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OCR for page 123
CHAPTER IX
THE SHAPE ANT) ASIDE OF THE 1SARTI:T
N)T D . LA M BERT
U.- S. CYO~'Si and Geodetic Surgery
The idea that, apart I'ron~ the hills and valleys that d;versi1';y its
surface, the earth is essentially flat is of course ogle that natty ally presents
itself to all primitive peoples. The notion of a flat earth would, however,
present ciifficulties to ~ though.t-t'ul observer of the stars who had not.icecl
that as he traveled no~thwarcl or southward the aspect of' the heavens
changecl; it would present difficulties also to a thoughtful seaman who
had noticed the gradual disappearance of' a receding vessel, hull first
and then the masts alla jigging.
In China, Egypt, and Babylonia astronomy was cultivated by priests
and astrologers centuries and millenniums be-fore tile beginning, of our era,
but it does not appear that their speculations on the subject led there
to formulate the idea of the rotundity of' the earth. At any rate no such
speculations have come down to us. These phenomena previously me
tioned, being but roughly obser~:ecl, would, of course, prove merely that
the surface of' the earth is rounded and convex, not that it is exactly
spherical.*
The Greeks were both a seafaring and speculative people, and it is id
Greek thought that we first find the idea of' a spherical earth. It is not
likely that these earliest advocates of' the idea of' a spherical earth made
measurements to find out how closely the earth approximates to a sphere.
The exact sphericity was considered as proved, or as made plausible, by
metaphysical considerations. The first clear and unequivocal statement
of the sphericity of the earth is due to the Greek philosopher Pythagoras
(about 340 B. C.) or to his followers. No doubt the evidence in favor
of the sphericity of' the earth found increasing favor among, thou`,hti'ul
men, but even too centuries later Aristotle (384-322 B. C.) thought
it worth while to refute the arguments against' the sphericity of' the earth
and to set forth those in favor of its (See numbered references at end of
chapter). Some of his ar~,u~r~er~.ts are very :t'oreign to our ways of thinli-
ing; others sound very modern.
* Theoretically at least, the difference in curvature of the sea. surface in different
directions could be measured by measuring the varying dip of the horizon in
different azimuths.
123
OCR for page 124
12!
F7~ OF r~ ~6 r~
Evidently the astronomers Off Aristotle time h~4 some idea of the size
of the earth. The concluding passage of the Depone took of his truth se;
Ma Ago, aeons:
~ Forever those m~thPm~ici~ns ago try to romputc the cir~nmference
of the each say that it is i00~000 strain, which initiates got only Ant
the earthy mass is sphrric~1 in shape hut Vito that ~ is of no grant size
as compare] gab the be~venly todTes/, four hundred thong Athc
static Could he 74~000 kilometers. This is rert~i~T~ Lot ~ p~rticul~ly
good ~pproxim~Hon to the rectum hauler 40~000 kilometers; but it is at
least of the right order of magnitude. Hog this (aura ~ reached does
lTOt appears but it eras prol~hl~: by some such p] in] 11S a\' t1.S 111.tOl! 11SP]
by Lr~toatbenes.
About ~ bUndre] Bears later ~ better dete~in~tio~ of the aide of tar
each gas made by Fr~tosthe~s Plied about I93 B. C.~, libr~ri~ at
Alert and ~ Max of gibe attainments TO Various herds of learning.
Fr~tos~enes assume] Act Spend; Me modern Assuage on the Upper
Age lay OU the Myopic of cancer ~D] that therefore the sun at the summer
solstice Could cast ~ vertic~1 shadow. He assumed furler that Syene
~] Alexandria mere OU the same meridian: tbo~gh their dT~erence of
longitude is some two degrees. The distance between them be supposed
to Be S;000 static. Some commentators bade thought that this figure
derive] from the computations of the o~ci~1 surveyors who kept
track of property hag, ~bieb were likely to become distinct owing to
the ~nDu~1 inundations of the gild tUt more pivotally it represents
merely an cstim~te Based on the accounts of trsvele~s.
Abed Er~tostbenes observe] at Alexandria the zenith distance of the
sun at the summer solstice and found it to he one-hitietb of ~ ci~cumfer-
once ~° IS ~bicb is therefore [~fn~ffAnA~, vn]~P fan 1~ ~1]~r
~ ^ ~ ~ v ~ ~ ~ v ~ v ~ . ~ _ ~ _ ~ _ _ ~ _ ~
meridion~1 distance between suede and Alexandria. Since one-hItietb
of the Circumference corresponds to J:000 studio, obvionsl~ the Whole
rircU~Terence is 30 x 5~000 strata Or 230~000 static. To make the length
of ~ degree come out in round numbers the result gas taken as 252~000
stat or 700 attain to He degree. It is not certain that those mere Attic
studio of 183 meters each. However, with any re~son~tle value of the
stadiums Fr~tostbenes, result is closer to the truth tan the one cited
LY Eli stotle Rnd slthoUgb methods of oBserv~tioD mere improve] in
.,
course of fume, the simple priDriple Use] ty Er~tostbenes regained the
Stogie] nntM the eipUteentb century of OUI elan Ruben the necessity
arose for disLblgUishing between ~ spberic~1 ~] ~ s~beroida1 ebb.
Abe Greek geographer; Str~bo; ago wrote at the Beginning of OUP elan
and the Gr~ero-Egypti~ astronomer Cladding Ptolemy; hobo live] in
the second census> of one elan both give the circumference of the ebb
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SBrAPE AND SIZE OF THE EARTH
125
as 180,000 staclia. Willis figure cl.oes not appear in. either case to represent
an independent deter~nina.tion; its chic-i' interest for us is the use made
of it by Columbus.
The Arabs of the ~.~;n.th. century are reported to have ~llarl.e a deter~ni.na-
t.ion of the size of' the earth. by direct measurement manacle ;~ level country
over two degrees of latit.uLle. Being ignorant of the exact Forlorn equiva-
lents of the units userl, we cannot judge the accuracy of their work, but
it was probably greater than the accuracy attained by lTra.tosthenes. They
had fairly accurate astrolabes and the meridional distances are spill to
have been Pleasured with cords.
Before col~siderin~, ally modern determi.natiolls it may be well to say
sol~lethi~l`, concerning, the cloctri~le of the sphericit;y of' the earth. It is
sonnetizes said that the Glen ol' the Middle Ages believed the earth to
be flat and that Columbus proved it to be rouncl. No doubt ignorant
need of' the Middle Ages, and of other times as well, believed the earth
to be flat, but the idea of' the sphericitsr of' the earth never died out amoral
the learned. It its true that during the early centuries of the Christian
cra there were those among, the fathers o-E the Church who, basing their
objections on a lit.era.1 interpretation of scriptural la.n~,ua`~,e, denier1 the
sphericity of the earth. Put there were always some among them who
put the matter aside as not involving, any vital Christian doctrine and
still others who franl~ly accepted the view, handed down from classical
times, that the earth is spherica.1.
At least the doctrine of a spherical
earth was latent alive by the attacks upon it, and gradually it gained
ground. In the thirteenth and l'ourteenth centuries it may be considered
as the generally accepted doctrine among, the lear~ecl.3 For example,
the idea of a spherical earth is a fundamental part of the scheme of'
Dante's Divine Comedy.
Columbus, nearly two centuries later than Dante, tool: the idea of a
spherical earth for granted. Columbus' chief' point was that, given the
sphericitv of the earth, and assuming as its most probable circum-i'erence
the small figures of' Ptolemy and Strabo, and assuming, further the great
eastward extent of' Nsia--whi.ch, from the accounts of land travelers, it
was easy to exaggerate, so that Asia would wrap itself pretty well around
this small world of' ours then it was clear that the distance westward
from western Europe to eastern Asia could not be so very great and that
Asia could readily be reached by sailing westward.
After the time of Columbus we next note the world of Snell, who is
also letdown for his discovery of the sine law of refraction. Calling himself'
the " Batavian ~,ra.tosthenes," he published in 1617 at Leyden the results
of' the length of' a cle~,ree obtained by a then novel methocl. Instead of
obtaining his distance along a meridian by estimate or by direct measure
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126
FIGURE OF THE EARTH
ment as his preclecesso~s hacl done, he laid out a chairs of triangles between
Alkma.ar in North Holland and Bergen-op-%oom near the present Belgian
frontier, and measured their angles; he also measured a comparatively
short base from which all lengths in the triangulation could be determined.
The principles usecl. were in no way different from those of modern geo-
cletic operations. The length of a cle¢,ree obtained by him was, however,
too small by mole than three pet cent.
The work of Picard in France, begun in 166O, was also ba.secl ore the
method of triangulation and was notable for two other things: it was the
first geodetic work done with telescope and reticule; and its results were
used by Newton' in his publish.erl calculation to prove that terrestrial
gravity controls the motions of: the moon.*
Picard was followed by the Cassinis who gradually extended his arc
and improved upon his methods. With the acceptance of the Copernican
theory and the development of mechanics at the hands of Newton,
Euyghens and their contemporaries, there appeared reasons for thinking
that the earth is cot spherical but approximately an oblate spheroid, i. e.,
flattened at the poles. This would make the length of a degree Increase
from the equator to the poles,! but from the work of Picard and Jacques
fin.~sini the length of a degree having its mid-point in latitude 45° 4l'
~ ~ . . .
was b?,097 toises and that of a degree with its mid-point in latitude
49° 56' was only 56,960 toises,; a decrease instead of: ax increase. To
us there is nothing surprising in this, as the combined deflections of the
vertical necessary at the four astronomical stations two for each arc-
even after allowing for the normal increase in length, is only some 10",
quantity that we now know to be often found in practice. More-
ove~, the errors both in triangulation and in the astronomical determina-
tions were by no means negligible in comparison with this discrepancy.
These fact were realized to a certain extent even then, but not so strongly
as today, and of course the physical reasons given by Newton and others
* Newton probably used Picard's de-termination merely as the latest and. best.
It has been said that Newton was not satisfied of the validity of his conclusion
until Picard's work was available. This seems to misrepresent the development of
Newton's thought. See reference :No. 9 at end of chapter.
tIt should be noted that the length of a, degree of latitude on the elliptical
meridian of a spheroidal earth means the length of a degree of astronomical lati-
tude and is therefore governed, not by the direction of the radius vector, but by
the direction of the normal. The normal changes direction most rapidly within
a given distance along the meridian where the meridional curvature is greatest,
that is, at the equator of an ablate spheroid, and it is here that the length of one
degree is shortest.
~ A toise was composed of six French feet, or a little less than 6 feet 5 inches
of English or American measure.
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SHAPE AND SIZE OF THE EARTH
127
for the flattening did not then carry nearly so much weight as they do
now, so that for a time there was a lively' controversy between the earth-
flatteners and the earth-elon~ators.
The major axis and the ellipticity of the earth, supposed to be an ellip-
soid of revolution, may be determined from the lengths of a degree of
latitude in two different latitudes and it is fairly obvious that the greater
the difference in latitude of the mid-points of the two degrees the less
will the values of the major axis and the ellipticity be a.ffectecl by the
inevitable errors of observation and by the errors of the underlying
assumption that the earth is an ellipsoid of revolution, these last-named
errors being the deflections of the vertical under another namers With
this in view and in order to settle the dispute between the earth-flatteners
and the earth-elon~,ators, the Royal Academy of Sciences of Paris or~,~.n-
ized two geodetic expeclitions, one to Lapla.nd arid the other to " Peru.":
The n~riv in TJ~nl~nd ~ndn.r the o,nn~rn1 Blare Of 1\/~n~rh~i.~ T.o
~ Ar · ~ ~ · c~ ~ ~ ~ · r. r. ~
lVlonuler, anti as representing ~weclen, cels.lus, professor ot ast.ronom:
at Upsala, ~neasurecl in the years 17'3(~-~737 an are of a? mi-r~utes in 1~id-
latitude 66° 20' along the river Tornea, which empties. into the Gulf' of
Bothnia. From this they concluded that the length of one degree at this
point is b7,437.9 tosses, a value which we nowl~now to be rather inac-
curate, but since this ler~',th is greater than the lengths previously found
by Picarcl and Cassini in Glance, it was co~.~clucled that the earth is
flattened at the poles.
The party bound :t'or " Peru " set out ahead of the Laplan~l party in
37la.y, 1735; it was composed of three members of' the Paris Academy,
Bouguer, Godin and de la Conclamine and their assistants. Two Spanish
officers representing the Spanish government joined-it in Colombia. On
the way to the scene of their future operations the party made some
studies of' refraction and determined the position of the Equator, chiseling
in the rocks an appropriate inscription. It was not until June of 173G
that the entire party was together in Quito. Base measurements, trian~u-
lation and astro~:~on~ical world were not completed until 1743.
The total amplitude of the are was 3° 0?' with its micl-point in
latitude 1° 3l' S. The length of' a degree as derived by different members
of the party varied :t'rom bC,7'46 toises to 56,768 toises. Even the greater
:figure is less than the length of a degree in France and still less than the
1 ~1 1
.. . .. .. . . .. .
* The arc is always called the Peruvian arc and the bar with which bases were
measured is called the " toise of Peru," but the territory within which operations
were conducted is part of the present Ecuador, then-a portion of the Spanish
province of Peru The " t.oise of Peru," being a more definite standard than other
units then in current use. was used as the standard for much of the geodetic worl;
down to the time of the introduction of the meter. and even later.
9
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128
FIGURE OF THE EARTH
length of' a de'~,ree in La.pland, so that the question of whether the earth
is flattened or elongated at the poles could be considered as definitely
settled by observation in favor of' the flattening,, but the amount of the
flattening was far from being accurately determined and so remained
i'or a long time thereafter.*
Other meridional arcs were measured; among, them may be mentioned
that of La.caille in South Africa in 1762, the seco~cl arc in ~no~lern times
outside of Europe, and that of Mason and Dixon in America in 176S,
the third arc outside of' Europe. The line used for the latter was not
the east-and-west line now forming, the boundary between ~IarvlanUl
and Pennsylvania and ordinarily thought of as Mason and Dixon's lines,
but a line containing as its principal part the line running nearly north
arid south and forming the boundary between Maryland and Delawarc.
Although this line was chosen by the Royal Society at the instance of'
Mason and Dixon because of its supposed freedom from deflections o-t'
the vertical, it was evidently the immediate surroundings that were ;I1
the Society's mind. It seems to have been assumed that the effect of' more
clista.nt topography such as the Alleghenies and the Atlantic could be
neglected. But when the question was examinecl by Ca.vendish it was
found that these geo~,raphic features might have a very perceptible effect
There was no question of isostatic compensation The name was not
vet invented and the thintr itself' was hardly suspected. To be sure
., , ~
~ 1.__ ~ 1 ~ ~ ._ 1 ~ _ ~ 1 __.; ~ 1~ ~ '.; ~ ~ .- ~ 4~ an; ~ A 1~ A, 1
~ou~uel~ in comparing, ~ravlty at sea-leve1 wlln gravely al ~UIIo Ll~t(l
found that the 10,000 feet of matter between sea-level and Quito seemed
to have strangely little edect on ~,ravity there, but there had been loo
systematic study of the effect of topography, although the subject of local
attraction was much in the minds of' ¢,eodesists and the mathematical
knowledge of the day was quite adequate for the necessary computations.
It might be supposed that the explanation of this was a lack of knowledge
of' the topography and a lacl; of scientific institutions capable of the
necessary organized effort to malice the computation; yet nearly a. huncl.recl
years later Pratt, with only a rather vague l~nowledt,-e o-t' the topo<,raph:'
of the Himalayas and central Asia and working, sint,-le-handed, was able
to bring I'orward convincing, evidence that t.opo~raphic features should
not be talker at their face Value, that is, that there is compensation, encl.
probably some sort of isostatic con~pensation. The state of' mind of'
geodesists of the time in question, and for a generation later, may be
illustrated by a passage -i'rom Delambre's Grancleur et figure de la Terre.
The book is a mine of' information on the history of ei~,hteenth-century
* By combination of the Lapland arc with talc arc between Paris and Amiens,
the flattening comes out as 1/178;; by talking the Peruvian arc with the Lapland
arc, the flattening collies out 1/213. The French and PeIUN ion arcs giN e 1/304.
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SHAPE AND SIZE OF THE EARTH
129
<,eodesy. Delambre died in 1827 but the latest geodetic world passed in
review was published in 1808. On page 199 Delambre says:
It was at first believed that this degree (Mason and Dixon's) could not have
been changed by local attractions. But Cavendish after examining the question
more carefully found that the excess attractions of the Allegheny mountains on
one side and the deficient attraction of the Atlantic Ocean on the other might
hare shortened this degree by 60 or 100 toises. He likewise found that; these two
causes might have appreciably affected the Italian degree and that of the Cape
of Good Hope.
In our :F'ranco-Spa,nish arc the greater attraction of the continent must have
attracted the plumb line to the northward at D!unl~irk as well as in Spain; these
two effects must have to a great extent canceled one another. All this is quite
possible but rather vague; and the most certain conclusion is that it, is not well to
rely on anything. There are few degrees that are not liable to greater uncertain-
ties than has hitherto been believed. We may assume the average degree to be
57.000 toises in round n~mbeis arid change the others in aGeordance with a, flat-
tenint, of 1/308, Lehigh today seethes; to havoc the support of sure eyors and
~,
astronomers.
Cavendish~s work has not been published in full. Some account of it
is given in The Scientific Papers of the Honourable Henry Cavendish,
Cambridge University Press, vol. I, p. 401-407 (1921~.
The number of meridional arcs measured now began to increase rather
rapidly; geodetic work was begun in England in 1787 and in India in
:1790, but it seems hardly worth while to go into details. Methods were
of course gradually improvecl. In geodesy, as in astronomy, the work of
Bessel in Germany introduced standards of accuracy comparable with
those of the present time. The introduction of the method of least squares
marked a great advance; it substituted a rational method for the arbi-
trary procedures hitherto adopted in reconciling inevitable discordances
and it increased to some extent the real accuracy of the final ~ esults. It
provided a procedure for determining, the dimensions of the spheroid best
adapted to represent as nearly as might be all the observations under dis-
cussion, whether they included many arcs or few. Bessel's ellipsoicl
published in lS41 and the Clarke ellipsoids of 1806 and 1880, all of which
have been widely used in geodetic world, were derived bar the method
of least squares.
Latitude determinations combined with measurements of meridional
arcs give only the meridional component of the deflection of the vertical.
Theoretically the prime-vertical component might be determined from
a comparison of the astronomic azimuth with the geodetic azimuth, and
the resulting information used to improve our knowledge of the figure
of the earth. In practice, however, the geodetic azimuth is liable to such
an accumulation of error as the length of a chain of triangulation in-
crec~ses, Plot little confidence call be placed in the pritne-:-ertical con~-
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130
FI GURE OF THE EAR TH
ponent of the deflection obtained from a comparison of astronomic and
geodetic azimuths unless some control for the latter is provided.
This control comes from the possibility of the astronomic determination
of differences of longitude. Before the invention of the electric tele-
graph, longitude determinations by the transportation of chronometers
or by lunar methods were so laborious and inaccurate as to be unsatis-
factory for geodetic purposes. The invention of the electric telegraph
introduced the possibility of obtaining deflections in the prime-vertical
comparable in accuracy with those in the meridian. There are two ways
in which the advantages of the telegraph may be utilized to give the
prime-vertical component. First, by means of Laplace stations ~ see
Chapter X)T) the ~,eocletic azimuths may be controlled and the undue
accumulation of error prevented; the prime-vertical component may then,
as previously stated, be obtained from a comparison of <,eocletic and
astronomic azimuths, the formula being
prime-vertical component of cleflection= (GAG-ally cot ¢.
Here a denotes azimuth, the subscripts A and G serving to distinguish
the astronomic from the geodetic azimuth, and ~ is the la.titucle.
The prime-vertical component may also be determined directly from
a comparison of the longitudes, the formula bein`,
prime-vertical component of' cleflection= (AA-AG) COS ¢,
A denoting west longitude, and the subscripts A and G and the symbol
having the same meanings as before.
In mountainous regions clift'erence in longitude may be determined
by recording the local times of the same event, such as the flash of' a
prearranged explosion on a high peak visible from both points. This
was done in a few cases, and differences of longitude were also determined
by the transport of chronometers. Eclipses of the moon and the phe-
nomena of Jupiter's satellites provide signals visible over a much greater
extent of the earth's surface than an explosion of gunpowder on a
mountain peal<, hut the phenomena in question are so ill-clefined as to
be nearly useless for marking a definite instant of time. The method
involving the flash o-i' an explosion is available only ill the neighborhood
of mountains, where local deflections are likely to be large.
There is another method for determining differences of longitude,
which requires for its effective use the existence of two intervisible points
with a considerable difference of longitude between them, a condition
again usually realized only in mountainous regions. The astronomical
azimuths are determined at both places and the astronomical difference
in longitude is determined frown the cli-tt'erence in azimuth by Dalty's
theorem, on which is based the ordinary routine computation of geodetic
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SHAPE AND SIZE OF THE EARTH
131
~.~osit.ions from triangulation. The deflection in the prime vertical is
cletermined by a comparison of the astronomic difference in longitude
with the geodetic differences, nibs IS The method gives poor results near
the Equator.
E;~,hteen geodetic ares were discussed by Airy ~ but the four arcs of
parallel were not used in his final determination, and some of the
meridional arcs were rejected for one reason or another. Bessel ~ nude
a calculation of the figure of the earth that is still the standard for many
countries and with about the same material at his disposal as Airy he
again rejected the available arcs of parallel. In 1858, in the Account
of: the Principal Triangulation of Cleat Britain a.ncl Ireland, Clarke lab
uses longitudes, though not telegraphic ones, and azimuths in deriving
a spheroid fitting as closely as possible the region covered by the triangu-
lation-he expressly repudiates the idea that he is here determining the
figure of the earth and also derives a figure of the earth as a whole from
meridional arcs only, but later in his Geodesy * ii~ he mentions tele-
~,raphic longitudes and uses them in deriving a figure of the earth-the
Clarke spheroid of 1880.
Under the auspices of the International Geocletic Association, Ilelmert,
Borsch and Krtiger 5 computed the deflections of the vertical in Europe
and the partial derivatives of these deflections with regard to the major
axis and the eccentricity of the ellipsoid. These partial derivatives
could be used in determining the dimensions of the ellipsoid that would
reduce the deflections in the region under discussion to a minimum ~
and hence this ellipsoid might be said to be the one that best represents
the figure of the earth for the particular region under discussion. The
deflections of this or that arc or this or that region have been discussed,
but no ellipsoid derived from the deflections of the whole of Europe
has become either well ltnown or generally accepted, much less an ellipsoid
derived from European deflections and deflections outside Europe.
The only ellipsoid since Clarke's time based on deflections both in
the meridian and in the prime vertical that has found any general ac-
ceptance is Hayford's. The deflections on which lIayford based his work
were confined to the continental United States but since the actually
observed deflections were corrected for topography and isostatic compen-
sation (see; page 108), the spheroid that best fits these corrected deflec.-
tions in the United States probably approximates closely to the spheroid
that best fits. the earth as a whole.
* P. 3~4.
~ The actual criterion is that the sum of the squares of the deflections-or per-
ha.ps the sum of the squares multiplied by certain preassigned weights-shall be
. .
;1 mlmmum.
.
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132
FI CURE OF THE EAR TH
With this idea in mind the Section of Geodesy of the International
Geodetic and Geophysical Union in 1924 adopted the Hayford ellipsoid
of 1909 as the basis of the International Ellipsoid of Reference. Heis-
kanen ~ b has shown that European observations of the deflection of the
vertical, when isost.atically reduced are in excellent agreement with
IJayford's dimensions. Some dimensions of the International Ellipsoicl
of Reference are:
Equatorial radius (semi-major axis)..
Polar radius (semi-minor axis).......
Quadrant of meridian...
Quadrant of equator....
Ellipticit',r (flattening) .
...... 6,378,388' meters
...... 6~356,912 meters
...... 10,002,288 meters
..... 10,019,148 meters
. . 1/297.00 = 0.00336700
~.. ...
Square of eccentricity of me~i~li:~n ellipse 0.00672267
Deflections are found by assuming, the latitude and longitude of some
point, usually a point near the center o:t the t.ria.n~ulation where the
deflections are to be clet.ermined, and the azimuth of ~ line of the triangu-
lation radiating, from the point. These quantities together with the
dimensions of the ellipsoid used constitute a Geodetic dating. For instance,
the latitude and longitude of the triangulation point Meades Ranch in
Kansas, the azimuth of the line from Wades Ranch to the triangulation
point Waldo, and the dimensions of Clarke's spheroid of 1866 taken. to-
~ether constitute the North American Datum, which is the basis of the
geodetic positions in the United States, Canada and Mexico. In theory
it would be necessary to go back to the fundamental geodetic observations
in the field and express each observation by means of an observation
equation with correction terms depending, on the still undetermined
elements of the new geodetic datum. Such a fundamental process would,
however, entail considerable labor. In actual practice the best. ellipsoid
-for a giver region, or the so-called -fi¢,ure of the earth, has beets determ:inerl
by a less rigorous method.
A change in the dimensions of the assumed spheroid changes the compu-
tation of the triangulation ab Ritz, and hence the length and azimuth
of a geodesic line connectin, two points of the triangulation. In the
less rigorous method, however, these changes are ignored. A central point
is taken (as O. see Figure 1 i and geodesic lines OA, OR, OC, etc., are
drawn to points of the triangulation at which astronomic observations
have been made and deflections of the vertical are therefore available.
The length and azimuth of OA, OB, etc., are computed and it is assumed
that these lengths and azimuths will remain invariable when the lines
are transferred to a new ellipsoid. On this assumption the change in the
deflections due to Mien changes in the ellipsoid may readily be computecl.
By a least-squares adjustment it is possible to determine the changes in
the ellipsoid that will reduce the suns of the squares of the deflections
OCR for page 133
SHAPE AND SIZE OF THE EARTH
133
to a minimum. This is by definition the best ellipsoid of reference for
a given region or for the earth as a whole (that is, for all the regions
considerecl), as the case may be.
The difficulty with this method of adjustment, is that the deflection
obtained at a given point depends on the route followed to reach that
B
/ I
l
l
F ~
H
\
\
A/ /1 \
\
\
E
FIG. 1.
-
-
point, that is, the deflection at ~ -i'ound by going clirectly from O to A
is clid'erent -'roan that IGUnCI by going, via the lines OB and BA, and diff'er-
ent also front that found by goin, via OC, CB and BA, and so for each of
tile other points reached by the radial lines OA, OB, OC, etc.~9 sob
The comparison of the framework of' the triangulation to a network of
wires may make the matter clearer. This network is necessarily curved
to fit the curvature of' the spheroid :t'or which it was originally designed.
OCR for page 140
140
FIGURE OF THE EARTH
Art example entirely hypothetical will make the process clearer. The
formulas for the various functions are given for completeness, and in
several forms in view of practical applications. No especial geomet-
ri.cal or mechanical meaning can be given to the several terms, am
more than a similar meaning can be given to the various terms in the
formulas for the attraction of various bodies. The terms are simply the
result of integration and other mathematical transformations.
Let ~ denote in general the angular distance of a point from the
center of the system of circles, that is, for a point where the elevation
or depression of' the Void is sought. With subscripts 2 or 1 it may
denote respectively the outer or inner radius of the zone under con-
sideration.
In its most general forte Stokes' Theorem may be written
N= i~gf(~)dS
DIG
(~9a)
where dS is an element of' solid angle, fig is the gravity anomaly for the
element concerned., the integration covering the entire sphere, and
feel =~4cosec lb+!-6 sin -by-~ cos
-3 cos .b loge Sin :b ~ ~ + sin A) ~ t. ~10 ~
In practice the integration will be effected by mechanical quadrature.
IT we specialize the element of solid angle dS as sin (dads, where
is an azin~uthal angle, we have
1\~= ~ day l~(7f(~) Sill Fd§
27rG ~0
(it)
If we uIlcle-lstalld b;y A~1lt1 the mean value of' /`y around ~ circle of:' a~,ula
adius §, the -l'ormula may be written
1V= G tj ~mqf(~) sin fd§.
(~12)
Suppose we divicle the ir~ter~-al from O to ~ into zones, of' which a typical
one has the radii 4, and §,, and let /`'mg denote the mean value of the
anomaly over the entire zone, and If' sin 522 the mean value between
the limits H.' and §, of fact sin §. Then since id practice we substitute
summation for integration, we write
~7\' = G ~ /` rug [f ( ~ ) sin H] ~ ( ~ ~-§1 )
(13)
The summation covers all zones from §=0 to the antipodal point where
§=7r. The quantity b'-§, is to be expressed in radians.
The mean value in question is given by
~~f(~) sir1 ~ = h i ~ ~ fib) sill (cay it'
OCR for page 141
SHAPE AND SIZE OF THE EARTH
141
If, disre~,ardiIl~, constallts of inte~,ration we write if(~) sil1bd§=~(~)
we find for d>(~) the expression
'A i b) = ~; 4 sir1 2 + 2 sin' ~ -(; sift ;: - 4 sill 1 ~
-Silly blow |sin ~ (~+Sill 6 ~ ~ ~ (ill)
~2
`~.~.~.`l As ~ wo~lL;n~, for no Viol Stoli~es"Theo:ren~ that. is I:articula~ly- co~ve~ie~.t
Beheld tile solves are wicle, we get
1;-- G `'~'~ d)(b ~-~(~1.)
The summation extends groin .§ = 0 to ~ = Or.
The -following, table gives an idea o:L how fact, ft §) sin §, and ~)
fiery. It is given at closer ;~te~vals near the center of the zones on
account o:t the panics variations there of tl~e first two -fU~letiO~lS.
V.~1 I,ES OF THE FUNCI'1ONS USED IN STOKES THEOREM
~f (I) f (I) bits ~'t, (I) ~f (I)f (I) Sit, ~'i, ~ d)
0° GO 1.000 0.0000 80° - 1.10- 1.082 +0.0138
2° 32.6 1.139: + 0.0378 90° - 0.911- 0.914 - 0 -1626
4° 17.2 1.200 + 0.0787 100° - 0.63- 0.623 - 0.2981
6° 11.7 1.227 + 0.1212 110° - 0.0291- 0.270 - 0.3714
S° S.84 1.230 +0.1641 120° + 0.09+0.077 - 0.392S
10° 6.99 1.215 + 0.2068 130° + 0.417+ 0.358 - 0.3534
20° 27.5 +0.941 +0.3989 140° +0.82+0.526 - 0.2745
30° 0.95 a- 0.474 + 0.5241 150° + 1.12+ 0.559 - 0.17~8
40° - 0.08 - 0.054 + 0.5608 160l° + 1.35+0.461 - 0.0873
50° - 0.70 - 0.537 + 0.5078 170° + 1.49+ 0.259 - 0.0230
60° - 1.03 - 0.896 + 0.3806 180l° + 1.540.0100 0.0000
70° - 1.15 - 1.082 + 0.2053
Let us world out an entirely hypothetical example, merely to illustrate
the use of the tables ancl formulas. Divide the entire globe into zones
<30° wicle centering, at tile place -here lie wiser to Ludlow the elevatior~ of
the ¢,eoicl arid assuage the lol]ow.i~, values of of,,, for the several zones:
Rail; i ot a',,, ~
zone cm/see'
0 ° _30 ° 0.000
30° 60° + O 050
60°-90° + 0.0~60
90 °-120 ° + 0.030
20°-150° - 0.010
00°-180° - 0.040
Tl~; r ty den ecs = =/(; i: Clips = 0. ~23(; = ~onstal:~ t value of §,-As,
c~=(;....3,'1 x 10'; metes, G=981 em.isec.2
OCR for page 142
142
PI CURE OF THE EAR TH
(~9 alla G must be in the same unit, which, however, need not be the linear
unit as for a; the unit for a will be the unit for 7V). Tapir the constant
value of (~-b~) outside the summation in (13) we have
N= 6 3 ~ i x i0 x 0..6236~/i'~,,g if (~) sin b j1= 3400~!` myrf (~.
Let us use the simple trapezoidal rule to find the mean values for
fall sin b. Then we have the following, using, for uniformity only the
10° interval even in the first zone. Since fig is zero for this zone, the
error is of no consequence here. In other cases a more accurate evaluation
is desirable, since near the origin fall sin ~ changes rapidly.
zone Mean valises of f(~) sin ~
30 .; [ ~ (1.000) + 1.215 + 0.941 + ~ (0.474) ~ = + 0.964
30° - 60° I- _(0.474) - 0.0~04 - 0 537 - --(0.896)] = - 0.267
60 _ 90 1~ [ _ -1- (0-896) - 1-082 - 1.082 - 4- (0.914) ~ = - 1.023
!)0° - 120° 1` [ _- (0.914) - 0.623 - 0.~:'0 + -in (0.077) ] = - 0.437
120 - 1.50 -~1 +3,-(0-O:~) +iO.~08+0-,)26+ (0 S59) 1 +0-401
150° - 180° -.3 [ +- (0-~69) +'0 461 + 0259 + -1- (0.000) ] = + 0.333
+0.964X
- 0.267 X
0.000 = 0.000
+ 0.050 = - 0.013
- 1.023 X T 0.060 = - 0.061
- 0.437 X +10.030 = - 0.013
+ 0.401 X + 0.010 = + 0.004
+ 0.333 X ( - 0.040) = - 0.013
Sum= - 0.096
Sum X 3400 = N = - 326 Inetei s,
or the geoid is 326 meters below the spheroid corresponding to the
gravity formula on which the gravity anomalies are supposed to be based.
If we use the function Ah), which leads to a little simpler :form of
computation, we get:
G = 64~30
.~=6490~ ~n1~) ~]
= 6490 ~0.000 X ~ 0.,0241 - 0.000 ~ + 0.0,30 ~ 0.~806 - 0.,5241 ~
+ 0.060; ~-0.1626 - 0.3806) + 0.030 ~-0.3928 + 0.1626)
+ 0.010 ~-0.~78 + 0.3928) - 0.040 (0.0()0 8- 0.1778) l
= 6490 x ~-0.0~16) = - 33o meters.
The slight discrepancy is due to the fact that the process of taking mean
values of fib) sin d in the first method is not particularly accurate for
such wide zones. The second method is here better and slightly easier.
OCR for page 143
SHAPE AND SIZE OF THE EARTH
143
The values of ~/q7',7 are not such as would be given by a properly ad-
usted gravity formula; their average value, when the areas of the zones
to which they apply is consiclered, is not zero, as it should be, but 0.0308.
Pv subtractin, this irons the >,;ve~ anomalies Ale get for the various
zones-0.0308, + 0.0192, + 0.029?, - ().0()063, - ().208 a-cl-0.0708. If
we apply these values in tl~e second met]~o~l. we get
Y = 6400 ~~ - 0.0308 x (). ~241 + 0.0192 x ~-0.:1435 )
+ 0.0292 x ( -0.~432? -0.0008 x ( -0.2302) -().0208 x 0.2150
-0.07()8 x 0.177 8] = - :~:~.~ Dieters, as before.
There is another fault to lie found with the ave~a~,e ~,r.a~ity anomalies as
Mien. If we consider the anomalies as thy;. sumac leavers of positive or
negative netter with thicknesses p:ro~'ortio~.~al to the anomaly, then the
center of gravity of this surface shell o-t matter should coincide with the
center of the sphere over which the shell is spread. This would be true of
actual gravity anomalies with re:terence to the earth, treated (with suffi-
cient approximation) as a sphere, but it is not true of these hypothetical
anomalies. This defect could be corrected but, just as in the case of the
correction to reduce the average anomaly to zero, this second correction
would leave the value of N unchanged. The simplest. type of ideal distri-
bution of gravity anomalies that is correct from the start is one that is
s\m~netrical (as to average value) Title respect to tile equator for which
the center of the zone distend and its antipode are the poles.
LUNAR ~II',T'HO])S OF DETERMINING THE FLATTENING
OF TH]3 EARTH *
Thee e are two lunar telethons of determining the flattening of the
earth: the first, a mixed geometrical arm dynamical method dependent
on the parallax of the moon and going back in principle to Newton's
-fan~ous calculation that identified terrestrial ¢,ra.vitatio~ with the force
that retains the moon in its orbit; the second method, purely dynamical
and depenclen.t on the lunar perturbations. This seco:~cl method might
lie subdi.vicled. ;~:~t.o several methods according, to tile particular pertur-
bation usecl.
Newt.or1 calculated what the parallax of the moon would be if terrestrial
¢,ravita.tion, dim;~;shin~, inversely as the square of the distance, were
the controlling, force, and on comparing this with the observed parallax
he found them to a<,ree within the uncertainties clue to' errors of ob-
servation.T The so-called observed parallax, lo-ever, that is, the equa
*Professor E. W. Brown gave valuable assistance in the preparation of this
portion of the text. He is not, however, to be considered responsible either for
the statements made or for the opinions expressed.
~ There are terms involving also the mass of the moon and the effect of the sun
which require no more than mention in this connection.
10
OCR for page 144
I~14
FIGURE OF THE EARTH
torial horizo~ta1 parallax. r eall\ involves an assun~pti.on as to the
ellipticity of the earth, acid still other. assumptions as to the local
deflections of the vertical and as to the elevations of the ~eoicl above the
spheroid at the observatory ies where the parallax is cletermirred. The
usual and almost i.nevitahle assun~l:,tion l~;therto has been that these
deflections ancl elevations are `~11 zero. The calculation bv which ter
restrial gravitation is extencle`1 out to the moon and the parallax o L
the latter thus louncl is also a-tTe~te~l l.'v the assumed flatte.~in~, of the
earth, though not so much as the obserwecl parallax. The flattening is
found by assuming, di.~.-~:ere~t \!alues for it until the calculated equatorial
horizontal parallax comes out equal to the obse~vecl parallax. This
happens for a flatte::~i.r~ equal to -1/~.'3~3.4, according, to do Sitter,~4;1 on
the assu~-nptio~. that the ~leflectio~s o:t the vertical at Greenwich and
Cape o:t Good TIope observatories are zero. To make the two parallaxes
equal for a flatten;, of 1/2'37 would requite a deflect;.o~. of about 11"
at both observatories. A calculation of these deflections on the theory
of isostasy, with uniform compensation down to a depth of' 120.D lam.,
gave deflections of' 1"16 at Greenwich and 3",, at the Cape. These ha:1
the proper signs to shift the reciprocal of' the flattening -l'rom 293. L
~1 ~ A JO 1
1 ~
towarcts ;~J s IOU: are not. large enough to change it all the wa,v.~iC
They could be increased somewhat by adopting, some other clist.ributio~-~
of' isostatic compensation, but the most probable explanation of much of'
the discrepancy in the flattening is observational error in the deter~ina-
tion of' the parallax. De Sitter concludes that the parallax calculated on
the theory o:t' gravitation is more reliable than the parallax direct!:!
observed and the parallax used in Brown's new lunar tables is likewise
derived 'row theory. These conclusions of' de Sitter and letdown and the
uncertainty due to id orance of' the actual deflections o:t' the vertical are
thus unt'avorable to the accuracy of' the flattening, obtained from it.
The purer,\ cl,nI~>lmical lu.n.lr methods do Mot cleterm-ine tile flattell:in<`r
l.i.rectlv but ;~ste`~.l. tl~ev `'let.crm:i:~e the qu..~titV
.i .. ..
J C--~-( 1+]~)
Eat
where A, i? anal C are the pr'incipa,1 moments of' inertia of the earth Hi
ascencli~lo order ol' ~na~itucle, E is the mass 0:; the earth and a, its equa
~ ~ , , ~
· ~ ~ ~ ~ · ~ · r ~ ~ 1 , 1 I L' I 1_
tor~a.1 radius. Lams quallt,]ty J occurs in the developments in series of tne
earths gravitational potential. It' the earth were composed of homo
,eneous concentric spherical shells, J would evidently be zero. Groin J
the flattening,, f, is deduced by a simple I'ormula.
*The deflections and elevations are understood to refer to the unknown ideal
ellipsoid that fits the earth as ~ whole as closely as possible.
OCR for page 145
SHAPE AND SIZE OF THE EARTH
145
The fact that the earth is not spherical, that is, that J is not zero,
causes a number of perturbations in the moon's motion, both periodic
and secular. From the observed values of these perturbations the value
of J can be deduced and hence the flatt.enin`,.
The periodic perturbations due to the fi.~,ure ~ of the earth have been
used to determine the flattening by various astronomers, I'rom Laplace
down to the present. These periodic perturbations are, however, rather
difficult to disentangle by observation from the perturbations due to other
causes and hence the flattening deduced Tom them is not entitled to
much weight.
The principal secular perturbations due to the earth's figure are those
of the node and perigee of the lunar orbit. These, being cumulative with
the lapse of time, can be determined very accurately. Uni'ortuna.tely,
however, for our present purpose much the larger part of these perturba-
tions is due directly or inclirectly- to the action of' the sun and this must
be determined and subtracted from the observed effect in order to
determine the portion due to the figure of' the earth.! Tl~ere is also a
portion, minute but by no means negli,,ible in the present problem, due
to the figure of' the ~noon. Fortunately the portion due to the sun depends
mainly on the ratio of the month to the year, a quantity known with great
precision, so that apart from possible but not very probable errors in the
theory ol' the smaller terns representing the sun's effect 4; and the un-
certainty due to the figure of the moon, the perturbation due solely to
the figure of the earth may be considered as having almost the same
observational error as the entire observed secular perturbation, whether
of perigee or node.
In connection with the preparation of new lunar tables the flattening
was determined by Brown,7 who found 1/293..~ from the combined results
for the perigee and node. His adopted result 1/294.0 comes from includ-
i~g his value 1/294.4 determined from the Greenwich-Cape of Good Hope
parallax observations. By changing the assumed data for the figure
of' the swoon do Sitter 34a found 1/296. Tl~e data for the figure of the
moon depend partly on observation o:t' the lunar librations, partly on
* The word Azure is used in the extended sense of figure and constitution, more
specifically its figure and constitution as they affect the value of J.
t Of the total annual motion of 146,435" in the perigee, and 69,679" in the node,
all but about six or seven seconds of each are due to the action of the sun.
1: The formulas for this occupy many lines of the quarto page of Delaunays
" Lunar Theory." Even so, they are not sufficiently accurate for the present pur-
pose and the more modern theory, such as Brown's, does not use a complete and
explicit formula, but is based on a. process of numerical approximations. Delaunay's
formulas, however, bring. out the fact that the solar effect depends essentially on
quantities known to ~ high degree of precision.
OCR for page 146
146
FIGURE OF THE EAR TI-I
assumptions as to the way in which the density of the moor varies from
center to surface.
The process of deducing the flattening, from lunar perturbations is
thus seen to be far from simple. The possibilities for reconciling the
flattening of 1/294 :lound by this method with the 1/297 of the inter-
nat.ional ellipsoid lie: 1) in changes in the secular variations deduced
from observation; 2) in data for the Inure of the moon; ancl 3) in
possible corrections to a very complex mathematical theorem.
Jones has used occultations at the Cape o:l Good I]ope to correct the
motions of the perigee and node, and his corrections tend to diminish
the disagreement between Brown's value of the flattening and the new
international value. The older lunar observati.o~.s, however, are not so
well represented and it is questionable whether the full amount of his
corrections can be accepted without further consideration.
The figure of the moon, as de Sitter has shown, gives one way of
reconciling, partially at least, the two values of the flattening,. Because
the moon is smaller and cooler than the earth, its figure might depart
relatively more from hydrostatic equilibrium than the figure of the earth
can, and there is evidence to show that it does. In the present state of
our knowledge, or ignorance, a wide range of suppositions is admissible.
In the absence of direct evidence on this point the disagreement between
Brown's value of the flattening and the value obtained by other methods
might be used as evidence to show what the figure of the moon actually is.
The theory of these secular variations, like the lunar theory in general,
is exceedin<,ly long and intricate, but it has been so much worked over
that it is probably now correct and must be assumed to be so, at least
until some one with the necessary ability, energy and inclination discusses
the matter further and finds an error. There seems to be room, however,
for some slight improvement in a much simpler theoretical matter, the
calculation of the flattening,, I, from J. Astronomers, as a rule, have
omitted certain second-order geodetic terms in the equation connectin,
these two quantities.* The inclusion of these terms increases the re
* The approximate relation often used is
3 1 ma
f 2 JO 2 ge
(for meaning of w", a, and go see paragraphs dealing with Clairaut's formula). This
corresponds in accuracy to the simpler form of Clairaut's equation. The relation
correct to terms off the second order is
~_ .T ~ -An. -as.'= 15 m.,7 ~9 '7'
2 ' ' 2
cola
where m is written for-
g
.
r) 6 2 S s
OCR for page 147
SHAPE AND SIZE OF THE EARTH
147
ciprocal of the ellipticity by 0.4 or 0.5, or perhaps more, the exact amount
depending on the exact form of the approximate equation between J and
f, which may be stated in more than one way when correct to the first
order only. These second-order terms would thus help to diminish the
discrepancy still further.
The only conclusion that can safely be drawn at the present time is
that the discrepancy between the value of the flattening from lunar
observations and the new international value is probably not quite so
large as it at first appeared to be, since various considerations all working
in the same direction tend to bring the two values together. The ideal
and the problem of astronomers and geodesists is, of course, complete
reconciliation of the flattening as determined by geodetic methods with
the values determined by all the various astronomical methods.
We can also get a hold on the flattening of the earth by studying
another mechanical effect also due to the moon and in lesser degree to
the sun and planets also but of somewhat different character from the
perturbations of the moon's orbital motion just considered. This is the
effect of the moon and the sun and the planets on the direction in
space of the earth's axis of rotation, in short the precession of the
equinoxes. The moon, the sun and the planets,: but chiefly the moon,
exert a torque on the earth's equatorial protuberance tending to bring
flee terrestrial equator into coinciclence with their own orbital planes,
that is, approximately with the ecliptic. But since the earth is rotating
rapidly its equator never is brought into coincidence with the plane of
the ecliptic. Instead, the pole of the equator describes about the pole
of the ecliptic a small circle of approximately constant radius equal to
the obliquity of the ecliptic.! The pole o:t the equator completes its
circuit about the pole of the ecliptic, which retains a nearly constant
(Erection in space, in a period of about 26,000 years. Obviously, the
greater the equatorial protuberance of the earth, that is, the greater the
flattening, the greater will lie the torque exerted on the earth and the
faster will be the motion o1: the pole of: the equator about the pole of the
ecliptic, so that by working, Awards :from the observed rate of pre-
cession the flatte~-~ir~, o:t the earth may be cletermined, subject to certain
assumptions.
* The pull of the planets on the earth's equatorial bulge is very minute. The
so-called planetary precession is the effect of the planets on the position of the
plane of the earth's orbit.
~ The obliquity of the ecliptic is not constant but its variations are slow and
relatively small and a consideration of their causes and periods would lead us far
afield from our subject.
OCR for page 148
118
FIO~ 0F T~ ~
In the lunar perturbations the quantity determined is-0 <~+ )
(for notation see page 144~; to pass from ~bicb to Battening no ~ssu~p-
tion Shout the interior of the e~rtb is necessary. From the precesaion
the quantity determined ie 1~+ ), from Ibid we cannot pass
to the Battening without making ~ssnmptions.
Let US assume abut the density distribution vitbin the earth is con-
sistent with bydrost~tir equilibrium, any since me know isost~Uc ~djust-
ment to he almost complete, this supposition seems Unlikely to lead
to serious error. Theoretically; in order to derive ~ flattening I~om the
precession, me must assumes not merely b~rost~tic equilibrium in gen-
er~l, bat ~ JeRnite 1~v of V8liRtiOD from center to surface. Ibe eject
of cb~nging the 1~v is, hoverers surprisingly smelt in fact, Afloat
negligible. Any reasonably plausible law of density any some laws
not so pl~nsitle-will serve.
It was at one hme suppose] that the agreement of the Battening de-
rived Idiom the observed precession and the assumption of I;~pl~re?
(otherwise Legendre\) 1~v of density Aim the Battening derived in other
Alas was ~r argument in Indoor of the [~pl~Ge-Legendre 1~; then Inch
in vane; hut when other laws were tang ~ was found that the agreement
was just as good. Abe explanation was given by H~d~u ~ and mount
to this: ~ certain m~tbem~tic~1 expression on which the Battening de-
pOD4S ~D] whim in turn depends OD the 1~v of denser is in form
v~ri~tle ~] in Act Jocs vary, but so Mighty that, Then the density is
restricted to Abut is pllysi rally 1jossible, the Battening deduce] Irma
the precession is practically independent of the 1~v of density, if hydro-
st~tic equiLbrium prevails within the earth.
Y6ronnet derive] Irom the precession ~ Battening of 1/29~.12~0.38.
Here We ~ indicates; not probable erroT in the ~di~ry senses but iDe
estimated range of UDce~taint~ fine to -r ignorance of the 1~ of Jensi#,
no ~llov~Dce being made Ior the Uncertainty Of the observed value of the
constant of precession nor of other quantities involved. [e Sitter 14~ 14c
Kiter ~ c~reIu1 JiSCUSaion finds even narrower Watts. Ibis result is
1/296.92~0.IS6 or in ~ later revision 1/29~.96. Most of this small
provable error he finds to be dUe to the Uncertainty in the mass of the
moon, We Uncertainty due to oar ignorance of the law of density Being
hut ~ comp~r~t~ely small parts Aid Ho U~rert~inty JUe to the probable
error in the precession~1 coust~Dt being negligible.
The elUpticit~ OF ~HtteniDg of the ebb seems then to be Wont 1/297,
whatever method of determination may be nseJ.
.. . . . .
OCR for page 149
S~ ]\~ ~Z~ OF T~ ~T49
I. Any, C. B. Figure of the eurtb. Article in Encyclopedia ~e~opol~n~, [on-
don, 18~, vol. 3. (An Decedent account of the Bole subject, including
its m~tbem~tio~1 aspects, with much bistoric~1 info~ion.)
2. Aristotle. He cello. Eng. trains. by J. L. Bomb Oxford, 1~. (See especially
Book II.)
3. Be~zley, C. R. The din of modern geography. 3 vole London, 18g?-1~5.
(An interesting add v~uablo Coopt of the prowess of ieogr~bic kooky
edge, including ~ dis~s~on of speculation about the figure of the fib.)
4. Beset F. W. B~sEmmun~ tier Axed des elEpti~cben Botudo~bEroids,
Cubes den vorbundenen ~essungen am fisted entspricht. Astron.
Cache vOl. 14, no. 333: 333 (1887); or, Abh~ndl. (ea. by R. En~el~unn)
Leipzig, 3: 41 (1876) .
. l}eber einen Fehler in tier fr~zO~iscb~ Or~dme~ng and seinen
FinOuss Of die Beshmmuni der Hour der Erde. Ahb~ndL, 3: SS. (Con-
t~ins the dimensions of the ~pberoid gener~Uy known us BesseLs)
S. BGrscb, As and KrUger, L. (also F. R. Helmert for Heft 1). Lot~eichungen
Repts. to Internal Oeod. Assoc. Reran, 1 (1~6); 2 (1002j.
On Genie, Ha. Coodetic o~er~ion~ in Abe United Stoutest Washington, 1027 (Ce S
Coat Ceod. Rev Spec. PuhL ~^ 134).
7. Brown, F. W. The determination of the oon~nts of the node, Me iicEn~tion,
the e~rtb's eUiptic~y Ad the Inequity of the eoLptic from the Oreen-
wicb meridian observations of the moon, 1847-1901. Roy Astron. Boas
Ho. Notices, 74: S~ (1914).
8e Wilds A De ~ bigotry of the determine of the figure of the eurIh
from arc measurements. Worcester (amp, 19~.
9. Cujort P. Charter in On Sir Isaac Kenton, 1727-192~ bicenten~^ ev~ln~-
tion of his Folk, ad. ~ F. F. Br~scb. Baltimore, 1928.
lOe Club~nt, Ae ThEprie de 1~ figure de 1~ Beige. Maria 1743.
lie Clarke Ae R. Ceodesy. Oxford, 1380. (A good genera treatise, ~1 standard
in spite of its dam.)
- . Ordnance tri~onometricu1 survey of Ct. Britain and Irelund
~ccou~ of observ~i~ and cation of the princip~I tri~gul~on ~nJ
of the fire, dim~eciEc Amity derive] Ceremony.
London, 1~8. (Compiled under the dkechon of Lt. Colonel H. James)
12. Darwin, C. H. Abe theory of Me hgurc of the earn curried to the second
order of small quantities. Roy. Astron. Soc., Ho. Notices, 50: ~ (1900)
or, ScL Furlers C~m~d~e (Fng.), 3: 78 (1910).
13. Lelumbre, J. B. J. Histoire de l'ustronomie uDcienne. 3 vols Paris, 1S17.
(Lseful for the early history of determinuti~s of the azure of the eurtb.)
. Fd. by Bigourd~, C. Orundeur et hgure de 1~ Terry. Perk. 1912.
(Published nearly ~ century aft outbox de~tbJ
14. de Sitter, W. On the Aeon ladies of the earth, the intensity of gravity, and
the ~oon's Purdue. G. Akin. Wetted Amsterdam, Proo. 17:12~ (1013).
. On the Fattening and the ConshtutioD of the earth. G. Eked. Weten-
scbu~pen Amsterdam, Proc. 27: 244 ( 1924) .
. For ~ rig r~76 ~6 8~6~y ~r~6, ~ Astron. Inst. ~ether-
lunds, BulL v. 4, no. 1~: 60 (1927).
OCR for page 150
130
FIGURE OF THE EARTH
15. Fiske, John The discovery of America. Boston, 1892, vol. 1. (Contains an
account of the clevelopr~ent of geographical knowledge of the prevailing
ideas about the figure of the earth.)
16. Hayford, J. F. Supplementary investigation in 1909 of the figure of the earth
and isostasy. Washington, U. S. Coast Geod. Surv., 1910, 80 p. (Not to
be confused with work published in 1909, entitled The figure of the earth
and isostasy from measurements in the United States.)
17. Heiskanen, W. Untersuchungen uber Schwerkraft und Isostasie. Veroffent-
lich. des finnischen Geodatischen Institutes, no. 4 (19;24)- To be sum
plemented by
. irber die E,rddimensionen. Vierteljahrsschr. Astron. Gesellschaft, 61.
Jahrgang, p. 215, and
. Ist die Erde ein dreiachsiges Ellipsoid? Gerlands Beitr. Geophysil<.,
19:356 (lD28).
18. Helmert, F. R. Die mathematischen und physikalischen Theorieen der hoheren
Geodasie. 2 vols. Leipzig, Teubuer, 1880-84. (Still a standard work.)
For deflections of the vertical see
9. Hunter, J. G. The earth's axes and triangulation. Survey of Indi<~, Prof. Paper
No. 18 (19~18).
20. Lambert, W. D. The figure of the earth and-the new International Ellipsoid
of Reference. Science, (i3: 242 (1926).
-. The effect of variations in the assumed figure of the earth on the
mapping of a large area. Washington, 1924. (U. S. Coast Geod. Surv.,
Spec. Publ. No. 100.)
. The figure of the earth and the parallax of the moon. Astron. J., ~rol.
38, no. 908 :181 ( 1928) .
21. Muller, Q. Ges;chichte der Gradmessungen bis zur peruanische Gradmessung.
(Doctor's thesis) Rostock, 1871. (Convenient historical summary.)
22. Pizetti, P. Sopra il calcolo teorico delle devia%ione del Geoide dall' Ellissoide.
Atti Acc~d. sci. Torino, 46: 331 (1911).
23. Radau, R. Sur les lois des densites ;~ l'interieur de la Terre. Compt. rend.,
100: 972 (1885); o' ir~ ~absta~ce in Tisserand's Traite de mecanique celeste,
Paris, 2: 221 ( 1891 ) .
24. Rudzki, M. P. Physik der Erde. Leipzig, 1911. (A very convenient book.
Contains the forn~ulas (p. 46) for gravity outside an exact ellipsoid of
revolution. On p. 62, in the calculation of the rate of change of ~'r~vity
with elev:~tion, the factor 1.0006821 has been used instead of 1.00682;1. By
using tl~e correct value, the forrn;ul;1 for l becomes -0.000003088
dr~
~ 0.000000004 sin B. which is consistent with the formulas commonly
used.)
25. Stokes, G. G. On the variation of gravity at the surface of the earth. Cam-
bridge Phil. Soc. Trans., 8:672 (1849); or, (Stokes') Mathematical and
Physical Papers, Cambridge (En<,r.), 1883, p. 13.
26. Wiechert, E. Ueber die Massenvertheilung im Innern der Erde. Nachrichten
kg. Ges. Wiss. Gottingen. Math. Phys. K1., 1897, p. 221.
Representative terms from entire chapter:
der erde
