Indischer_Kreis2

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In der Feldmessung bestimmte man die Nord-Sud-Richtung mit Hilfe von Gnomon und

IndischemKreis. Dabei wird die Nord-Sud-Linie zur Halbierenden des Winkels zweier

gleich langer Schatten am selben Tag. Das Verfahren wurde erstmals von Vitruvius I, 6

erwahnt, ist aber sicherlich alter; zu seiner Geschichte ausfuhrlich Schmidt 1, S. 197

202, zur Namensgebung Wiedemann, Kreis 2, S. 252: ,,Bekanntlich ist zuerst nichtder Almagestdes Ptolemaus den Arabern durch eine Ubersetzung zuganglich gemacht

worden, sondern die indische Siddhanta unter al-Mansur (754 775), wahrend das

zuerst genannte Werk erst unter al-Mamun (813 833) in das Arabische ubertragen

wurde. Hierin mag der Grund liegen, dae das (...) Instrument (...) den Namen indischer

Kreis tragt, obgleich er den Griechen wohlbekannt war." Auch der Astronom Oinopides,

ein Zeitgenosse des Horodotos, experimentierte mit dem Gnomon. vgl. Heath 3 I, S. 78

und Boehme 4.

Karlheinz Schaldach, Die antiken Sonnenuhren Griechenlands, 2006, S. 21 Anm. 3.

Figure 2.1 Determining the east-west line with shadows cast by a stake.

The preliminary step for altar constructions is the drawing of a baseline running east

and west. We do not know for sure how this was accomplished in the time of the early

Sulba-s utra authors, but the later K aty ayana-sulba-s utra prescribes using the shadowsof a gnomon or vertical rod set up on a flat surface, as follows:

Fixing a stake on level [ground and] drawing around [it] a circle with a cord fixed

to the stake, one sets two stakes where the [morning and afternoon] shadow of

the stake tip falls [on the circle]. That [line between the two] is the east-west line.

Making two loops [at the ends] of a doubled cord, fixing the two loops on the

[east and west] stakes, [and] stretching [the cord] southward in the middle, [fix

1 Fritz Schmidt, Geschichte der geodatischen Instrumente und Verfahren im Altertum und Mittelalter,

1935 (ND Stuttgart 1988).2

Eilhard Wiedemann, ,,Uber den indischen Kreis, in: Mitteilungen zur Geschichte der Medizin undder Naturwissenschaften 10 (1912), S. 252 255.

3 Thomas L. Heath, A History of Greek Mathematics, 2 Bde., Oxford 1921 (ND New York 1981)4 Harald Boehme, ,,Oinopides Astronomie und Geometrie, in: Mathematik im Wandel 2 (Mathe-

matikgeschichte und Unterricht III, Hildesheim/ Berlin 2001), S. 40 54.


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2 Der indische Kreis

another] stake there; likewise [stretching it] northward: that is the north-south

line. (K aSS1.2)

The first part of the procedure is illustrated in figure 2.1, where the base of the

gnomon is at the point O in the center of a circle drawn on the ground.5

At some timein the morning the gnomon will cast a shadow OM whose tip falls on the circle at point

M, and at some time in the afternoon the gnomon will cast a shadow OA that likewise

touches the circle. The line between points A and M will run approximately east-west.

Then a cord is attached to stakes at the east and west points, and its midpoint

is pulled southward, creating an isosceles triangle whose base is the east-west line.

Another triangle is made in the same way by stretching the cord northward. The line

connecting the tips of the two triangles is a perpendicular bisector running north and

south.

Kim Plofker, Mathematics in India, Princeton and Oxford, 2009, p. 19.

Figure 1. Finding the cardinal direction (Neugebauer 1971).

The procedure to determine the cardinal directions is illustrated in Fig. 1. G is the foot

of the gnomon. The path of the end of the shadow enters and leaves a circle, center G, at

W and E. Then the line EW is in the eastwest direction. With E, W as centers, circular

arcs are drawn intersecting at N, S. Then NS, the perpendicular bisector of EW, is in

the northsouth direction and intersects the circle at N and S, the north and south points.The east and west points, E and W can be found by the same procedure since they are

on the perpendicular bisector of NS.

This method depends on the symmetry of the shadow path about the north-south line.

It does not take into account the small change in the declination of the sun during the day.

Brahmagupta prescribed a correction for this error in theBrahmasphuta Siddh ant.a. This

method of finding the cardinal directions, described in the Pancasiddh ant.ik a (written

in AD 505 by Varahamihira), is found in a much earlier treatise, the Sulbas utra,

which contains mathematical topics related to the construction of sacrificial altars. The

Pancasiddh ant.ik a also has an approximate method for finding the meridian direction

from any three positions of the shadow. This method assumes that the path of theshadow is a circle, whereas in India, it is a hyperbola.

5 Note that the text itself is purely verbal and contains no diagrams. This figure and all the remaining

figures and tables in this chapter are just modern constructs to help explain the mathematical rules.


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Der indische Kreis 3

Neugebauer, O. and D. Pingree: The Pancasiddh ant.ik a of Varahamihira. Copenha-

gen: Munksgaard, 1970.1972.

George Abraham, Gnomon in India, in: Helaine Selin (Ed.), Encyclopaedia of the History of Science,

Technology, and Medicine in Non-Western Cultures, Heidelberg: Springer, 22008, p. 1035f.

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