^'P'' P^^^"^' (^ -t ^) -d the angle
iff) that this radius makes with the middle meridian
■^
i
1!^
■J^*lf'im
i
Fig. 41
(I)
(2)
PyojecHuns.
Take the origin at L»
(Fig. 41) the point of in-f
tersection of a piirallel ||
with the middle meridian;
the middle meridian as
the axis of _y; and ilie per-
pendicular throngh L as
the axis of x. Tlien we
shall have for any point P
whose latitude is / and
longitude from the meri-
dian n°
^=-Y Psin /?=-N cot /sin ^
:v— Y P versin ^=N cot / versin d
being, of course, some function of «.
To find the relation between d and n, since the parallels
are developed with their true lengths the distance LP
:phetid T^f 'T ^^'■'°" LP^f ^'-parallel :: \l
spheio.d Therefore the angles at the centres of the two
arcs will be mversely proportional to the radii, and
N^ot^/_ «•
N cos r~~d~' °^ ^^"° s'" ^ (3)
These three equations are sufficient to projerf any
point of the spheroid when we know its latitude and its
longitude from the middle meridian. If we take n con
stant we can project the successive points of any meri-
frn^^l.'' the distance on the elliptical middle meridian
from the ongm to the point where the parallel through
tion (2) will become, ^=N cot / versin ^±S.
j '1
HI
174
Projections.
From the abov^ equations
tables may be formed for the
construction of charts.
Fij^. 42 sliovvs the geometri-
cal relation between the angles
^ and n.
This projection, when the
amplitude in longitude does not
exceed three degrees from the
middle meridian, has the fol- Pig- 42
lowing properties.
It distorts very little, and has great uniformity of scale.
It is well adapted to all parts of the earth, but best to
the polar regions.
The meridians nrik." practically the same angles with
each other and wiu; ;[.!. parallels as on the sphere. Angles
are projected with .i:tde change.
The great circle . r geodesic line is projected as a
straight line practically equal to itself.
■
1.
•■
CHAPTER VI.
scale,
est to
i with
Angles
I as a
TliMONOMETRlCAL LEV EL LI NO.
TO FIND THE HEIGHT OF A POINT B ABOVE A STATION A.
In the accompanying figure
O is the centre of the earth,
AC is • mgential tf» the earth's
snrfa. at A, B' is the apparent
position of B, owing to refrac-
tion. CC is the correction for
K»
curvature, or- „ , wliere K is
2 R
the horizontal distance of B
from A, and R is the radius of ^''> 43-
the earth ; both in feet. BB' is about 0.16 CC
ACB may be taken as a right angle; and AC*, the arc
AC, and the straight line AC, ar e"alt equal?' We shall
have then, if - th e dint finr" K ii mi l ii 111 {j i'- U t.
BC=K tan B'AC + CC-BB'
^K tan B 'AC + 0.00000002 K^
where B'AC is the observed angle of elevation of B. This
formula supposes that AC'B is practically o^' If the dis-
^.O-A.'*'*/
\
IMAGE EVALUATION
TEST TARGET (MT-S)
/.
.?
C^
^
/
A
1.0
I.I
|50 "^B
^ b£ 12.0
2.5
11-25 II 1.4
A-KX^l-V^
rtograpnic
Sdences
Corporation
1.6
23 WEST MAi.: STREET
WEBSTER, NY. 14580
(7)6) 872-4503
//
^A,
.v^^
4 ^, V
^0
K<^^
^1^
i \
176
Trigonometrical Levelling.
tance is so great that this is not the case we shall have
in the triangle ACB
,sin BAG
BC=K-
sin B
To find the angle B, we have in the triangle AOB,
B=i8o=-(0+BAO)
=180— (0 + 90° + BAG)
=- 90— (O + BAG)
Hence, sin B=-cos (0+BAG)
sin BAG
BG=Kx
cos (b + BAG)
And BG'-=BG + GG'-BG +
2K
(<vhere R4s the radius of the earth in. feet.)
The angle O, in minutes, is 0.0001646 K, and
K2
^ is 0.000000023936 K2
REFRAC-
RECIPROCAL OBSERVATIONS FOR CANXELLING
TION,
If we measure the reciprocal angles of elevation and de-
/ pression of two stations— in other words, if at each
^ we observe the ;;cnith distance of the other— we shall get
nd of the effects of refrac-
tion. Let a he the angle
of elevation of B at A and
A? the angle of depression
of A at B.
Then
BG'— Kx
sm I (« + /5)
cos A {a+l^-fo)
If the zenith distances
are observed call them «Jand
d', and we shall have (since
^-^90° — a and <J'=-9o°+^J)
^'A''- i4'
'lE?ftn
shall have
AOB.
Trigonometrical Levelling.
177
REFRAC-
n and cle-
at each
shall get
cos^id'—8+0)
If O is very small compared with the other angles we
may neglect it, when we shall have
BC'=.K tan ^ (a+/9)
==K tan I (d'—d)*
REDUCTION TO THE SUMMITS OF THE SIGNALS.
Suppose there are two stations, a and b, which cannot
be seen from each other, so that
signals have to be e.'ected at
each. Lei A and B be the sum-
mits of the signals, f/. and ^5 the
true angles of elevation [and de-
pression of a and b respectively.
At ci the angle B <{ C is observed
and at b the angle A b D. Call
H^'C,^^; AbD,,/';Aa,h; and
B b, h'. Then, to find the re-
duced angles « and /? we shall
have
„ ti h cos tp
Ksm I
,3 Ji' cos d)
''^^+Ksinr
P'g- 45-
the differences being in seconds.
If zenith distances A and A ' are taken we shall have
*Clarke gives the formula
h'~h=K tan i {8:—d) {i-|-^^l
where h and h' are the heights of the stations, and r the
radius of the earth.
I
n
ill
178
Trigonometrical Levelling.
for d and d'
5= A +
d'=A'-i
h sin A
K sin I"
h' sin A '
K sin i'
Reciprocal observations ought to be simultaneous in
/ order that the effects of refraction may be as nearly as
possible the same for both.
In problems of this kind we ought, strictly speaking,
instead of using the mean radius of the earth, to take the
normal for the mean latitude of the stations.
The following geodetical formulce are used for more
exact determinations. In addition to the letters used in
the foregoing problems we have a the known altitude of
the lower station ; N the normal for the mean latitude ;
M the modulus of common logarithms; and r the co-
efficient of refraction.
1. TO FIND THE DIFFERENCE OF LEVEL BY RECIPROCAL
.''ENITH DISTANCES.
Log. diff. oflevel=log jKtan J (d'—i
^
2. TO FIND THE DIFFERENCE OF LEVEL BY MEANS OF A
SINGLE ZENITH DISTANCE.
Log. diff. level = log.
M
K
tan
f « I — 2 r
V 2 N sin I"
K
N 2 N
K
M
tnr. ^^ i~2 r „ ) ~i2 N»
tan 1 a -— , — „ K ,-
i 2 N sm I J
K'
The third term is positive if A is less than 9^
Ml
r.T.itiCTra; ffleaw
Trigonometrical Levelling.
179
3. TO ASCERTAIN THE HEIGHT OF A STATION BY MEANS OF
THE ZEMTH DISTANCE OF THE SEA HORIZON.
In this case, wh-n possible, different points of the hori-
zon should be observed on different days and the mean
of the whole taken, the state of the tide being also noted.
The formula is
Log. altitude=log-^{-^^|"+iog. (^-go")^
, M / sin I* ) • .
The angle d— 90* is in seconds.
The last term may generally be neglected.
The following is an example of finding the difference of
level by a single zenith distance.
The altitude of the lower station (a) wtf :ooo yards,
and h or the height of the instrument 5.
The horizontal distance between the stations (K) was
57836 yards. The zenith distance of the upper station
(A) 88* 24' 40".
First, to find the value of the angle d.
Log h :o. 69897
Log sin A =9.99984
Co-log K -115. 23780
Co-log sin i"=5.37443
Log
h sin A
K sin V"^^ -^5 104= log I7-.8
Therefore 5=88° 24' 57".8
Next, to find the value of the angle
1—2 r
Log
I — 2 r
2 N>rY" =^'^3252
Log K =4.76220
2 N sin i'
K
I — 2 r
2.89472=log 784''7=o° 13' 4".7
2 N sin I " ^-^^° ^'' 53".i
I
i8o
Trigonometrical Levelling.
Thirdly, value of the difference of level.
Log K =4.7621984
Log tan 88" ir 53".! =1.5022427
Log ist term=3.2599557
2nd terni= 691
3rd term= +627
4th term= 9
Second Term.
Log^= 28393
Log a =3
Log 2nd term = 5-8393
2nd term = 0.000069 1
Log. diff. Ievel=3-26oo884=log 1820.07 yards.
Third Term.
Log ^^3.538
2 N
log I St term = 3 2599
log3rdterm=57982
3rd term =00000627
REFRACTION, die.
Fourth Term.
34387
Log _M
I 2 N'J
log Ks =95244
log 4th term = 2 9631
4th term =0-0000009
TO FIND THE CO-EFFICIENT OF TERRESTRIAL REFRAC-
TION BY RECIPROCAL OBSERVATIONS OF 2EN1TH
DISTANCES.
Let A and B be two sta-
tions, and let their heights
(ascertained by levelling) be
h and h'. Consider the earth
as a sphere, and take O' its
centre. Call the radius r and
the angle AOB v. Let Z be
the true zenith distance of B
at A, viz., ZAB, and Z' that of
A at B or Z'BA. The dotted
curve shows the path of the
ray of light. A' and B' are the
apparent positions of the sta-
tions.
P'g- 46.
/ The co-efficient of refraction is the ratio of the differ-
U ence between the observed and real zenith distance at
Ml
1
07 yards.
Term.
=34387
=95244
1 = 29631
= O'000OOO9
REFRAC-
2ENITH
differ-
nce at
Trigonometrical Levelling.
181
either station to the angle v. Thus, if A is the co-effici.^
and z 2 the observed zenith distances, we have k equal to
L — Z Z' — zi
But these are not always the same.
-or
V V
In the triangle AOB we have
2
tan —tan ^ (Z— Z)=
h' + 2 r+h
These equations give 2 and Z.
If we substitute for tan^ the first two terms of its ex-
pansion in w«^s; the second equation may be put in the
form
h'—h=s tan i (Z'—Z)
2 r
— [
izr*)
where s is the length of AB projected on the sea level.
The co-efficient of refraction may also be obtained from
the si.nultaneous reciprocally-observed zenith distances
ot A and B without knowing their heights. Thus :
Z=z + k V, and Z'—.z' + k v
•: e+z'+2 k i'^i8o°-f y
or 1-2 k='±'-J^^l
V
The mean co-efficient is .0771. For rays crossing the
sea It 13 .o8og, and for rays not crossing it .0750.
The amount of terrestrial refraction is verv variable
and not to be expressed by any single law. In flat, hot
countries where the rays of light have to pass near the
ground and through masses of atmosphere of different
densities the irregularity of the refraction is very great-
so much so that the path of the rays is sometimes
<-onvex to the surface of the earth instead of concave In
Great Britain the refraction is, as a rule, greatest in the
early mornings ; towards the middle of the day it de-
creases and remains nearly constant for some hours, in-
creasing again towards evening.
:
CHAPTER VII.
THE USE OF THE PENDULUM IN DETERMININO THE
COMPIiEtiSION OF THE EARTH.
The spheroidal form of the earth causes the force of
gravity to increase from the equator towards the poles,
and this force may be measured at any place by means of
the oscillations of a pendulum.
If we had a heavy particle suspended from a fixed
point by a fine inextensible thread without weight we
should have what is called a simple pendulum. If this
pendulum were allowed to make small oscillations (of not
more than a degree in amplitude) in vacuo, and in a ver-
tical plane, the time of oscillation would be given by the
formula
Iff )
Where / is tho number of seconds, / the length of the
pendulum in feet, and g the force of gravity.
Therefore, taking^ as constant, if there were another
pendulum /' feet long and vibrating in t' seconds, we
should have
t:t".:\/l: 4//'
or, if the time were constant and g changed to g',
Pendulum Observations.
or HI were constant and ^ variable
ft'" -. '
If « and n are the number of osci
i^mti, then, »': M::^.^'
From (2) we have, ir'=-^/r
_n2
„8^
183
nations in tfee time y^
(3)
rnelLremluhetlhX"^' T'/ ^'T^ *'^^^^^^^^" ^^
tain number o os^ £^^^ P'"'"'""! '^^' ^^^^^^ ^er-
what ar/called "c;rou^d>f^° H 7' """"'^ ''^'^'' "=
sible .0 calculate theleZh I ^ """f' """ " '^ P"'"
would oscillate in .he same it a-".'lT ""'"'"T "■="
by finding the position ^0^ "cLt ! 0?™''°^''- ™''
wou',^.h:;:::d'^rr 7? ^^^^^—^^
and if a pent Z t '''r?" "'= '"'"^''angeable,
'--eco^™r.trri::r:z,^'----e
is "er-r„:r.^t^,rdr^^^^^^^^^^^^^ -j ^ -
have, by the formula ^n^olt ctva^t^T^rm " "'
S~g [i+(|w-0sin2yj
and, since ;^' "'
n"
fi. if « is the number of oscillations in
i ^ I
i
184
Pendulum Observations.
a given time at the equator and »' the numbert at the
station.
n
r
'a—ns [iMi '"— ^) sin" <p)
(4)
Also,^' J- g
:. if we take the lengths of the seconds pendulums in-
stead of the number of their oscillations, we have
/'=/ [I + (I ,n—c) sin>] (5)
I being the length of the pendulum at the equator, m
being known, and » n', or / /', being found by experiment,
we at once get the value of c from equation (4) or (5).
Borda's pendulum, which was used by the French
astronomers to find the length of the second's pendulum
(that is, a pendulum oscillating in a single second) at dif-
ferent stations, consisted of a sphere of platinum sus-
pended by a fine wire, attached to the upper end of
which was a knife edge of steel resting on a level agate
plane. The length of the simple pendulum corresponding
to Borda's was obtained by measurement and calculation.
In 1818 Captain Kater determined the length of the
seconds pendulum in London (39-13929 inches) by means
of a pendulum which had two knife edges facing each
other— one for the centre of suspension, the other at the
centre of oscillation— so that, provided the two knife
edges were at the correct distance apart, they could be
used indifferently as points of suspension ; the pendulum
being, of course, inverted in the two positions. The
pendulum was made to swing equally from either point of
suspension by adjusting a sliding weight. The distance
between the two edges gave the length of the simple pen-
dulum.
The advantage of such a pendulum is that it contains
two in one, and that any injury to the instrument is de-
tected by its giving different results when swung in the
two positions. This pendulum was afterwards super-
seded by another of similar principle, in which, instead of
}er» at the
(4)
dulums in-
ave
(5)
qaator. m
ixperiment,
) or (5).
he French
5 pendulum
;oiid) at dif-
itinum sus-
per end of
level agate
rresponding
calculation,
igth of the
s) by means
facing each
5ther at the
! two knife
2y could be
e pendulum
tions. The
her point of
'he distance
simple pen-
it contains
ment is de-
ATung in the
ards super-
I, instead of
Pendulum Observations.
185
using a "^liding weight, one end of the bar of which it
consisted was filed away until the vibrations in the two
positions were synchronous. In using the pendulum it is
swung in front of the pendulum of an astronomical clock,
the exact rate of which is known. By means of certain
contrivances the number of vibrations made by the two
pendulums in a given time can be compared exactly, and
the number made by the clock being known that of the
experimental pendulum is obtained. Certain corrections
have to be applied. One for changes in the thermometer,
which lengthen or shorten the pendulum : a second for
changes in barometric pressure, which by altering the
floatation effect of the atmosphere on the instrument,
affect the action of gravity on it ; a third for height of
station above the sea level, which also affects the force of
gravity, the latter diminishing with the square of the
distance from the centre of the earth ; and a fourth for
the amplitude of the arc through which the pendulum
swings, which, in theory, should be indefinitely small.
The number of pendulum oscillations in a given time
has been observed at a vast number of stations in various
parts of the world, and in latitudes from the equator to
nearly 80". The most extensive series of observations
was one lately brought to a close in India, the pendulums
used in which had been previor; ;, tested at Kew. The
general results of all the pen iiilum experiments gives
about 292 : 293 as the ratio of the earth's axes, which is
the same as that deduced from measurements of meri-
dianal arcs.
L
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