Episode 36: Aryabhata & the Siddhantas

November 30, 2023

In our final episode on ancient Indian astronomy, we tour the five astronomical Siddhantas, and then meet some of the astronomers whose names and works survive to us, most importantly, the great Aryabhata.


Good evening, and welcome to the Song of Urania, a podcast about the history of astronomy from antiquity to the present with new episodes every full moon. My name is Joe Antognini.

Well last month we dived into the astronomy of the late Vedic period in India, which culminated in a text called the Vedanga Jyotisha. The text itself was written probably in the middle of the first millennium BC, but many of the astronomical techniques it presented were probably centuries older than that. Regardless, the astronomy of the Jyotisha held sway as the state of the art of Indian astronomy for centuries after it was formalized, well into the first millennium AD.

Over all these years, however, certain flaws in the Jyotisha came to be harder to ignore. Of course, the big one is setting the length of the year to 366 days, but other periods of time were also slightly inaccurate, and over the years these inaccuracies grew and grew and became more apparent. Furthermore, the Jyotisha was also fairly limited in its scope. It was explicitly written as an aid in calculating the appropriate times to perform the requisite rituals in the Vedic calendar, and so it just didn’t have anything to say about other aspects of astronomy like the motion of the planets or how to predict when eclipses would occur. It’s also in these first few centuries of the first millennium AD that contact with the west began to expand, and knowledge of new techniques in the astronomy of Greece and Babylonia starts to reach India.

One of the earliest indications of this change is from the astronomer Vrddha Garga, who was probably active early in the first century AD around the turn of the millennium. Incidentally, this Vrddha Garga is not to be confused with a sage who composed one of the hymns of the Rigveda and is simply known as Garga. Vrddha Garga is mostly remembered for a text that is attributed to him, which is usually called the Gargiya-Jyotisha. As with the other texts from this era and earlier, we still can’t definitively attribute this work to a single individual, and it was more likely to have been composed from the writings of a number of astronomers over several generations and then ascribed to one individual. But the book contains several references to the Greeks. Now, during the explosive expansion of the Macedonian empire under Alexander the Great in the late 4th century BC, Alexander famously claimed to have conquered the known world out to India. Now, this was perhaps a bit of propaganda. Alexander made it some of the way into the Indus river valley, and it should be noted that he made it further than Persia had ever managed. But it was nevertheless essentially a toehold on the subcontinent and didn’t even reach to the modern-day borders of India. But, Alexander’s conquests began to strengthen trade networks between India and the West, and ultimately facilitated the exchanged of information between east and west. Well, based on some of the references to the Greeks in the Gargiya-Jyotisha, it seems that the invasion of the Greeks was a fresh experience for at least one of the astronomers who contributed to it. But Vrddha Garga himself seems to have been quite admiring of the Greeks, at least scientifically. He wrote:

The Greeks are indeed barbarians, but amongst them this science of astronomy is well established; therefore even they are honored as rishis [sages].

A more comprehensive piece of evidence for the influence of Greek astronomy on Indian astronomy came in the next few centuries with a work called the Yavanajakata. This was a work that was directly based on a Greek text that several Indian astronomers translated into Sanskrit and converted into verse. The original work was probably written in Alexandria in the early 2nd century AD and was first translated and versified just a few decades later. The text is almost entirely astrological and taught the reader how to read horoscopes since by this point in time, that is what Hellenistic and Roman society was almost exclusively interested in in their astronomy. The name itself, Yavanajakata, comes from the Yavana kingdom, which is also known as the Indo-Greek kingdom. This was a kingdom of mostly Greek people in what would today be modern-day Pakistan. You may recall from the early episodes on Greece that Greece underwent a period of expansion, where groups of Greeks would travel to distant lands and establish colonies there, over time spreading from the Iberian peninsula in the west all the way out to the Black Sea. Well, some of these settlers traveled east to Persia, but the Persian state was generally not happy to see the growth of foreign settlements in their empire. So periodically the Persians would forcibly relocate the Greeks to the eastern side of their empire. Eventually, Alexander’s conquests extended Greek rule out to these peripheries and reconnected these Greek settlements to the broader Hellenistic world. After Alexander’s death, his short lived empire fragmented, and eventually the Greco-Bactrian kingdom developed out of the eastern portion, and later on the eastern portion of that developed into the Indo-Greek kingdom, which lasted from about 200 BC until the turn of the millennium. But the older name for this kingdom, the Yavana kingdom, derives from the word “Ionian.”

At any rate, despite the fact that modern day Greece and India are really quite far apart, in Hellenistic times, Greek culture was right at India’s doorstep, so the major influence of Greek astronomy on Indian astronomy is perhaps more plausible than it might at first glance seem. Well, as new developments in astronomy from Greece made their way into India, Indian astronomers could no longer ignore the shortcomings of the Vedanga Jyotisha. A few centuries into the first millennium it was clear that a new approach to astronomy was needed. So by around the 4th and 5th centuries AD we start to see a new series of works produced that codified these new developments. These works are called “siddhantas.” The term “siddhanta” translates from Sanskrit as something like “settled doctrine,” or “final conclusion,” so it was meant to be a kind of summing up of the centuries of studies on a particular topic. The siddhantas as a genre are not exclusively astronomical, in fact, only a few of them deal with astronomy. Traditionally there were 18 of these Siddhantas, but as with the texts of ancient Greece, most of these were lost, and also similar to ancient Greece, in many cases later authors impersonated the famous sages of yore who wrote the original texts in order to steal a bit of prestige for their own works. Most of the siddhantas deal with topics in theology and philosophy in Hinduism and Buddhism, but five of them deal with astronomy.

One of the general features of the astronomical siddhantas is that the astronomy has by this point clearly become more sophisticated, and more than that, it has become interested in a much wider range of phenomena than the Jyotisha. Which, to be fair, was not terribly difficult to do since the Jyotisha was pretty short and couldn’t cover too much material.

The first of these five Siddhantas is called the Paitamaha Siddhanta and isn’t really of too much interest except to show that despite advances in other astronomical schools, there was still a contingent of astronomers who held to the ways outlined in the Vedanga Jyotisha. The astronomy of the Paitamaha-siddhanta is more or less identical to the astronomy of the Jyotisha and has the same focus on the calendar and bases its calendrical system on the same yuga of 5 years.

A kind of intermediate work is the Vasishtha-siddhanta, which was named after the ancient sage Vasishtha, one of the authors of the Rigveda. The Vasishtha-siddhanta has a number of real developments over the Jyotisha, but compared to the next three siddhantas it’s the most inaccurate. The first really new feature we see in it reflects the influence of Babylonia and Greece on Indian astronomy by this point. The Vasishtha-siddhanta introduces the 12 signs of the zodiac as a way to measure ecliptic longitude, and divides each of these signs into degrees and minutes just as the Babylonians and Greeks did. Now, the Vasishtha-siddhanta really only does this as a supplement to the more traditional Indian system of 27 or 28 nakshatras that I described in the last episode, but we see that by this point Indian astronomers were no longer developing their ideas in isolation from the western world.

In addition to this obviously western influence, the Vasishtha-siddhanta expands the scope of the astronomy it’s interested in relative to the Jyotisha. The Vasishtha-siddhanta describes the concept of the anomalistic month, which is the time it takes the moon to go from perigee to perigee. The moon’s orbit is slightly elliptical, so when it is closest to earth at perigee, it moves faster than average, and when it’s farthest from the earth at apogee, it moves more slowly than average. This variation in its speed over the course of the orbit is called the anomaly, and so the time it takes the moon to go from its maximum apparent speed to maximum apparent speed on the sky is called the anomalistic month. The Vasishtha-siddhanta actually gives two separate lengths for the length of the anomalistic month. The first is somewhat simpler, 248 / 9 days, or 27.555 days, which is off by about 1 and a half minutes. But in another place it gives another, more accurate value of 3031 / 110 days, which works out to be 27 days 13 hours 18 minutes and 32.7 seconds and is off by just half a second.

The last development of the Vasishtha Siddhanta that I’ll mention is its description of the planets. The treatment isn’t terribly sophisticated, but it’s nevertheless a fairly monumental development just by sheer virtue of being there at all. The text names the five planets and describes their motions both qualitatively and quantitatively. It says that the planets in general have a prograde motion from west to east, but that they undergo periods of retrograde motion as well, with stationary points in between. And then it gives the values for the synodic periods of the five planets, along with the ratio between the sidereal period and the synodic period.

Well, the next two siddhantas seem to have been of a more explicitly western origin. The first of these is called the Paulisa Siddhanta, and it’s believed that the name, Paulisa, refers to a Greek astronomer whose name was some variant of “Paul.” During the Islamic Era, a Muslim astronomer named Al-Biruni was more specific and wrote that this author was a Greek from some city called “Saintra,” that Al-Biruni was not himself familiar with. But he speculated that the this “Saintra” have have been a Sanskrit version of the city Alexandria.

Now, one thing I should mention at this point is that by the time that the siddhantas had been written in the middle of the first millennium, Indian astronomy had evolved to become heavily mathematical. In fact, as in ancient Greece, there wasn’t a strong distinction between mathematics and astronomy. Mathematics was simply a part of doing astronomy, and many of the great Indian astronomers were also the great Indian mathematicians. If you were to listen to a podcast on the history of mathematics, the episode on Indian mathematics would turn up mostly the same list of names and texts.

So with the Paulisa Siddhanta many of the new developments are from a conceptual standpoint, rather unremarkable. But from a technical standpoint, they represented a huge advance in the mathematical techniques of the astronomers of this period. The Paulisa Siddhanta gives you, for instance, methods to calculate the position of the Sun and Moon. First by calculating the position of the mean Sun and mean Moon, and then applying the corrections you need to make to account for the fact that the orbits are elliptical, so the Sun and Moon do not move uniformly on the sky — sometimes they move faster, and sometimes they move slower. This technique for correcting for the anomaly was called the “kendra,” and unfortunately the text just gives the steps that you need to do to perform the calculation but it doesn’t actually show how this method was derived. In another place, the Paulisa Siddhanta has an equation which is today called the “sunrise equation,” because it can be used to figure out when sunrise and sunset will be, and from that the length of day and night. Rather than the simple linear zig-zag function that the Vedanga Jyotisha used, along with the ancient Babylonians and Egyptians, where each day gets longer by a constant amount until the summer solstice, and then gets shorter by a constant amount until the winter solstice, the method in the Paulisa Siddhanta is actually trigonometric, and accounts for the spherical geometry of the Earth. To make these kinds of trigonometric calculations possible, the text also includes sine tables, which list out the values of the various trigonometric functions at different angles.

The other Siddhanta with a clear western influence was called the Romaka Siddhanta, the name “Romaka” being a Sanskritization of the word “Roman.” Traditionally it was attributed to an author named “Srisena,” but later scholarship has found that he compiled the work from earlier sources, and apparently did so rather poorly. Even not too long after the work was written, there were those who doubted Srisena’s competence. The astronomer Brahmagupta, about whom I’ll have more to say later, wrote that Srisena had “taken a heap of jewels and turned it into a patched rag.”

At any rate, the astronomy of the Romaka Siddhanta is quite similar to the astronomy of Ptolemy, and many of the measurements it presents are very close to the Ptolemaic values. Now this could, on the one hand, simply be a coincidence because the Ptolemaic values were fairly accurate, but they seem to be too similar to just be a coincidence. That said, the numbers aren’t identical either, which suggests that the Indian astronomers had taken Ptolemy’s techniques and used them to make their own measurements rather than copying his numbers wholesale. And, it should be said, as a reminder here of episodes 21 and 23, that when I say Ptolemy’s numbers and Ptolemy’s techniques, it’s really Hipparchus’s numbers and Hipparchus’s techniques and Ptolemy was mostly recording Hipparchus’s work for posterity. So, for example, in the Romaka Siddhanta, the length of the tropical year is given to be 365 days, 5 hours 55 minutes and 12 seconds, whereas Hipparchus had gotten a value of 365 days, 5 hours, 55 minutes and 15.8 seconds. So, these two measurements were within 4 seconds of each other, but they were both about 6 minutes longer than the true value. So, clearly the author of the Romaka Siddhanta wasn’t just cribbing off of Ptolemy, but it’s also the case that these two values are too close to just be a coincidence.

One other place where the author of the Romaka Siddhanta clearly adopted Greek techniques to Indian astronomy was in the Metonic cycle. You may recall from Episode 17 that Meton of Athens had discovered that phases of the moon along with solar and lunar eclipses repeat in a 19 year cycle today called the Metonic cycle. Once you have established this cycle, predicting eclipses becomes much easier because you just have to look at what happened 19 years ago to predict what is going to happen today. The trouble is that the Metonic cycle is not an even number of days, or, in particular, since Indian astronomy recognized several different kinds of days, it wasn’t an even number of savana days, or civil days, from sunrise to sunrise. But by multiplying the cycle by 150 they got a cycle which that was quite close to being an even number of savana days.

The last of the five astronomical siddhantas is the Surya Siddhanta, and was the most sophisticated of the bunch. As was generally the case with classical Indian literature, the Surya Siddhanta was written in verse. You might recall from Episode 27 on the Astronomica by Marcus Manilius that that work was also written in verse, and we discussed back then how writing an astronomical work as poetry poses a certain set of challenges, namely that there are just a lot of numbers that you need to convey, and it’s just difficult and frankly boring to do that in verse. Well this was also a problem for the Surya Siddhanta and the way that its authors dealt with this problem was by using what is called the Bhutasamkhya system. Rather than representing numbers with distinct symbols as we do today, numbers were represented with words. So, for instance, the number 2 was represented with the word for “eye,” since people have two eyes. The number 5 was represented with the word for “arrow,” since the god Kamadeva was traditionally depicted as holding 5 arrows made of flowers. But this system was not unique. To make it easier to conform to the poetic structure multiple words could represent the same number. So, for instance, the number 1 could be represented with the word for “earth,” since there was only one earth, but it could equally well be represented by the word for “moon,” since there is also only one moon. Some larger numbers had their own representations. The number 32 could be represented by the word for “tooth,” since a person has 32 teeth, and the number 33 could be represented by the word for “the gods,” since at least in some traditions there were 33 gods in the Hindu pantheon. But for larger numbers the strategy was to string the numbers together in a base-10 numbering system. One of the most interesting features of this number system is that in order to do this effectively, you need the concept of the number “zero,” since otherwise you can’t distinguish between the numbers 11 and 101. The Bhutasamkhya system used the word “sky” for this purpose since the sky is empty. This seems to have been the first step toward the development of a unique symbol to represent the number zero which first developed in India in the subsequent few centuries.

Well, being the most sophisticated of the astronomical siddhantas, the Surya Siddhanta is highly mathematical, and its innovations are rather technical, so its contents are somewhat difficult to describe in a podcast. But I will at least try to give a sampling of some of the topics it covers. At its foundation is a quite complete description of spherical astronomy. Spherical astronomy is a way of measuring the positions of different celestial bodies by imagining that we are surrounded by a great sphere and the different celestial bodies are points on this sphere. When we do this, there are a couple of important landmarks. There is the zenith, which is the point right above our heads, and the nadir, which is the point directly below our feet. The Surya Siddhanta calls these the “urdhva swastika,” and the “adha swastika,” respectively, “urdhva” meaning “upward,” and “adha” meaning “downward.” The word swastika means exactly what you think. Despite its later association in the West with Nazism, originated in India and the symbol had a variety of meanings, often as a good luck symbol since the name literally translates into something like “well-being.” But the symbol also seems to have long had an association with the universe as a whole. A possible source of this association is from the north celestial pole. If you imagine drawing the Little Dipper four times over the course of a day, six hours apart, each time it will be 90 degrees rotated and at the end of the day you will have drawn something that looks a little bit like a swastika. Incidentally, the Surya Siddhanta has another similarity to the Astronomica of Marcus Manilius in that it posits that just as there is a pole star at the north celestial pole by symmetry there is also a pole star at the south celestial pole as well.

At any rate, halfway between the zenith and the nadir is a great circle which represents the horizon, and we can’t see anything that is below the horizon. The Surya Siddhanta then marks the cardinal directions of north, south, east, and west and gives a method for finding them. Plant a stick in the ground and draw a circle around it. Then watch the shadow of the stick over the course of a day. Sometime in the morning and the afternoon the end of the stick’s shadow will exactly touch the circle. Mark those two points. Then you draw a line between them and that tells you the direction of east and west. Then it defines the meridian, called the yamyottara-vrtta, which is a great circle in the sky that goes from north, through the zenith above you, to south. The ecliptic, or the path of the Sun on the sky over the course of the year, is also defined, and its inclination is given to be 24°, only about half a degree off from its true value. The Surya Siddhanta also uses a number of concepts that aren’t often used in modern day astronomy. There’s a circle called the “prime vertical,” which is like the meridian, except that it goes east-west instead of north-south. And there’s also the “6 o’clock hour circle,” which also starts out at east and west, but goes through the north celestial pole rather than the zenith.

The Surya Siddhanta applies spherical geometry to these various concepts to derive a number of useful relations. For instance, if you measure the angle of the shadow of a gnomon on an equinox, it tells you how to calculate your latitude. That is a fairly straightforward calculation to make, but it has more complicated relationships, too, like the declination of the Sun as a function of its ecliptic longitude. This, then, can be used to calculate when the Sun rises and sets over the course of a year. The last thing I’ll mention about the Surya Siddhanta is its description of the planets. By the time we get to the Surya Siddhanta, Indian planetary astronomy has really come a long way. In the last episode on the Vedanga Jyotisha, my discussion of the planets was mostly about how they didn’t have much to say about the planets. There are a few plausible references in the Vedic literature, but they’re fairly oblique and don’t really have much to say except that the planets exist. By contrast the discussion of the planets in the Surya Siddhanta is quite comprehensive.

To start off with, the Surya Siddhanta actually identifies the five planets, which is more than can be said of the astronomical literature of Vedic period. And more than that, it orders them based on the speed of sidereal motion, with the Moon being closest to Earth, and then Mercury, Venus, the Sun, Mars, Jupiter, and Saturn being most distant. It also identifies the retrograde motion of the planets. Compared to the Greeks and Babylonians, the Surya Siddhanta uses a finer grained classification system. You may recall that the Babylonians had five points they were interested in, called stations: there was first appearance, when the planet emerged from behind the sun, the first stationary point, when the prograde motion stopped, opposition, and then the second stationary point, when the retrograde motion stopped, and finally the last appearance, after which the planet went back behind the sun. Well the Surya Siddhanta divides the orbit into eight kinds of motion. Three of these are periods of retrograde motion: there is vakra, or retrograde motion; anuvakra, or somewhat retrograde motion; and kutila, which is transverse, or stationary. Then there are five categories of prograde motion: manda, or slow; mandatara, very slow; sama, even; sighra, swift; and finally sighratara, or very swift.

The Surya Siddhanta’s model of planetary motion was more than qualitative, though. It was of a similar level of sophistication as Ptolemy. Now, the challenge with modeling the observed movements of the planets is that you have several complications you have to account for. First off, since the Earth is revolving around the Sun, we are observing the planets from a moving platform. Then secondly, the observed motions of the planets have to be corrected for the fact that both the planet’s orbit and the Earth’s orbit are not perfectly circular. As we learned from Episode 20, there are two ways you can model this, which turn out to be mathematically equivalent. One is with eccentric circles. You assume that the planet is moving in a circle, but the center of this circle is somewhat offset from the Earth. This center point then revolves around the Earth as well. The other way you can do it is with epicycles. With epicycles, we take the planet’s orbit and then we add on a second, smaller circle and put the planet on that. Well, the Surya Siddhanta uses both techniques to model the planetary motion. The motion of the Earth’s orbit is captured by the epicycles, but the planets are also placed on eccentric circles to account for the fact that their orbits were also eccentric.

In addition to detailing this highly sophisticated planetary model, the Surya Siddhanta provides the various parameters that go into it, based on, most importantly, the synodic periods of the planets. It then combines these various periods to find a universal cycle that all the planets participate in. This grand yuga is about 180,000 years long, 65,746,575 days, to be precise. At the beginning of this period, all the planets are in conjunction and they do not return to conjunction all together until this 180,000 year cycle is completed. Finally, the Surya Siddhanta combines this grand yuga into ever longer periods called the mahayuga. The mahayuga gets divided into four ages, much like the five ages of man that Hesiod described. The first of the mahayuga’s ages is called “Krta,” or the golden age, and lasts 1,728,000 years. During this time, mankind was governed by the gods and all that man produced was ideal and pure. During the age of Krta the Vana Parva says,

Men neither bought nor sold; there were no poor and no rich; there was no need to labour, because all that men required was obtained by the power of will; the chief virtue was the abandonment of all worldly desires. The Krita Yuga was without disease; there was no lessening with the years; there was no hatred, or vanity, or evil thought whatsoever; no sorrow, no fear. All mankind could attain to supreme blessedness.

Krta is followed by “Treta,” the Silver Age, which lasts 1,296,000 years. One of the symbols that appears in depictions of these ages is of a bull who represents Dharma, which can very roughly be translated as “virtue,” or “morality,” or “religious duty”. During the age of Krta, this bull stands sturdily on the four legs of religion, symbolizing the four vedas, but in Treta, one of these legs has disappeared. In fact, etymologically, the name “treta” just means “three,” so this is the “Age of Three.” During this time, mankind was no longer ruled completely by the gods and had to turn to sacrifice to acquire the favor of the gods. Mankind had to engage in labor and commerce. Thus it is during Treta that the castes emerge, with the Brahmans devoted to performing the sacrifices, the vaishyas to trade and commerce, and the sudras to work the land and perform other labor. But during this time there was no conflict between the classes.

Treta is followed by “Dvapara,” the Bronze Age, which lasts 864,000 years. During this time the third of the bull of Dharma’s legs has disappeared, and the bull is left standing more precariously on only two legs: compassion and truthfulness. And in fact the word “Dvapara” means two, so this is literally the “Age of Two.” Now, conflict and envy enter the world. Wars break out and officials become corrupt. The commoners distrust their rulers.

And finally the current age is called “Kali,” and is the “Iron Age,” and we are 432,000 years into this present age. Now rulers have abandoned virtue entirely and people struggle simply to survive. If we look at the lengths of these four ages, we find that the ratio of the length of these ages to each other is then 4:3:2:1.

Well, not long after these five Siddhantas were written, we at long last begin to see the names of the great Indian astronomers survive down to us. The first of these is Aryabhata, and he is perhaps the greatest of India’s astronomers, at least until Chandrasekhar in the 20th century. Now, even though we have his name and the names of other later astronomers, one common feature you’ll find here is that, as with the early Greek astronomers, not too many biographical details survive down to us. In the case of Aryabhata it seems that he was born in 476 AD. This detail comes to us from his magnum opus, a comprehensive astronomical text appropriately called the Aryabhatiya. At least if Aryabhata’s claims are to be believed, he says that he wrote this text in the year 499 AD at the age of 23. Now, this does seem to be absurdly young for a work of its sophistication, but if we look at the other greats that science has offered us, it is perhaps not without precedent. After all, Richard Feynman developed the path integral formulation of quantum mechanics at 23 as well. Évariste Galois developed what is today known as Galois theory at the age of twenty. By contrast Albert Einstein was a bit of a late bloomer, waiting until the age of 25 to develop the theory of relativity. So we shouldn’t reject out of hand the possibility that he really did write this work at 23. Aryabhata’s location is somewhat more uncertain than the year of his birth. He writes that the science of astronomy is held in high esteem in Kusumapura, which is in northeast India in the Ganges river plain, but doesn’t directly say that that is where he is from. Nevertheless, because of this, he is associated with this area. But later astronomers frequently say that he came from Asmaka country which is in the far south. And in support of this, the textual record indicates that his writings were frequently studied and commented on in the south of India, so he seems to have had strong ties there. It’s possible that he was born in one of these locations but moved to the other, probably being born in Asmaka and moving to Kusumapura which was closer to the locus of political power in India.

The Aryabhatiya is broken into four sections. The first of these is the Gitikapanda. This is a kind of introductory section which describes the importance of astronomy and lays out some of the basic techniques that will be used. Already here we see an important innovation. Here Aryabhata describes a new numbering system, which is today just called the Aryabhata numbering system. The older Bhutasamkhya system that I described earlier was, as you can probably imagine, somewhat cumbersome. Each number got an entire word and there were multiple words that could correspond to a particular number. This was useful for poetry where one word might fit the verse if another wouldn’t do, but is really pretty annoying if you’re trying to compute something. Aryabhata associated a number with each possible syllable that is allowed in Sanskrit phonology. Each syllable is one consonant followed by one vowel. There are 33 consonants and 9 vowels. Each consonant gives the numeral, and the vowel gives the place in the number. The first 25 of the 33 consonants just give the number 1 through twenty-five. After that they go in increments of 10, so the next one, “y-“ is 30, then “r-“ is 40 and so on up to 100. Because the consonants get you up to 100, the vowels then let you move up in factors of 100 rather than factors of 10 as we have in our unit system. So numbers ending in the vowel “-a” are multiplied by a 1, numbers ending in “-i” are multiplied by 100, numbers ending in “-u” are multiplied by 10,000. And since there are 9 vowels, this gets you all the way up to 10 to the power of 16, or 10 quadrillion, which is large enough for all practical purposes. The numbers were denoted the opposite way they are today, so the smallest digits were written first and the most significant digits came last. So, for example, the number 215 would be represented as “ṇa khi”, “ṇa” for 15, and “khi” for two hundred. Now, some of the numbers aren’t directly included. So, for instance, there are unique sounds for numbers up to 25, but after that it skips to 30. So for a number like 26, you’d just join two separate numbers, 20 and 6. This system had a couple of big advantages over the earlier Bhutasamkhya system. Firstly, it was completely regular. Rather than needing to remember which of the various words corresponded to which numbers, the word itself told you what the number was. You just had to remember the order of the alphabet and apply the rules. The next big advantage was that it was very easy to get a sense of a number’s size. Now, the Bhutasamkhya system did have one nice feature, which is that like our modern numbering system, you could just string numbers together and implicitly get larger and larger numbers since each digit in the number represented another factor of 10. But the nice thing about associating each place in a number with a unique vowel, is that just by looking at the vowel of the last digit, you can immediately tell how large the number is. You don’t need to go through the pesky task of counting the digits in the number to figure out if it’s 10 billion or 100 billion. The number just tells you itself.

The next section of the Aryabhatiya is called Ganitapada. This section is purely mathematical, and develops all the necessary mathematical machinery for the astronomy in the later sections. As I mentioned earlier, at this time there wasn’t a large distinction between mathematics and astronomy. Mathematics was generally developed for the purpose of astronomy and the great astronomers were by the nature of what they were doing also great mathematicians. That said, as with the Greeks, although the ancient Indians did closely associate the two subjects, they also did recognize that they were distinct, which is why these topics get their own unique sections rather than mushing them together with the astronomy. At any rate, this section has things like arithmetic and geometric sequences and a derivation of the quadratic equation.

The third section is called Kalakriyapada and this is where the astronomy really gets going. Here Aryabhata deals with techniques for keeping time, how to measure the solar year and the lunar year and how to insert intercalary months into the calendar. Then he gets into planetary models. Here he takes the model of the Surya Siddhanta and makes it even more sophisticated. The Surya Siddhanta model was an eccentric-epicycle model. That is, it had both eccentric circles and epicycles. In modern terms, the epicycles accounted for the Earth’s motion around the Sun, and the eccentric circles accounted for the eccentricity of the planet’s orbit. But of course the Earth itself is on an eccentric orbit, so this model could not account for that eccentricity. To deal with this, the Aryabhatiya added a modification where the revolution of the epicycles varied between even and odd quadrants. This ended up producing a planetary model that rivaled Ptolemy in its accuracy. Incidentally, this section makes clear that Aryabhata was not writing his work for a general audience. Not only is the material highly technical, but he also skips some of the important basic components, most importantly how to calculate the mean motion of the planets. In essence, he only tells you what the corrections you need to apply are. He seems to have assumed that anyone reading his book will be learned enough to already know how to do these simple things. In this way, the Aryabhatiya is almost like a modern day research monograph, where the author assumes that the reader already knows the fundamentals of the field and doesn’t need to have them rehashed.

But that is not all. In this section Aryabhata also calculates the distances to the various planets. For this he uses a similar technique to Hipparchus, by effectively measuring parallax during a lunar eclipse to get the distance to the moon and the Sun. Now, as I discussed back in Episode 21, this is quite a difficult measurement to make, and Aryabhata was off by many orders of magnitude from their true values just as Hipparchus was. This, incidentally, is a common theme in the history of astronomy. Astronomers throughout the ages have continually been astonished at just how large the universe is, and their measurements have historically almost always been too low. This bias is probably due to two factors. The first is psychological. We are all human and are familiar with human scale things. We can comprehend the distances between towns and we know that there are other countries in places far away that it takes a very long time to get to. But beyond that, our intuition starts to break down. Even the size of the Earth itself is hard for us to comprehend. But once you start looking at the distance to the Sun, the distance is just so much vastly larger than anything we are familiar with. So astronomers of the past tended to prefer measurements of the distances to various celestial bodies that were very large, certainly much larger than anything on Earth, but not too much larger than that, because it was just hard to comprehend those scales. But there was probably another factor which is somewhat more technical. And this is just that when you are measuring distances in astronomy, in general, any errors in your measurements have a tendency to bias you towards smaller distances. In the case of these ancient measurements, astronomers were essentially looking for an effect called parallax — that an object viewed from two different locations on Earth will appear to be in slightly different locations relative to the background stars, just as when you’re looking out the window in a car, nearby trees seem to move more rapidly than mountains in the distance. The trouble is, these objects are quite far away, so the parallax effects are pretty small. Smaller distances produce larger parallaxes and larger distances produce smaller parallaxes. But if there is any error in your measurement, this will tend to make the object appear to move more than it really did. It’s just very unlikely that your error will happen to be of the same magnitude and in the exact opposite direction as your effect so that the two cancel. Consequently, any observational errors tend to produce a larger apparent parallax than there really is, and that introduces a bias towards smaller distances. If the size of your error is much larger than the size of the true parallax effect, as it was in ancient times, the distance you obtain is essentially just a measurement of the error in your measurement rather than the actual distance to the object. So it took quite a long time, really until the Enlightenment with the development of more sophisticated measuring apparatuses, before these kinds of measurements got distances that were approximately correct.

Well, the last section of the Aryabhatiya is the “Golapada,” from the word “gola” which means sphere, and this is essentially a treatise on spherical astronomy. In modern times, the section on spherical astronomy comes at the beginning of a textbook, but Aryabhata puts it at the end, which seems to suggest that he viewed this kind of astronomy as the culmination of the science. Now, many of the concepts developed here are similar to the Surya Siddhanta, with perhaps more sophistication, but the main innovation in this section is Aryabhata’s controversial claim that the Earth rotates. Aryabhata seems to have been the first Indian astronomer to unambiguously argue that the reason that the stars, and planets appear to rise and set through the night is because it is really the Earth that is rotating. He makes an analogy to a man in a boat moving down the river. He seems a tree on the shore that appears to be moving backwards. But of course the tree is not really moving backwards, it just appears to be doing so because the boat is floating down the river. Likewise the stars do not really revolve around the Earth, but it is the Earth that is rotating which produces the appearance of their motion. Of course today this idea seems quite plausible, but it was by no means accepted by later astronomers in India and was one of the more contentious aspects of Aryabhata’s astronomy for centuries to come.

The next astronomer I wanted to mention is called Bhaskara I. He is given the number “the first” to distinguish him from another mathematician and astronomer who was also named Bhaskara but lived in the 12th century, so we’ll have to wait a while until we start to get into the astronomy of the middle ages well after the arrival of Islam. This first Bhaskara was probably active in the late 6th century and early 7th century, and as with Aryabhata we aren’t entirely certain of his location. He seems to have had some association with Asmaka in south India just like Aryabhata, but also Surastra in Western India. So it’s speculated, as with Aryabhata, that maybe he was born in one and then lived and worked in the other. Bhaskara wrote three texts, but the most important is probably the Mahabhaskariya. In many ways, this text filled in the gaps left by Aryabhata in his work. As I said earlier, Aryabhatiya is almost like a research monograph, and seems to assume that the reader is familiar with the state of the art of astronomy of the day. Even fairly technical details like how to compute the mean motions of the planets are omitted. And the text is fairly terse, so if you’re just reading it in isolation, it’s hard to really learn from it. It’s an ancient version of the stereotypical abstruse mathematical textbook that just lists the theorems and says that the proofs are left as an exercise for the reader. The Mahabhaskariya is a gentler introduction to Aryabhata’s astronomy. Bhaskara I includes the details that were beneath Aryabhata to include like how to compute the mean motions of the planets. In fact, Bhaskara is generally credited as being the one to invent a particular technique he calls the “kuttaka,” because Aryabhata’s mentioning of it is so impenetrable that it’s not really obvious that he really figured it out. The kuttaka literally means “pulverization,” which is interesting because Bhaskara’s instinct to name it this seem to have paralleled Isaac Newton’s instinct to choose the name “calculus” for his discovery which literally means “pebble.” Calculus and kuttaka are really quite distinct things, but both rely on a general mathematical technique of taking a larger thing and then repeatedly breaking it up into smaller and smaller bits that can be more easily analyzed. Well, the kuttaka is an algorithm for finding solutions to Diophantine equations, and at the time Bhaskara was writing it was the most sophisticated algorithm developed. A Diophantine equation is an equation where all the coefficients and unknowns are integers. So in the simplest kind of Diophantine equation you might have something like a times x plus b times y equals c, where a, b, and c are integers that are given to you but you need to figure out the numbers x and y and they have to be integers as well. These kinds of equations come up in astronomy when you’re trying to find the length of the overall period it takes two cycles to line up. For instance, if you know the synodic period of Mars this will tell you how long it will take Mars to go between opposition and opposition. Likewise if you have the synodic period of Saturn, this will tell you the time it takes Saturn to go between opposition and opposition. But let’s suppose Mars and Saturn are in opposition at the same time and thus are in conjunction with each other? How long will it be before this happens again? Figuring this out, at least in terms of an integer number of days or months as the ancients were interested in doing, requires you to solve a Diophantine equation.

Bhaskara I’s other claim to fame is that he was the first to correctly describe the physical mechanism of solar and lunar eclipses. I mentioned in the last episode that in the late Vedic period, the prevailing theory of eclipses is called the “rahu-ketu” theory of eclipses. Now, to maybe make a bit of pedantic distinction, astronomically, this older theory was correct. “Rahu” and “ketu” were essentially the two nodes of the Moon’s orbit, and so it identified the fact that an eclipse could only occur when the moon was full or new as it passed through one of these nodes. But the “rahu-ketu” theory had no physical mechanism for why this occurred. It seems to have been understood somewhat mythologically, as the Moon or Sun getting devoured by a beast at these points. Bhaskara I provided an astrophysical model of eclipses in Indian astronomy. He argued that lunar eclipses were caused by the moon entering into the shadow of the Earth. He wrote:

The orbit of the moon is below that of the Sun. Just as a cloud moving from behind covers the sun so does the moon moving faster cover the sun from behind. That is why the western part of the sun is eclipsed by the moon first and the eastern side released last. Owing to differences in the latitude eclipses are sometimes seen and sometimes missed.

Well, the last astronomer I will mention in our tour of ancient Indian astronomy is Brahmagupta. Brahmagupta was born at the beginning of the 7th century, probably about one or two generations later than Bhaskara I. Here we at least are somewhat more confident about his location, coming from the north of Gujarat in the Indus river valley. Brahmagupta’s principal work is the Brahmasphuta Siddhanta, which he wrote at age 30 in the year 628. The work consists of 24 chapters, most of which are a comprehensive summary of the known astronomy of the time. But probably the chapter everyone remembers best is chapter 11, in which Brahmagupta gives a no holds barred critique of all the other astronomical systems that had been proposed. Religious dogma was no barrier for Brahmagupta and he started with the Vedanga Jyotisha and argued that the 5 year yuga is a terrible system for timekeeping. But his critiques are not limited to the Hindu religion. He also brings the Jains into the mix and says that their notion that there are two suns and two moons is patently absurd. He even attacks the great Aryabhata for his various innovations, but above all believes that Aryabhata made a fool of himself for proposing the ridiculous idea that the Earth rotates. The idea that lunar eclipses are caused by the Earth’s shadow is also attacked.

Well, Brahmagupta seems to have been a rather strong-headed youth when he wrote the Brahmasphuta Siddhanta, but he apparently mellowed with age. 37 years later, at the age of 67 he wrote another work called the Khandakhadyaka. By this point in his life it seems he had more or less come around to Aryabhata’s astronomical system and this book is heavily based on it, and he even provided a few improvements of his own. In the end, Brahmagupta turned out to be one of the most highly influential astronomers in history. Both of his works were translated into Arabic not too much later in the 8th century, very early in the development of Islamic astronomy. Consequently, they were tremendously influential on the Islamic astronomy of the middle ages, which we will eventually get around to on this podcast.

Well, perhaps the next major astronomer in India is Bhaskara II, and as I mentioned earlier, he is in the 12th century, so by the time we get to him we are well into the Islamic era in India. So his story will have to wait until then. With that, we will leave India behind, at least for the time being, and move to the northeast, to another great civilization that has existed continuously since the ancient world: China. I hope you’ll join me then. Until the next full moon, good night and clear skies.

Additional references

  • Rao, Indian Astronomy
  • Kak, Birth and Early Development of Indian Astronomy
  • Ghosh, Descriptive Archaeoastronomy and Ancient Indian Chronology
  • Subbarayappa, A Concise History of Science in India