VCAA6: Another Speculative Reconstruction

Since the chord table of Hipparchus has not survived (if it ever existed), it is remarkable that such slender evidence can be offered as the basis for "inevitable" conclusions about it. Yet, as we have seen, such speculative reconstructions are not unusual in the field of the history of science. Here we will give one more example. This is provided by the mathematician B. L. van der Waerden, who traces back Hipparchus' trigonometry to the Greek mathematician Apollonius of Perge (VW, pp. 211-12). Van der Waerden's reasoning goes as follows:

(1) The Indian sine tablets accompanied by a complete theory of trigonometry, as shown by the writings of Aryabhata. This too must have come from the Greeks, but Hipparchus, in van der Waerden's estimation, was not a good enough mathematician to have invented it.

(2) This mathematician could not have been Archimedes, since he used 22/7 for pi. Therefore it must have been an able Greek mathematician living between the times of Archimedes and Hipparchus.

(3) There was exactly one excellent mathematician living in this period, namely Apollonius of Perge. Now, Eutokios, in a commentary on Archimedes, says that Archimedes' estimate of pi was intended for "the needs of daily life," and that Apollonius had given more accurate estimates.

(4) On this basis, "we are bound to conclude" that the value of pi used in Indian trigonometry is due to Apollonius (VW, p. 212).

(5) In fact, the Indian astronomer Bhaskaracarya gives 3927/1250 (3.1416) as a good estimate of pi, and also gives [22/7] as an estimate "adopted to practice." Since this statement is very similar to Eutokios' statement about Archimedes and Apollonius, "we are bound to conclude that they go back to a common source, and hence that the estimate of pi is due to Apollonius" (VW, p. 212).

One should note here that van der Waerden does not cite a reference giving Apollonius' estimate for pi, and he also gives no reference that specifically attributes studies of trigonometry to Apollonius. Thus we do not know what Apollonius' estimate of pi was, nor do we know whether he actually knew any trigonometry. Nor do we know whether Apollonius was the only able mathematician living between Hipparchus and Archimedes. And even if he was, we do not know who invented the basic theory of trigonometry, when this was done, or in what country that person lived. Van der Waerden's argument is simply a chain of suppositions.

We have discussed the arguments of Pingree, Toomer, and van der Waerden in detail to show the kind of foundations that underlie scholarly conclusions about the origins of Indian astronomy. The main characteristic of these foundations is that they are composed almost entirely of unsupported assumptions, biased interpretations, and imaginary reconstructions. It is unfortunate, however, that after many scholars have presented arguments of this type in learned treatises, the arguments accumulate to produce an imposing stratified deposit of apparently indisputable authority. In this way, supposedly solid facts are established by the fossilization of fanciful speculations whose original direction was determined by scholarly prejudice. Ultimately, these facts are presented in elementary texts and popular books, and accepted on faith by innocent people.

The arguments of Toomer and van der Waerden are clearly very weak. But the objection might be raised that the division of the circle into 21,600' in Indian trigonometry is itself evidence of Greek influence. In answer to this, we should first point out that according to modern scholars, the division of the circle into degrees, minutes, and seconds was borrowed by the Greeks from the Babylonians. We therefore ask, Did the Babylonians invent this division, or might they have borrowed it from some other source?

In fact, there is evidence that the division of a circle into 360º is very old, and is related to the number of days in a year. In the Srimad-Bhagavatam the number of days in a year is given repeatedly as 360 (see SB 3.11.10-12, for example). The same number is given in the Rg Veda, which is accepted even by Western scholars as dating back to 1000-1200 B.C. (HA, p. 8). For example, in the Rk-samhita, it is stated,

This verse speaks of a year as having 360 days, and it can be compared with a similar statement in SB 5.21.13, in which the year is also described as a wheel. There are many statements in the Vedic literature comparing the year to a wheel or circle.

The 360-day year was kept in alignment with the seasons by periodically inserting an intercalary month. This is described in Srila Prabhupada's purport to SB 5.22.7.

The time accepted by scholars for the Rg Veda antedates the known period of Babylonian astronomy. According to Neugebauer, Babylonian astronomy dates back no further than about 600 B.C.:

We would suggest that the division of the circle into 360º was an ancient feature of Vedic civilization. In Egypt and Mesopotamia it may also date back to times when the civilizations of the Near East were part of a larger Vedic world system. As far as we are aware, this is neither demonstrated nor contradicted by known historical evidence. As the above statement by Neugebauer indicates, we have practically no historical evidence regarding the early history of astronomy in the Near East.

If the division of the circle into degrees corresponds to the 360-day year, then its division into 12 signs of the zodiac, each with 30º, may correspond to the 12 Vedic months of 30 days. Likewise the Greek bathmoi, or 15-degree intervals, may correspond to the 15-day bright and dark fortnights of the moon. Going further, we note that among the many Indian time divisions there is the ghatika, which is one sixtieth of a day. Also, the pala is one sixtieth of a ghatika, and the vipala is one sixtieth of a pala. Next comes the prativipala, which is one sixtieth of a vipala (BJS, part 2, p. 13). Do these correspond to the divisions of a degree into minutes, seconds, and so on? We can only speculate about the ultimate origins of such divisions.

As a final point, we should note that the assumption of the Western historians of science seems to be that no one in India could have exhibited mathematical or scientific inventiveness, and thus all Indian mathematical astronomy must have been due to Western creativity. However, the available historical evidence seems to contradict this. For example, the 14th-century Indian mathematician Madhava gave the following approximation for pi: