VCAA4: The Theory of Observation

The Theory of Observation

We used standard computer programs published by Duffett-Smith (DF) to calculate the mean longitudes of the planets and the moon's ascending node. We can also calculate the longitude of Zeta Piscium by looking up its position in the Astronomical Almanac and modifying this for a given date in accordance with the modern rate of 50.29 seconds per year for the precession of the equinoxes.

Let us assume, for the sake of argument, that the mean longitudes computed according to modern astronomy are correct. Then the error in Aryabhata's mean position for Jupiter on a given date must be equal to Aryabhata's mean position minus the position of Jupiter relative to Zeta Piscium by modern calculation. In Figure A2. 1, these errors are plotted for the eight planets for dates ranging from 10 B.C. to A.D. 1007. The vertical axis is located at noon of March 21, A.D. 499.

We can see that for the seven planets Saturn, Jupiter, Mars, Venus, the sun, the moon, and the ascending node, the errors converge sharply to a value of about 1.5º at a date near A.D. 499. (Actually, the point of closest convergence is at roughly A.D. 540.) The planet Mercury, however, is an exception to this pattern.

In Figure A2.2, similar error graphs are plotted. For these graphs we plot Aryabhata's mean positions minus the corresponding differences between Ptolemy's mean positions and Ptolemy's longitude for Zeta Piscium. Here we also see a convergence at about A.D. 499. However, this convergence is much less sharply focused than the convergence in Figure A2. 1. It is good for Saturn, Mars, the sun, and the ascending node, but it is poor for the other planets in comparison with Figure A2. 1.

Figure A2.1 Comparison between Aryabhata's system and modern astronomy. The horizontal axis represents time in unirs of 40 years. The origin corresponds to noon of Mar. 21, A.D. 499. The vertical axis represents the difference in degrees between modern mean planetary positions relative to Zeta Piscium and Aryabhata's mean planetary positions. These differences are plotted for the seven planets and Rahu (the ascending node of the moon). Note that for all planets except Mercury, the differences between Aryabhata's calculations and modern calculations converge sharply at about A.D. 539.

What is the explanation of these patterns? Pingree's argument is that the convergence in Figure A2.2 is due to the fact that Aryabhata calculated his parameters so that his mean motions would agree with a Greek astronomical table at this date. But if this is so, we must ask, Why is the convergence in Figure A2.1, representing Aryabhata's deviations from reality, so much sharper than the convergence in Figure A2.2, which represents his deviations from Ptolemy?

We propose the following simple answer to this question: The convergence in Figure A2.1 is due to the fact that Aryabhata observed the planetary mean positions in the period between A.D. 499 and 540. The lesser convergence of plots in Figure A2.2 at this time is due to the partial agreement that exists between the Ptolemaic system and modern calculations. The convergence in A2.2 is not as sharp as that in A2.1 because there are errors in Ptolemaic mean positions relative to those computed by modern methods.

Figure A2.2 A comparison between Aryabhata's system and Ptolemaic astronomy. The horizontal axis represents time in units of 40 years. The origin corresponds to noon on Mar. 21, A.D. 499. The vertical axis represents the difference in degrees between Ptolemy's mean planetary positions relative to Zeta Piscium and Aryabhata's mean planetary positions. These differences are plotted for the seven planets and Rahu (the ascending node of the moon). In this case there is a sharp convergence only for the sun, Mars, Saturn, and the ascending node. This figure should be compared with Figure A2. 1.

This interpretation is borne out by a comparison of Ptolemaic and modern calculations. Figure A2.3 shows plots of the difference between Ptolemaic and modern calculations of mean positions relative to Zeta Piscium. We can see that Ptolemy's errors for Saturn, Mars, the sun, and the ascending node are consistently small; the error for Jupiter is somewhat larger; and the errors for the other planets are much larger. In fact, the convergence in Figure A2.2 was strikingly good precisely for Saturn, Mars, the sun, and the ascending node.

This confirms our interpretation that the partial convergence in A2.2 is simply a by-product of the greater convergence caused by Aryabhata's observations-that we see in A2.1. By Pingree's hypothesis, the scatter seen for Jupiter, Venus, and the moon in A2.2 just happens to be such that these planets converge along with the others in A2.1. This, however, seems unlikely.

Figure A2.3 A comparison between Ptolemy's system and modern astronomy. The horizontal axis represents time in units of 40 years. The origin corresponds to Jan. I, A.D. 161, a date in Ptolemy's lifetime. The vertical axis represents the difference in degrees between modern mean planetary positions and Ptolemy's mean planetary positions. Both the modern and the Ptolemaic mean longitudes are relative to Zeta Piscium (using modern and Ptolemaic calculations for Zeta Piscium, respectively). The differences are plotted for the seven planets and the ascending node of the moon. Ptolemy does fairly well for the sun, Mars, Saturn, Jupiter, and the ascending node, although he does make a systematic error for these planets. A much greater difference arises between Ptolemaic and modern calculations if they are both made relative to the vernal equinox. This suggests that Ptolemy's observations were initially made relative to a fixed star, and then converted to the tropical Zodiac.

At this point the argument may be raised that the convergence of error graphs in Figure A2.1 does not take place at the origin, but is about 1.5º above it. One might ask whether this can be readily explained on the hypothesis that this convergence is due to Aryabhata's observations. One answer, of course, is that Aryabhata may have made an error in observation that had an equal effect on all the planets. But we can go further and suggest the particular error that he may have made.

To do this we must consider the sun, which Pingree did not mention in his reconstruction of Aryabhata's parameters. According to Aryabhata's system, the sun is required to have a longitude of zero after 3,600 years of Kali-yuga have elapsed. (This is due to the fact that 4,320,000 is evenly divisible by 3,600.) If Aryabhata found that the sun had a non-zero longitude, it would be natural for him to take this as an error and revise all his longitudes so that the longitude of the sun would come out to zero. Or, knowing that the sun should have a longitude of zero, he might have simply measured the longitudes of the other planets relative to the sun. This would automatically cause the errors in his observed longitudes to be roughly equal to the actual mean longitude of the sun at the time of his observations.

Let us suppose that Aryabhata did this, and that he then computed his parameters using his observed longitudes rather than longitudes copied from a Greek table. This leads to a reconstruction of his parameters based on modern calculation of the differences between mean longitudes and the sun's mean longitude. The longitudes and resulting parameters for this reconstruction are listed in the last two columns of Table A2.3, and the errors in this reconstruction are listed in column (5) of Table A2. 1. As we can see, these errors are zero, except for Mercury, where the error is equal to that in Pingree's reported reconstruction (see columns (1) and (2)). Thus, the hypothesis of observation yields better results than the hypothesis of copying from Greek tables.

A few final points will help to round out our discussion of Pingree's theory. The first is that in Figure A2.3, we can see that the Ptolemaic error graphs for several planets converge at about A.D. 161. This makes sense, since Ptolemy is thought to have written his Almagest at about this date. However, the convergence point is about 1.25º above the time axis. It would appear that Ptolemy too may have made some systematic observational errors.

Indeed, to properly evaluate Ptolemy's errors, we should plot the differences between Ptolemaic longitudes and modern longitudes (without making these relative to a fixed star, such as Zeta Piscium). This is because both Ptolemaic and modern longitudes are relative to the vernal equinox. If this is done, all the error curves in Figure A2.3 acquire a decided positive slope, indicating a systematic error affecting all the planets equally. (Possibly, Ptolemy's calculations were first worked out relative to a star, and then made relative to the vernal equinox using an erroneous value for the precession of the equinoxes.)

The second point is that there is no actual evidence showing that Greek astronomical tables were being transmitted to India around A.D. 500. Indeed, Neugebauer's discussion of the post-Ptolemaic period suggests that the quality of Western astronomy declined sharply after the time of Ptolemy. Thus he remarks that the astronomical material "extant from the later time of Roman Egypt is rather sad" (NG, p. 5). Of the second century work of Vettius Valens, he says, "The intervening less than 150 years succeeded not only in introducing several numerical errors into the basic parameters but also in obscuring almost completely the meaning of the prescribed operations" (NG, p. 793).

Persia is the natural link between India and the West, but of this country Neugebauer says:

We know of Pahlavi translations of such first and second century astrological writings as Teucer and Vettius Valens and the presence of "Indian books" as well as of the "Roman megesti" around A.D. 250 under Shapur I. Under Khosro I Ī was revised, around A.D. 550, the famous Zij ash-Shah, which has been shown to be greatly dependent on Hindu sources (NG, p. 8).

Here "Roman megesti" may refer to Ptolemy, but the phrase "Indian books" suggests that Indian astronomy existed at A.D. 250 and was being exported.

Our final point is that, given the highly fragmentary nature of the surviving historical evidence, the process of speculative reconstruction is likely to create nothing more than illusions reflecting the opinions of the reconstructors. We therefore do not insist that our reconstruction of Aryabhata's parameters is correct. We merely offer it as an alternative that is in better agreement with the available facts than Pingree's reconstruction.

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