VCAA3: A Preliminary Critique of Pingree's Argument

However, one can indeed find other ways by which Aryabhata could have arrived at his parameters. The Brahmapak-a parameters are expressed in revolutions per kalpa of 4,320,000,000 years, whereas Aryabhata wanted parameters in revolutions per yuga cycle of 4,320,000 years (see Table A2.3). What happens if we simply divide the Brahmapak-a parameters by 1,000 and then round them off to suitable integers of the form 4n or 4n + 2? Column (4) of Table A2.1 shows the differences between the parameters computed in this way and Aryabhata's original parameters.

We can see that for Saturn, Jupiter, Mars, and Venus the differences are not much greater than those produced by Pingree's reconstruction. For these planets we come within 4 units of Aryabhata's parameters, and for Mercury, the moon, and the ascending node we come within 20, 36, and 88 units, respectively. (Pingree neglected the parameter for the sun, but we also obtain this parameter precisely.) This illustrates that Pingree's reconstruction at most accounts for the delicate fine tuning of Aryabhata's parameters; most of the significant digits in these parameters come from the Brahmapaksa parameters, which Aryabhata acknowledges as source material.

As we shall see, this fine tuning can be accounted for in ways other than the one advocated by Pingree. To do this, it is first necessary to examine Pingree's argument more closely.

As the first step in reconstructing his calculations, we consulted Ptolemy's Almagest (TM2) and wrote a computer program to calculate mean planetary positions according to Ptolemy's system. In this system, mean motions are computed by linear equations, starting with initial conditions at Ptolemy's epoch of noon on February 26, 747 B.C.-the first year of the reign of King Nabonassar of Babylon. To clarify exactly what we are computing here, we will give some definitions of mean planetary positions in Indian, Ptolemaic, and modern astronomy.

In Ptolemy's system the planets Mercury, Venus, Mars, Jupiter, and Saturn move in two cycles in a way similar to the motions of these planets in the system of the Sūrya-siddhanta (see Chapter 1). For Mars, Jupiter, and Saturn, the mean positions in Ptolemy's system are angles measured counterclockwise on the first cycle relative to the point on the ecliptic representing the vernal equinox. In the Sūrya-siddhanta, the mean positions for these planets are the same,except that the reference point is the position of the star Zeta Piscium rather than the vernal equinox. The system of Aryabhata is essentially the same as that of the Sūrya-siddhanta.

Ptolemy's system defines the mean anomalies of Mercury and Venus to be the angles measured counterclockwise on the second cycle relative to their mean positions, which are both equal to the mean position of the sun. In the Sūrya-siddhanta the sighras of Mercury and Venus are the corresponding angles, measured with respect to Zeta Piscium. For simplicity, we will redefine the Ptolemaic mean positions of Mercury and Venus to be their mean anomalies plus the position of the sun. This agrees with Pingree's implicit usage, and provides natural quantities to compare with the sighras of Mercury and Venus. We will also find it convenient to refer to these sighras as the mean positions of Mercury and Venus according to the Indian system.

In the SŸrya-siddhanta, the position of the ascending node of the moon, or Rahu, is defined relative to Zeta Piscium. Ptolemy's system does not directly define the motion of the moon's ascending node, but does define a related quantity called the mean motion of the moon in latitude. The position of the ascending node relative to the vernal equinox is 270º plus the difference between the moon's mean position and this quantity. In this way we can define the Ptolemaic mean position for the ascending node.

Using these definitions, we conclude that the mean positions of the planets in the Ptolemaic and Indian systems differ theoretically only in their choice of the reference point of zero longitude. In the two systems, this point is respectively the vernal equinox and the location of the star Zeta Piscium.

In modern astronomy, the mean longitudes of the planets are defined in a way that is comparable with the mean positions as we have defined them for the Indian and Ptolemaic systems. There, one measures the counterclockwise angle between the vernal equinox and the planet's heliocentric orbital position. The details can be found in texts on spherical astronomy such as SP. Here we would simply like to point out that the similarities between the Indian, Ptolemaic, and modern systems may arise as much from their describing the same planetary system as from cultural borrowing.

To find the Ptolemaic mean positions at a particular date, one determines the number of days between this date and Ptolemy's epoch and inserts this number into the equations for mean motion. For example, the traditional date for the beginning of Kali-yuga is February 18, 3102 B.C. Using Aryabhata's assumption that Kali-yuga began at sunrise, there are 860,172.25 days from the beginning of Kali-yuga to Ptolemy's epoch. (By convention, days begin at midnight, sunrise is .25 of a day, and noon is .5 of a day.) This figure can be used to obtain the Ptolemaic mean planetary positions at the start of Kali-yuga.

PlanetPtolemyPtolemy minus
Zeta PisciumPingree

Saturn45;5649;1948;40

Jupiter185;22188;44188;06

Mars4;247;477;08

Venus351;17354;39356;45

Mercury178;32181;55184

Sun357;160;39-

Moon279;46283;09283;30

Asc. Node-10;55-7;32-7;11

The star Zeta Piscium is important in Indian astronomy, since it is used as the starting point for measuring celestial longitudes along the ecliptic. We therefore wrote a program to calculate the position of this star by Ptolemaic methods, and we wanted to check the accuracy of this program.

This program is based on the following facts: According to Ptolemy's star table, Zeta Piscium had a longitude of 23º of Pisces on July 20, A.D. 137. According to Ptolemy's rule for the precession of the equinoxes, this longitude increases at one degree per century (of Egyptian 365-day years).

Pingree gave the Ptolemaic position of the star Zeta Piscium at the beginning of Kali-yuga. Calculation with our program confirms Pingree's statement that Zeta Piscium had a longitude of 320°37' at the start of Kali-yuga. We should note that in the Ptolemaic system such longitudes are measured from the vernal equinox at 0° of Aries. (These are called tropical longitudes.)

After we have checked our Ptolemaic calculations at the Kali-yuga starting date, the next step is to perform these calculations for noon of March 21, A.D. 499, the date of Aryabhata's alleged calculations. There are 454,759 days from Ptolemy's epoch to this date. If we compute the Ptolemaic mean positions for this date, a number of interesting points emerge. First of all, the Ptolemaic mean longitudes do not at all agree with Pingree's figures, as given in his Table 2 (PG, p. 116). This can be seen by comparing the rightmost and leftmost columns of Table A2.2.

The middle column of Table A2.2 lists the differences between our computed Ptolemaic mean longitudes and our computed Ptolemaic position of Zeta Piscium in A.D. 499. For simplicity, we will call such differences "distances from Zeta Piscium." If we compare these figures with Pingree's reported mean longitudes in the rightmost column, we see that there is rough agreement. They differ from Pingree's reported mean longitudes by 1.2º on the average (using a root-mean-square average). This rough agreement suggests that Pingree is really listing distances from Zeta Piscium, not Ptolemaic mean longitudes. But even if this is what he intends, the agreement is still rough and should be contrasted with the precise agreement that we found for Saturn, Jupiter, Mars, the sun, and the moon at the Kali-yuga starting date.

In his Table 1, Pingree lists distances from Zeta Piscium under the heading "Distance from Zeta Piscium," and mean longitudes under lambda, the Greek letter symbolizing these quantities (PG, p. 115). Yet in his Table 2, he lists quantities under lambda that are really distances from Zeta Piscium, and he refers to these quantities as mean longitudes. We have not been able to account for this discrepancy in nomenclature.

We have also not been able to account for the discrepancies between the middle and rightmost columns of Table A2 2, for it would seem that calculations for 454,759 days after Ptolemy's epoch should be even more precise than calculations for 860,172.25 days before that epoch. (We note that Pingree's Ptolemaic calculations apparently have not been corrected for the time difference between Ptolemy's city of Alexandria and Aryabhata's city of Ujjain; this possible correction does not account for the discrepancy.)

In column (3) of Table A2.1 we see the errors in reconstructing Aryabhata's parameters using actual Ptolemaic mean longitudes for the selected A.D. 499 date, and not the Ptolemaic distances from Zeta Piscium used by Pingree. Clearly these errors rule out this reconstruction. In column (2) we see the errors that arise if we reconstruct Aryabhata's parameters using our computed Ptolemaic distances from Zeta Piscium. These are the errors that Pingree's theory actually entails if we assume that he means distance from Zeta Piscium when he says mean longitude.

For Venus and Mercury the errors in Pingree's reconstruction of Aryabhata's parameters turn out to be worse than those reported by Pingree in his paper. (Compare columns 1 and 2 of Table A2.1.) This indicates errors on Pingree's part, but it might be argued that it does not detract very badly from his hypothesis. We therefore ask, Is there some reasonable way of reconstructing Aryabhata's parameters that produces smaller errors for all of the planets than Pingree's method? The answer is yes. To explain this, we must turn to a discussion of the mean positions of the planets according to modern astronomy.