VCAA5: Indian Trigonometry: A Speculative Reconstruction

In the remaining part of this appendix, we will give two more examples of the process of speculative reconstruction. These examples deal with the theoretical ideas and mathematical methods of Indian astronomy, which Western historians of science say were derived entirely from Greeks or Babylonians via Greek intermediaries.

Our first example concerns the trigonometry used in texts of Indian mathematical astronomy. Our modern trigonometry is usually traced back to the Arabs (PF, p. 260). However, in the Sūrya-siddhanta, as well as in texts by Aryabhata and other Indian astronomers, sines and cosines are used, and a table of sines is given. A modern sine is defined geometrically using a unit circle, and the corresponding Indian sine is defined in the same way, using a circle with a radius of 3,438. This means that each sine is 3,438 times as large as its modern counterpart. It also means that if angles are expressed in minutes of arc, then the sine of a small angle is nearly equal to that angle. This useful feature is achieved in modern mathematics by measuring angles in radians, a technique first invented in England in 1783 (PF, p. 270).

Another feature of the number 3,438 is that it represents a close approximation to pi. If the circumference of a circle is divided into 21,600' (i.e., 360º times 60 minutes/degree), then the circumference divided by 2pi is 3,437.746, or 3,438 to the nearest integer. Thus if one wishes to work with whole numbers, 3,438 is the best value for the radius of a circle of this circumference.

Here is what some prominent historians of science have to say about the Indian sine tables:

(1) Neugebauer: "The decisive step in proving that the Indian table of sines was derived from the Hipparchian table of chords was made by G. J. Toomer" (NG, p. 299).

(2) B. L. van der Waerden: "C. G. [sic] Toomer has shown that the chord table of Hipparchus was a table of chords in a circle of radius R=3,438. ...Toomer is justified in concluding that Aryabhata's table of sines was derived from Hipparchus' table of chords by halving the chords" (VW, p. 211).

(3) D. Pingree: "This Indian sine-table is closely related to Hipparchus' chord-table as reconstructed by Toomer, in which R also is 3,438" (PG, p. 114).

These statements certainly convey the impression that the Indian sine table was directly obtained from a related trigonometrical table used by the Greek astronomer Hipparchus. However, what do we find if we actually examine the paper by G. J. Toomer that these authorities are citing? Let us briefly consider this.

The first thing that we learn from this paper is that there are no surviving Greek documents containing Hipparchus' chord table, even in a fragmentary form. Indeed, "there is no explicit evidence about the nature of Hipparchus' chord table," and no real proof that such a table ever existed (TM1, p. 6). It is important to note that only one work of Hipparchus' has survived-a commentary on the stars-and this does not present his mathematical methods. As we have already noted, this is typical of the state of our knowledge of pre-Ptolemaic Greek astronomy.

(The chord of an angle is defined as follows: Extend the sides of the angle until they intersect a circle of unit radius centered on the angle. The chord of the angle is defined to be the length of the chord of the circle connecting the two points of intersection. The chord of an angle is therefore twice the sine of half the angle.)

Having admitted that he has no direct evidence regarding his hypothetical chord table, Toomer proceeds to construct the table from scratch. He does this using methods taken directly from works of Indian astronomy. Since in these works the sine of an angle is 3,438 times the corresponding modern sine, Toomer creates a chord table in which the chords are 3,438 times the corresponding modern chords. (These are computed using a modern sine table.) He also tabulates his chords at intervals of 7.5º or twice the interval of 3.75º typically used in Indian sine tables.

To justify his construction, Toomer uses it to show how Hipparchus might have arrived at two numbers describing the moon's orbit that are ascribed to him by Ptolemy. Since we do not actually know what computational methods Hipparchus used, Toomer takes it for granted that he used certain methods of Ptolemy. Using these methods, plus his hypothetical chord table, Toomer computes one of Hipparchus' numbers, but gets it wrong. He then argues that Hipparchus must have made a particular mistake in the complex procedure. When he computes the number again on this basis, it still comes out wrong (3,082[2/3] over 246[1/3], rather than 3,122[1/2] over 247[2/3]). But Toomer concludes that it is close enough to "prove" that Hipparchus did use a chord table of the proposed type, and that he made the proposed mistake (TM1, p.12). The second number also comes out wrong (3,134 over 338 rather than 3,144 over 327[2/3]), but Toomer again regards it as close enough.

By this reasoning Toomer maintains that "the nature of Hipparchus' chord table is conclusively established" (TM1, p.16). Since the table has the structure of an Indian sine table, it follows that Indian trigonometry must have been derived from the Greeks. The idea that Greeks may have been influenced by Indian developments is never even suggested by modern Western historians of science. But in this case, of course, we have no evidence for influence either way, since the connection between Hipparchus' two numbers and the Indian sine table is purely speculative.

Besides his interpretation of two numbers in the Almagest, Toomer offers only one other piece of evidence suggesting that the Greeks used a chord table with a radius of 3,438. This is a statement in Ptolemy's Geography mentioning for two cases the ratio between the length of a parallel of latitude and the length of the equator. For Rhodes, at 36º north, this ratio is 93/115, and for Thule, at 63º, it is 52/115. Toomer claims that these figures must have been derived from Hipparchus' hypothetical chord table, since in that table the diameter, expressed in degrees, rounds to 115. Also the chords corresponding to the two latitudes turn out to be 93 and 52 when read from that table by linear interpolation and converted from minutes to degrees. According to Toomer, "The conclusion seems inevitable that he [Ptolemy] is here using, directly or indirectly, the old chord table of Hipparchus" (TM1, p. 25).

Yet there are many ways in which Ptolemy might have arrived at these numbers. For example, he might have reasoned that it would be useful to use a degree of latitude as a unit of distance in geographical studies. (In fact, a degree of latitude was sometimes assumed by the ancient Greeks to have a length of 700 stades, or about 80 miles (NT, p.45).) In this case the diameter of the earth would be the circumference of 360 units, divided by pi. Using Archimedes' rough estimate Of 22/7 for pi, this diameter is 115 to the nearest unit. Using a compass, a ruler, and a protractor, it is easy to construct a circle of this diameter, marked with the parallels of latitude at 36º and 63º. Their lengths turn out to be 93 and 52 in round numbers.

In fact, we performed this construction in about 10 minutes; it is much easier to obtain the ratios in this way than by using linear interpolation in a table of chords. Thus there is no need to suppose that a chord table, with or without a radius of 3,438, was ever involved.