Establishment of Geodetic Coordinate Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 340

Gongyue measured the distances between Huaxian, Junyi (now Kaifeng), Fugou,
and Shangcai. He also measured the altitude of the North Pole in these four places
and the shadow cast by the sun at midday on the summer solstice. He concluded that
the length of 1 of meridian arc was 351 Li and 80 Bu (by Tang measurement, 1 Li
equals 300 Bu). Since 1 Tang Li is equivalent to 1,500 Tang Chi, and 1 Tang Chi is
equal to 24.75 cm, we find that 1 of arc length is 130.38 km. The ancient astron-
omers set a circumference of 365.25 , and it was converted to 360 ,so1 of arc
length was calculated to be 132.28 km. This value, although 21 km larger than the
given arc length of 1 being equal to 111 km, is extraordinary considering the level
of technology back then (Xiong 1985).
Since Newton first claimed that the Earth's shape was an ellipsoid and Snell
proposed the method of triangulation, the early eighteenth century ushered in a new
era of arc measurements. The concept of arc measurements has been extended to
determining the two elements of the Earth ellipsoid, i.e., the semimajor axis a and
the flattening f. Since the early 1800s, surveyors from different countries have been
engaged in a great number of arc measurements and have calculated many results
for the Earth ellipsoid. It can be seen from the formula for meridian arc length in
Chap. 5 that the meridian arc length is the function of a and e 2 (or f ). Based on the
surveying results of many segments of the meridian arc on the Earth, we can find the
solution of a and f (or e 2 ) by means of the least squares method. The currently used
arc measurement equation is derived from ( 7.23 ). In practice, the new ellipsoidal
elements are obtained using astronomical, geodetic, gravimetric, and satellite
surveying data based on the original old ellipsoid. As a result, the calculation of
the new ellipsoid elements is actually a process of successive approximation. Let
the elements of the old ellipsoid be a old and f old , and the elements of the new
ellipsoid be a new ﾼ a old +da, f new ﾼ f old +df. The problem now becomes to find da

and df.
It can be written from the formula for the deflection of the vertical that:

2 3 2 3

ʷ new
ʾ new
N new

ð ʻ L new Þ cos B new

4 5 ﾼ 4 5

ˆ B new

N new

2
4

3
5 þ

2
4

3
5 ,

ð ʻ L old Þ cos B old dL cos B old

ﾼ ˆ B old

N old

dB
dN

ð 7 : 27 Þ

where dN ﾼ dH. Substituting ( 7.23 ) into the above equation yields: