Ancient Indian Astronomy and Mathematics: A Brief History

he speaker examines ancient Indian mathematics and astronomy, beginning with the Jantar Mantar's Samrat Yantra in Jaipur—a 90-foot stone sundial accurate to two seconds. This paradox of monumental, low-tech construction achieving microsecond precision frames a broader argument: ancient Indian science operated on a completely different "operating system" than the West's linear, segmented tradition.

Central to this worldview was kāla, or time, conceived as recursive and granular. Texts like the Surya Siddhanta defined the truti—29.6 microseconds—derived not by measurement but by mathematical reasoning that time, like matter, must have an atomic unit. This theoretical ladder extended from microseconds to kalpas of 4.32 billion years, demonstrating a conceptual comfort with cosmic scales that Western cosmology lacked until modernity.

The speaker profiles key mathematician-astronomers: Aryabhata (b. 476 CE) proposed Earth's rotation, calculated pi to 3.1416, and developed the kutaka ("pulverizer") algorithm for solving indeterminate equations. Brahmagupta formalized zero as a number with operational rules and systematized negative numbers as "debts." Bhaskara II (12th century) approached calculus concepts, recognizing instantaneous velocity and solving Pell's equation via the chakravāla method. The Kerala School (14th century), led by Madhava, developed infinite series for pi and trigonometric functions—predating Newton and Leibniz by 300 years.

Crucially, this science was not secular but spiritual: mathematics was the language of dharma, the cosmic order. Ritual requirements for precise Vedic altars drove geometric discovery; accurate horoscopes demanded algorithms predicting planetary positions centuries ahead. The decimal system and zero transformed mathematics into a "dynamic machine," enabling complex computation impossible with Roman numerals. This tradition, the speaker concludes, proves that profound scientific insight emerges not only from instruments but from pure reason treating mathematics as discovered eternal truth.

Transcript

Unknown Speaker  0:00  
I want to start with

Unknown Speaker  0:02  
with a contradiction. It's a visual one. I want you to picture a structure in Jaipur, India. It's called the Jantar Mantar. And specifically, I'm looking at the Samrat Yantra, the supreme instrument, or the king of instruments. Yeah, right. And if you've ever seen photos of this, or if you've been lucky enough to be there. You know, it doesn't look like some delicate scientific instrument, not at all. It looks like a piece of, I don't know, brutalist architecture. It's 90 feet of stone and masonry. It's just massive. It's immovable, and it's completely silent. It just stands there under the blazing Indian sun, looking more like a like a stairway to nowhere than a clock. It creates a very specific atmosphere. It feels ancient, exactly. It feels low tech. I mean, it's made of the most basic materials on earth, stone, lime plaster. But, and here's the contradiction, this thing is a precision instrument, unbelievable precision. It measures time, accurate to two seconds. Two seconds, yeah. And the civilization that built it, and the, you know, the mathematical tradition behind it, they weren't just obsessed with seconds. They were obsessed with microseconds. And on the flip side, gig, years, billions of years, right? So that's the mission today. We are unpacking the mathematical cosmos of ancient India. We've got a whole stack of papers here excerpts from ancient texts like the Surya Siddhanta historical analyzes of these giants, like Aryabhata and Bhaskara the second. It's a deep, deep rabbit hole. And what jumps out at me isn't just that they were, quote, advanced for their time. That's such a cliche. It's that they seem to be working on a completely different operating system than the West. That's a great way to put it. We often look at the history of science as this, like linear relay race. Yeah, the standard narrative, Greece passes the baton to Rome. Rome drops it during the Dark Ages. The Arabs pick it up, polish it, and then they hand it over to Renaissance Europe, which leads to Newton. And, you know, the modern world, right, a straight line. But when you actually look at the source material, specifically the siddhantas, which are these astronomical treatises, you see a parallel track, a massive, sophisticated engine of discovery that was running independently for centuries and running on a completely different logic. I mean, in the West, we tend to separate everything. Math is in this building, religion is in that building over there philosophies down the street in this stack of research, those lines are they're blurry. If they exist at all, they don't exist. They're totally interwoven. For the ancient Indian astronomers and mathematicians, calculating the position of a planet wasn't just data entry. It was a way of understanding Dharma, the cosmic order. So science was a spiritual practice. It was the tool used to map the divine. It was a form of devotion. So let's boot up this operating system. Let's start with the most fundamental concept of all time, Kala. Because when I read about the Hindu concept of time in these texts, it felt less like a timeline and more like a, I don't know, a breathing lung. It's recursive. It expands and contracts, and it is incredibly granular. I mean, in the West historically, we've had a very human centric view of time. We think in days, years, maybe a lifespan, generations, right? And for a long time, Western history was bounded by, you know, biblical genealogies. The world was only a few 1000 years old, right? 4004 BC, October 23 at 9am was the creation date, according to Archbishop Usher, exactly a very specific, very small box. Yeah. But the ancient Indian texts, they were comfortable with time scales that are frankly astronomical. They didn't just measure time. They dissected it. They put it under a microscope. Okay, let's look at the micro scale first, because this is what really broke my brain. The research papers mention a ladder of time. It's a hierarchy of units, and at the very bottom of this ladder, the smallest possible rung is the true T, the base unit, the quantum of time. In a way, the definition given in the text is the time it takes a needle to pierce a lotus leaf, which is, I mean, it's a beautiful image, but we need to translate that into physics. We do, and they did mathematically. The Surya Siddhanta defines the true D as one divided by 33,750

Unknown Speaker  4:11  
so 133,700/50

Unknown Speaker  4:14  
of a second? Yes, which works out to roughly 29.6 microseconds. Okay, stop there. We have to unpack the why this is the Iron Age or the Vedic Period, depending on the text you're reading. You are herding cattle. You're lighting fires with sticks. You are living in a world of, you know, manual labor. Why do you need a word for 29 microseconds? You can't see it. You can't see it. You certainly can't measure it with any physical tool they had. It feels completely disconnected from their reality. It is if you look at it through a purely utilitarian lens, does this help me farm better? It makes absolutely no sense. You don't need microseconds to know when to plant wheat. So what's the purpose? Is it just intellectual flexing? I think it's more profound than that. If you look at it through a theoretical or philosophical lens.

Unknown Speaker  5:00  
It's brilliant. They weren't measuring it with a stopwatch. They were deriving it through mathematics. So they were reasoning their way to it exactly. They understood conceptually that time, like matter, must have a fundamental atomic unit. You can't just keep dividing something forever. Physically, there must be a point where you stop. So it's a limit, like in calculus, the smallest possible slice precisely. It's a conceptual limit. They started with what they could observe the day or the blinking of an eye, and they just kept dividing it mathematically. They sliced it in half and half again and half again until they reached a unit that felt fundamental. The true taint isn't a measurement you take. It's a recognition that reality is made of pixels. Time has a resolution that is a heavy concept for 500 BCE. It implies they viewed time as discrete, not a continuous, smooth flow, or at least that it could be quantized for the purposes of calculation. And they built a bridge, a ladder, from that invisible pixel all the way up to human scale time. It's incredibly systematic. Let's walk up that ladder, because the names of these units are just fascinating. Okay, so you start with the truti, the 29.6 microsecond unit. 100 trudis Make one Tappara that gets you to about three milliseconds. So we're still way faster than human perception. You know, house flies, wing beat right, still invisible to us. Then 30 tat paras make one namesha. Now, namesha is a word you see in poetry, it means the twinkling of an eye about 88 milliseconds. Ah, okay, so now we are in the realm of human perception. We can see a blink. We've crossed the threshold. Exactly. We have crossed from the purely theoretical to the biological. Then 18 nemeshas Make a kashta, 30 kashtas Make a Kala and 30 kalas make a gatika, and the gatika is the standard unit, kind of like hour, hour, roughly, yes, a gatika is 24 minutes. And this is really important, because their day wasn't divided into 24 hours, like hours. It was divided into 60 gatikas, a sexagesimal system base 60, which feels like a completely different rhythm for a day it is and two getika is make a Murta, which is 48 minutes, and you often hear about auspicious muhurtas in indie culture even today, for say, weddings or starting a business, it comes directly from this ancient system. It's interesting that you mentioned the blinking of an eye, because one thing that stood out to me in the reading was how much this time system was linked, not just to observation, but to the human body itself. Oh, absolutely, it wasn't abstract. It was physiological. It was embodied. There's another unit called the prana. One prana is the time it takes for one respiration cycle, one breath in, one breath out, for a resting adult, that's roughly four seconds. So your own breath is a clock, a metronome, your body is a clock, and that breath was mathematically linked to the rotation of the stars. In the text, they calculate how many breaths a human takes in a day, which is about 21,600

Unknown Speaker  7:52  
and they map that onto the movement of the heavens. So you have this incredible system where your own physiology, your very life force, is geared into the Grand movement of the cosmos, which reinforces that operating system idea we started with. You aren't an observer standing outside the universe looking at a clock. You are the clock. Your biology is a reflection of the celestial order. Beautifully said, That's exactly the world view. But they didn't just go small, they went huge. And this is where the Sci Fi feeling really kicks in for me, the macro scale, the yugas, the cosmic cycles. The yugas, this is where the western and eastern timelines just violently diverge. I mean, in the 17th or 18th century, the smartest people in Europe were seriously debating, if the earth was 6000 years old, what were the Indians saying? They were talking in billions with a B. The Hindu cosmology is cyclic. It's not a straight line from creation to apocalypse. It's an endless cycle of creation, existence and dissolution. And these cycles are the yugas, correct. You have these ages. Or yugas, the sources say we are currently in the Kali Yuga, the age of fice, which is 432,000

Unknown Speaker  8:57  
years long, which is already, I mean that alone, is longer than recorded human history as we know it. And that's the shortest one. Shortest one. It's the last and worst of the four a maya Yuga, which is a full cycle of all four ages, is 4.3 2 million years. Okay, my brain is starting to hurt, but then you zoom out again, 1000 mahayugas. That's a Kalpa. And a kalpa, which is just one day in the life of the creator god Brahma, is 4.3 2 billion years, 4.3 2 billion that is eerily close to the actual age of the Earth, which modern science puts at about 4.5 billion years. It is a stunningly close number. Now we have to be careful. We can't say they knew the age of the Earth in a modern geological sense. Sure, they weren't dating rocks with uranium lead decay, right? But philosophically, they were completely at ease with DEET time. Carl Sagan talked about this. He was amazed that this was the one ancient cosmology whose time scales were even in the right ballpark as modern astrophysics. That's the key, isn't it? It's not the exact number, it's the canvas size. Yes, they had the right canvas size while Europe was paying.

Unknown Speaker  10:00  
Painting on a post it note India was painting on a billboard that stretched to the horizon. They understood that for the universe to function the way it does, for stars to form, for spiritual evolution to happen, you need massive, almost incomprehensible scales of time, and it goes even bigger, doesn't it? What's the full lifespan of Brahma? The full lifespan of Brahma is 100 of his years, which, if you do the math, comes out to 311.0

Unknown Speaker  10:25  
4 trillion years. And after that, the universe dissolves for a period of equal length before a new Brahma is born and it all starts again. It's mind bending so we have microseconds and we have trillions of years. But let's bring it back down to earth, literally. How did they organize their actual day? You mentioned the 60 Gaudi day, right? 60 units of 24 minutes each. We start our day at midnight, which, if you really think about it, is completely arbitrary. You're asleep. Nothing is happening in the sky. Why midnight? It's a convention. It's useful for bookkeeping and timetables, but the ancient Hindu day started at sunrise. That makes so much more sense. Biologically, it does. It's tied to the solar cycle. Your day begins when the world becomes visible, but it introduces a complexity. What's that? Sunrise changes every day. It's earlier in summer, later in winter. So their start time was dynamic. It wasn't a fixed point. This meant their time keeping had to be constantly calibrated to the sun. They lived in a much more fluid relationship with time, which brings us to the hardware, because having a unit like a datika is one thing, measuring it accurately is a whole other problem. And this brings us right back to that 90 foot stone triangle I mentioned at the start, the Samrat Yantra, the king of instruments built by Jai Singh the second in the early 18th century, but based on these ancient principles from the setantis. I want to get into the engineering here, because the critique I always have of you know these ancient history shows is they say, Oh, it was precise, but they never explain how. Why does making a sundial 90 feet tall make it more accurate? Is it just about ego making the biggest thing? No, the size is the function. It's all about geometry and the nature of light. Think about a shadow. If you stick a pencil in the ground, you get a shadow. But have you ever looked really closely at the edge of that shadow? It's blurry. It's fuzzy. Exactly. It's called the penumbra. That's because the sun isn't a laser pointer in the sky. It's not a single point of light. It's a massive disk. So light comes from the left side of the sun, the right side of the sun, the top, the bottom, and where all those light rays meet past your pencil, they create a soft, fuzzy edge. Okay, so if I'm trying to read the time on a small garden sundial, that fuzzy edge might cover what five or 10 minutes of width on the dial exactly. You can't tell if it's 12.01 or 12.06

Unknown Speaker  12:40  
because the shadow is this blurry gray bar, you've hit a resolution limit. The fuzziness is too big compared to the scale of the dial. So how does making it huge fix that? I mean, the sun is still a disk. The sun is still a disk, but you change the scale of the projection by making the gnomon. That's the triangular wall that casts the shadow 90 feet high. And the quadrant where the shadow falls absolutely massive. You're effectively zooming in on time itself. You're stretching the timeline. You're stretching the distance the shadow has to travel. Yeah, on the samurai Yantra, the shadow moves at a speed of about six centimeters per minute. Wait, six centimeters. That's you can see that you can physically watch it move. It's roughly one millimeter per second. If you stand there and watch the edge of the shadow, you can actually see it creeping across the stone markings. It moved the width of a grain of rice every second. That is wild. You are literally watching the rotation of the Earth in real time. That's precisely what's happening. And because the scale is so large, that fuzzy edge becomes much less significant relative to the distance the shadow travels. The fuzz might be a centimeter wide, but the shadow moves six centimeters in a minute. So you can easily mark tick marks for seconds. Inside that shadows path so size equals resolution. Size equals resolution. That's why it had to be so big. It transforms a blurry five minute guess into a crisp two second reading. It's analog zooming. It's incredibly clever. And the engineering to build it, I mean, this is masonry. If that stone wall settles by even an inch or warps due to the heat, the whole calculation is off. The foundations had to be incredibly stable. The angles had to be perfect. The Gnomon, that triangular part, doesn't just point up. It points exactly to the North Pole. It's parallel to the Earth's axis of rotation, so it's basically canceling out the tilt of the earth in its reading. It is. It's a masterpiece, not just of time keeping, but of positional astronomy. But okay, you can't carry a 90 foot sundial in your pocket, and you can't use it at night or when it's cloudy. So what did they use for the rest of the time? For, you know, daily life, the gitiki Yantra, the water clock. Now, water clocks are ancient. I mean, everybody had them. Egyptians, Babylonians, Greeks was the Indian version, different. The core concept was similar, but the application was very specific to that 60 Gotti day. It's a sinking bowl design, which is pretty clever.

Unknown Speaker  15:00  
Explain the mechanism. How does it work? Okay, so you have a large vessel filled with water, and then you have a smaller, hemispherical bowl, usually made of copper, floating in it.

Unknown Speaker  15:09  
The key is that the small bowl has a tiny, precise hole drilled in the bottom, like a leak, a very slow leak, a calibrated leak. Water seeps in. The bowl gets heavier and heavier, and eventually, C, l, u, N, K, it sinks to the bottom of the larger vessel. And the time it takes for it to sink is exactly one gatika, exactly 24 minutes. And there would be an attendant, yeah and yes, this was a full time job for someone who would watch it, fish it out the moment it sank empty, it Bang a Gong or a drum to announce the time to the town and then float it again. It sounds simple, but I imagine the calibration part was an absolute nightmare. Ah, it was the constant battle of physics versus reality. The source material talks about this. Think about water. What happens to water when it gets cold, it gets thicker, more viscous. The viscosity changes. Yes, cold water flows slower than warm water. So on a cold day, your clock runs slow. In the summer, when the water is warmer and less viscous, it flows faster and your clock runs fast. So a winter hour and a summer hour would actually be different lengths of time they would be if you didn't adjust for it. They had to constantly calibrate the size of the hole, or maybe the weight of the bowl, and what did they calibrate it against the stars? Ah. So it all comes back to astronomy. Always they would use the transit of a specific star, say the star Chitra, which is Spica, to reset the clock every night. The water clock was just the stopwatch. The stars were the master clock. We have the time scale from microseconds to kalpas. We have the hardware from stone monuments to sinking bowls. But none of this works without the software. And by software I mean the language of math itself. And this is where we have to talk about the elephant in the room, or rather, the nothing in the room, zero. Shunya, I feel like we learned in school that India invented zero, and we just nod and move on. It's a trivia fact, but our sources really, really emphasized that this wasn't just inventing a new symbol. It was a complete conceptual revolution. It was a paradigm shift you have to understand other civilizations had a placeholder. The Babylonians, for instance, used a cuneiform system. If you wanted to write the number 101, you needed a way to show that there was nothing in the 10s column. So how did they do it? They would leave a space, or later, they put a little Double Wedge mark in there. But that marks wasn't a number. It was punctuation. It was punctuation, exactly. It was a separator. You couldn't take that wedge and add it to the number five. It didn't mean anything computationally. It just meant empty slot. Here. The Romans didn't even have that. So what did the Indians do differently? They treated zero as a number in its own right, a quantity, a value that had its own rules and its own identity. Just like one, two or nine, it wasn't just an absence, it was a presence, and it had rule and it had rules. This is where Brahmagupta comes in. He's one of the Titans we'll discuss in more detail in the seventh century. He's the first one we know of who systematically wrote down the operations for zero. He treated it like a mathematical object. What were the rules? He laid out? He explicitly says, When zero is added to a number or subtracted from a number, the number remains unchanged. He says, a number multiplied by zero becomes zero. It sounds so obvious to us now because we learn it in kindergarten, but writing that down as a formal mathematical law for the first time, that was groundbreaking. It's what allowed for the development of algebra. You can't balance an equation if you can't zero, it out. If you have the equation $5 plus five equals $10 you need to subtract 10 from both sides to get to five by five equals $10

Unknown Speaker  18:41  
if zero isn't a number, you can arrive at algebra breaks. The whole system collapses. But Brahmagupta did hit a snag, didn't he? There was a famous bug in the software, the glitch division by zero. He tried to figure out what happens when you divide by zero, which is still a problem for us. I mean, if you ask your phone to divide by zero, it gets sassy with you, it does. Brahmagupta, in his attempt, concluded that zero divided by zero was equal to zero, Swing and a miss, a forgivable one. It's an incredibly tricky concept. It wasn't until centuries later, in the 12th century, that another mathematician, Bhaskar, second, refined this. He suggested that a number divided by zero yields an infinite quantity. Whoa. He called it kahara, a quantity with zero as its denominator. And he said it was a quantity which remains unaltered, even if vast numbers are added to it or subtracted from it. That is essentially the mathematical definition of infinity. It is so the invention of zero as a number led them directly logically to the concept of infinity as a number. The two are two sides of the same coin and connected to zero, is the whole decimal system, the place value system, what we call the Hindu Arabic numerals, the simple idea that we can use just 10 symbols, zero through nine to represent any number in the universe, no matter how big or small. There's that famous quote by Pierre Simon Laplace, the great French mathematician he.

Unknown Speaker  20:00  
Called it an ingenious method and said it was a profound and important gift from India to the world. He said it was so simple that we fail to appreciate its true merit. But just think for a second about the alternative. Try doing long division with Roman numerals. What is mcmxlv divided by six? Yeah. No, thanks. My head would explode. It's practically impossible. You can't do complex algorithms with Roman numerals. They are just for recording static amounts on a monument the Indian system. Place value turned numbers into a dynamic machine. You could manipulate them, stack them, perform complex operations. It democratized mathematics. Okay, let's talk about the people who actually wrote this software, the titans of the Setanta period. We have a few names that keep popping up in all the research, Aryabhata, varahamihira, Brahm Gupta, the big three of that Classical Age. Let's start with Aryabhatta. This guy seems like the Einstein of the fifth century. Born in 476 CE, and he wrote his masterpiece, the Arya poppya when he was just 23 years old. 23 I was barely figuring out how to do my own laundry at 23 and he was calculating the value of pi and calculating it with incredible accuracy. His value for pi was 3.1416

Unknown Speaker  21:13  
which is precise to four decimal places, but the source says something even more interesting. He knew it was an approximation. He didn't think it was the final answer, yet, he called it Asana, which means approaching or nearby. This is a massive conceptual leap. He understood that Pi was what we now call irrational, that it couldn't be fully expressed as a simple fraction, that the digits go on forever without repeating. He sensed its transcendent nature. He also had some, some pretty radical ideas about the Earth at a time when the prevailing view, even in much of the world, was that the heavens rotated around a stationary Earth, a geocentric model. Aryabhata said, No. He proposed that the Earth rotates on its own axis, that the apparent motion of the stars is a result of our own spinning How did he explain that? Did he have observational proof? He used a beautiful analogy based on relativity of motion. He wrote, and I'm paraphrasing here, just as a man in a boat going forward sees the stationary objects on the bank as if they're moving backward, so are the stationary stars seen by the people on earth as moving towards the west. That's kinematic relativity. He understood that motion is relative to the observer's frame of reference Exactly. If you're the one spinning, the world looks like it's spinning the other way. It's an incredibly modern sounding insight. He also correctly identified that the light of the moon and planets is not their own, but is reflected sunlight. I want to get into his math, though, specifically this thing called the pulverizer, the kutaka. The kutaka, it sounds like a wrestling move. He's got him in the pulverizer, but it's an algorithm. The source describes it as a method for solving indeterminate equations. We need to translate that. What is the actual problem he is trying to solve here? Okay, let's make it concrete. Imagine you have a big pile of coins. You don't know how many there are, but you know that if you divide the pile into groups of eight, you have five coins left over. Okay, a remainder of five, right? And you also know that if you divide that same pile into groups of 13, you have nine left over.

Unknown Speaker  23:15  
The question is, what is the smallest possible number of coins in the pile? So it's a reverse engineering problem. You have the remainders, and you need to find the original number Exactly. Mathematically, it looks like an equation such as exhaust by C, l, o, y, you have two variables, x and y, but only one equation. In modern algebra, we'd say you can't get a single solution. There are infinite possible answers, but you're only looking for the integer answers, the whole numbers. You can't have half a coin, and just trying to brute force that guessing numbers until one works would take forever. If the numbers are large, it would so arabada figured out a systematic method to pulverize the coefficients. You essentially use the remainders against each other in a chain. You divide the larger number by the smaller, find the remainder, then divide the previous divisor by that remainder, and so on, grinding the numbers down in a recursive chain till you get to zero. That sounds suspiciously like the Euclidean algorithm for finding the greatest common divisor. It's very closely related to the extended Euclidean algorithm. It uses that same process of grinding down the numbers coupled with a clever method of back substitution, to find the values of the variables. But remember why he was developing this? He wasn't just counting coins for fun. He was tracking planets. He was tracking planets. A planet is just like that coin pile. You know, its cycle speed, its period of revolution, the divisor. And you know where it is now in the sky, the remainder. You need to calculate where it was 1000 years ago or where will be 1000 years in the future. That's a massive remainder problem. So he invented an algorithm, a piece of software, to precisely track celestial bodies 1000 years before, Kepler was doing his work with ellipses, effectively, yes, yeah, it was a huge leap in computational astronomy.

Unknown Speaker  25:00  
Okay, so Aryabhata is the brilliant theorist, the pioneer. Then we have varahamihira A century later. He strikes me as a different kind of character, more of an encyclopedist, an observer. Varahamihira, in the sixth century, was the great synthesizer. He was a polymath. He wasn't just doing abstract math, he was looking at the ground beneath his feet. The sources mention his work on groundwater exploration. This feels very earthy and practical compared to planetary orbits. It is in his great work, the brihasamhita, he links biology to geology. He meticulously documented that certain types of termite mounds are often indicators of deep underground water termites. How does that work? Termites need a lot of moisture to keep their mounds at a stable temperature and humidity, especially in arid regions. So if you see a thriving termite mound, it means they have found a reliable water source deep down, sometimes 20 or 30 meters, and are bringing that moisture up to the surface. So he was using insects as biological sensors to find water the internet of living things. Sixth century edition, he also linked specific plants, trees and even rock formations to the presence of water tables. It was observational science at its absolute best. He wasn't just looking up at the stars. He was looking down at the ants. He also seemed to be less, I don't know, less nationalistic about science. He was open to ideas from anywhere, very much. So he famously praised the Greeks. He called them yavanas, and he wrote that even though they were malekus, which means foreigners or barbarians in a cultural sense, they should be honored like rishis, like great sages, because their science was so advanced. That's a really refreshing attitude. I don't care where you're from. If your math works, I respect you. It shows that science has always been a global conversation, a global trade of ideas. His work, the Pontius of Antica, actually summarizes the knowledge of five different astronomical schools, including some based on Roman and Greek knowledge. Okay, moving to the third Titan, Brahmagupta, the analyst. We already mentioned him with the rules of zero, but he was also a bit of a contrarian, a critic. He was. He famously and very publicly criticized Aryabhata. Oh, a little academic beef from the seventh century. I love it big time. He really went after aryabhata's view of the Earth's rotation. Brahmagupta argued forcefully for a stationary Earth at the center of the universe. So he was wrong on that one, history proved Aryabhatta right. He was wrong. But in science, being wrong for the right reasons is a crucial part of the process. He demanded proof. And the physics of a rotating earth are really counterintuitive. If the Earth is spinning at 1000 miles an hour, why don't we all fly off? Why does a bird that flies up land in the same spot? That's a valid question, if you don't have a theory of inertia, exactly. But Brahmagupta gave us something else incredible, aside from zero. He formalized the rules for negative numbers. He was one of the first in the world to treat them systematically. He called them debts and positive numbers fortunes, which makes total intuitive sense, and his rules were perfect. A debt minus zero is a debt. A fortune minus a debt is a fortune. A debt multiplied by a debt is a fortune. He made negative numbers a workable part of the mathematical toolkit, and in the middle of his arguments against Aryabhata, He also hinted at gravity. He wrote, all heavy things are attracted to the center of the Earth. The Earth is the only low thing, and seeds always return to it in whatever direction you may throw them away and never rise upwards from the earth, attracted. That is a very specific, very powerful word. It is. It's not Newton's law of universal gravitation. He didn't have the inverse square law or the calculus to prove it, but it's a clear acknowledgement of a universal force of attraction that holds us to the sphere. Okay, so let's jump forward in time. We get to the medieval period in India. The Golden Age isn't over, not by a long shot. We have Bhaskara the second in the 12th century. And this is where things start getting really, really advanced, pre calculus, advanced. Bhaskara the second, or bhaskaracharya, is a giant. His main work, sitanta Shiromani, the jewel of accuracy, is a masterpiece that was a standard textbook for 700 years. I love this story about his book, Lilavati, which is the first part of that work, yes, the section on arithmetic. The legend, which is probably apocryphal, but lovely, is that he wrote it for his daughter, Lilavati, to console her after her wedding was called off. And is written in this very playful, poetic style. It's not like a modern math textbook. No, it's full of word problems addressed to her, Oh, little Avadi, intelligent girl, if you understand addition and subtraction, tell me the sum of the problems are broke about swarms of bees and beautiful maidens of flocks of geese. I remember one from the source, a flock of geese. 1/4 of the square root of the number went to the shore. Tell me, oh, beautiful girl, the number of geese. It makes math approachable, even beautiful. But don't let the poetry fool you. The math is hardcore. In his other work, bijaganita algebra, he tackles pell's equation. Okay, you mentioned that before. What is pell's equation? It.

Unknown Speaker  30:00  
A notoriously nasty type of equation. It has the form Unum by two throttlers plus one equals y2, two. And the challenge is to find integer solutions for x and y for some values of n, the smallest solutions are astronomically huge. And this was a known hard problem. It was so hard that Pierre de Fermat, the famous French mathematician, issued it as a public challenge to European mathematicians in the 17th century. He thought it was nearly unsolvable, and Bhaskara had already solved it. Bhaskara developed a general method for solving it, called the chakravalla, or cyclic method, 500 years earlier. It's an iterative algorithm that is guaranteed to find the smallest solution 500 years that's just a long head start. And then there's the calculus connection. This is really where Bhaskara touches the modern world. He was looking at the motion of planets, and he realized something profound about velocity. Walk me through this. Okay, imagine you throw a ball straight up in the air. It goes up fast, it slows down. It seems to hang for a moment at the very top, and then it falls back down right the apex of its trajectory, at that exact infinitesimal moment at the top, its velocity is zero. Bhaskara conceptualized this instantaneous motion. He realized that to calculate the true speed of a planet at any given point in its orbit, you have to look at infinitely small slices of time the true text again. The idea comes back. It comes back. He was effectively defining the derivative of what we call dedeck sendo, the rate of change at an instant. He didn't have the formal notation of Leibniz, but he had the core concept down cold. To understand motion, you must measure the change that happens in an instant. This is a foundational idea of differential calculus, and this leads us to what feels like the most controversial part of the source stack, or at least the part that requires the biggest rewrite in the standard history books, calculus we are all taught Newton Leibniz, late 1600s and apple falls on Newton's head, calculus is born. That's the standard narrative. Yes, but our sources are pointing to a school of thought in Kerala in the south of India that was flourishing three centuries earlier, the Kerala School of mathematics and astronomy founded by a genius named Madhava of sangamagrama in the 14th century. Madhava, why isn't this guy a household name like Newton or Archimedes? He should be. It's a matter of historical transmission, and frankly, Eurocentrism in the history of science. Madhava is the father of mathematical analysis, the father of infinite series. Break that down infinite series. Why do we need them? What problem do they solve? Let's go back to pi.

Unknown Speaker  32:31  
Ary pet, a new PI was roughly 3.1416

Unknown Speaker  32:34  
but how do you get it more precise? You can't measure a circle that perfectly. You have to calculate it. Madhava figured out that you could express pi not as a single division, but as an infinite chain of additions and subtractions of fractions, the famous one four minus 43 plus 45 minus 47 plus 49 and so on forever. That's the one that is the famous infinite series for pi. Now think about the logic of that. It's so beautiful. You are starting with a number, then subtracting a slightly smaller bit, then adding a tinier bit back, then subtracting an even tinier bit. You are oscillating around the true answer, getting closer and closer, with everything converging on the truth. If you could do it forever, you'd get the exact value of pi perfect precision. But you can't do it forever, no, but you can do it enough times to get to 13 decimal places of accuracy for pi, which is what Madhava and his school did. This is the bedrock of calculus. You are using the concept of infinity to solve a finite problem. And he didn't just do it for pi. He developed infinite series for trigonometric functions like sine, cosine and arctangent. And the source says this exact series for pi. It's now often called the Madhava Leibniz series appears in the work of Gottfried Leibniz in Europe almost 300 years later. Yes, and the series for sine and cosine predate Newton's work by a similar margin. Is that just a coincidence, or did the male just take a very long time to get from India to Germany? That's the multi million dollar question. It's highly unlikely to be a coincidence. We know that Jesuit missionaries were in Kerala in the 16th century. We know they were fascinated by the Brahmin sciences, translating Indian texts and sending detailed reports back to Europe so the knowledge could have traveled. There is no smoking gun letter that says, Dear Mr. Newton, here is the math for calculus regards Madhava. But the evidence for transmission of these ideas through trade, writs, through the Arab world and through these Jesuit reports, is very, very strong. It makes you wonder how many other discoveries in the European scientific revolution were actually rediscoveries or translations. It does. It suggests that the scientific revolution wasn't a purely European event that sprung out of nowhere. It was the culmination of a global relay race that had been going on for 1000 years. Let's zoom out to the so what of all this? Why were they doing this? We mentioned religion earlier, but usually we think of religion as opposing science. You know, Galileo versus the Pope. Here, it seems like religion.

Unknown Speaker  35:00  
Was the primary funding agency. Here's the R and D department. It provided the problems that needed solving. Look at the solva sutras, which are very ancient texts from around 800 500 BCE. These are essentially geometry manuals, but they aren't for building bridges or houses. They are for building fire altars, for Vedic rituals. Ritual altars, yeah, the Vedas prescribed very specific shapes for these altars, a falcon, a tortoise, a chariot wheel and the area of the altar had to be precise. It was spiritually important. So if the sacred instructions say, build a circular altar that has the exact same area as this pre existing square altar, you have a serious geometry problem on your hands. You've just squared the circle, which is famously impossible to do perfectly with a compass and straight edge, but trying to solve it forces you to get a really good understanding of pi and irrational numbers. So the drive for ritual precision drove geometric discovery. And then there's astrology, jeroticia, we make a hard separation between astronomy and astrology today, for most of history, they were the same discipline. They wanted to know where the planets would be in the future because they believed it influenced human destiny, the timing of the monsoon, the fate of the kingdom. So if you get the math wrong, you get the horoscope wrong, and the king fires you, or, you know, worse, exactly that need for accurate long term prediction drove the demand for planetary algorithms like the kutuka that were precise for centuries into the future. Superstition, or what we would call superstition, they would call spiritual belief, was the primary motivator for hard science. It's a beautiful irony, isn't it, the desire to know the future led them to develop the ability to calculate the celestial mechanics of the present with stunning accuracy. It brings us right back to that holistic worldview we started with. Math wasn't a dry, boring subject to be endured in school. It was the language of the Divine. It was the blueprint of the cosmos. It was how you saw God's architecture. As we wrap up this deep dive, I keep coming back to that Trudy, that 29.6 microsecond slice of time, and I'm picturing the Samrat Yantra, that 90 foot monument of stone, standing there in Jaipur. It's a journey from the completely invisible to the absolutely monumental. And here's my final thought, and it's a bit of a provoker for you to think about. The sources keep mentioning that they had these time units, microseconds, that they couldn't possibly physically measure. They derived them theoretically, through pure mathematics, right? It was a conceptual conclusion. My question is this, did they just get lucky with the math, or is there something fundamental about the human mind's ability to understand the structure of the universe that doesn't rely on technology? Were they through pure reason and meditation on numbers, tapping into a fundamental logic of reality that exists independent of our ability to observe it with instruments. That is the ultimate question of mathematics, isn't it? Is math discovered, or is it invented? The ancient Indians would have had no doubt. They would say it is discovered. It is the eternal truth, the Sanatana Dharma of the universe, and the numbers are just a window into it. They didn't need a particle accelerator to know that time was granular. At some level, they saw it in the patterns of the numbers themselves, and maybe that's the real lesson for us today. We get so obsessed with our tools, our supercomputers and our giant telescopes and our AI models, but the most powerful computer we have is still the one between our ears. And looking at what Aryabhata, Brahmagupta and mahava achieved with just that computer, it's deeply, deeply humbling. It certainly is. Thank you for taking this journey with us through the archives of time. My pleasure. It was fascinating and to you, the listener, next time you check your watch or look at a calendar, maybe remember the True Tea, remember the shadow on the stone, and remember the ancient thinkers who mapped the mathematical cosmos. Keep diving deep. We'll see you next time.

Transcribed by https://otter.ai