Mathematics has shaped every part of our world, from ancient tools to modern technology.
In this episode, we'll explore five revolutionary mathematical developments, from the invention of zero to the geometry of star forts and the vital role of double-entry bookkeeping.
[00:00:05] Hello, hello hello, and welcome to English Learning for Curious Minds, by Leonardo English.
[00:00:12] The show where you can listen to fascinating stories and learn weird and wonderful things about the world at the same time as improving your English.
[00:00:21] I'm Alastair Budge, and today is part three of our three-part mini-series on the theme of important scientific discovery.
[00:00:30] As a reminder, in case you missed them, part one was on the science of leap years and how humans found an elegant solution to the problem of the inexact orbit of the Earth around the sun.
[00:00:43] Part two was on the amazing story of the discovery of the structure of DNA, and how two men in particular tend to get all the credit, but the story is a little more complicated than that.
[00:00:57] And in part three, today’s episode, we are going to talk about how mathematics changed the world.
[00:01:04] So much of the world around us, arguably almost everything, can be explained through some form of mathematical discovery, and today we are going to talk about some of the most weird and wonderful.
[00:01:17] From Italian Renaissance forts to why the number zero was so revolutionary, we have a lot to get through, so let’s get right into it.
[00:01:28] The Greek mathematician Archimedes is once reported to have said, “Give me a place to stand on, and I will move the Earth”.
[00:01:39] It wasn’t a boast about his physical strength—there is no evidence to suggest he was any stronger or weaker than any other man of his time.
[00:01:49] Instead, it was a statement about a profound principle he had mathematically proven: leverage.
[00:01:58] Leverage is the idea that, by using a long enough lever and the right fulcrum [the point where the lever pivots], you can exert a small amount of force to move a much heavier object.
[00:02:15] Archimedes wasn’t the first person to “discover” this; anyone with a long piece of wood could see that they could put it under something heavy, and if they pressed down on the other end of it, the longer the distance between them and the thing they were trying to lift, the less force they needed to lift the object.
[00:02:36] Anyone could see that this was true, but Archimedes explained why it was true.
[00:02:45] He was able to prove how this works using the principles of geometry and balance, showing that the mechanics of levers were not just an observation but a predictable, universal truth governed by mathematics.
[00:03:03] Once he understood how it worked, there were practical implications.
[00:03:09] For example, he designed systems that allowed sailors to use ropes and pulley systems to lift up heavy objects more easily, thereby decreasing the time required to load and unload ships.
[00:03:24] It is just one example, but shows us the various steps that typically take place when mathematics interacts with the real world, or as a father and son pair of professors from the University of Stockholm and the University of Minnesota put it, “the calculating paradigm”.
[00:03:44] To start, there is a real-world problem to be analysed.
[00:03:49] In the case of the lever, anyone can observe the natural phenomenon; the longer the distance between the thing you are trying to lift and the force you are exerting on it, the less force you need.
[00:04:03] The first step is to measure this.
[00:04:07] What is the impact of changing the length of the lever, or adjusting the position of the fulcrum, the point on which the lever is balanced?
[00:04:16] When you change things, what happens?
[00:04:20] The second step is what these professors, the Malmbergs, called “manipulation”.
[00:04:27] In other words, you figure out the mathematical reason for what you observe.
[00:04:34] In the case of Archimedes and the lever, he observed that the relationship between the lengths of the lever arms and the force required was proportional.
[00:04:46] The longer the arm where force was applied compared to the shorter arm lifting the weight, the greater the mechanical advantage.
[00:04:54] This wasn’t just a rough guess—it was a precise mathematical relationship that could be calculated and predicted.
[00:05:04] This was what led Archimedes to famously quote that, with a lever long enough, he could move the Earth.
[00:05:12] It’s true, mathematically, at least, it is possible; he had proven it.
[00:05:18] And now the final step of this “calculating paradigm” is what the Malmbergs called “application”.
[00:05:27] In other words, how can this mathematical understanding be applied to the real world?
[00:05:34] In the case of Archimedes, it was by designing things like pulley systems, which worked on the principle of levers and allowed sailors to lift much heavier objects than they would have been able to had they just been trying to pull them up with ropes, without pulley systems.
[00:05:52] This is, of course, just one theoretical model to look at mathematical developments, but it is a useful one when it comes to thinking about the real-world application of mathematics, how this comes from real-world observations, and how mathematics is the science of transforming and explaining what we see so that we improve the world around us.
[00:06:18] So, with that in mind, let’s talk about some other ways in which mathematics changed the world.
[00:06:26] The first is a seemingly simple one. The invention of zero.
[00:06:31] Now, if you didn’t know about this before, you could be forgiven for thinking that it is a strange addition to this episode. Zero means nothing, any child can observe that you can have one apple, you can have two apples, you can have no apples, or “zero” apples.
[00:06:50] But while the concept of having “nothing” may seem obvious, the mathematical representation of nothing wasn’t always so clear.
[00:07:01] In fact, the invention of zero as a number was one of the most revolutionary moments in mathematical history.
[00:07:09] For much of human history, you see, people counted without zero.
[00:07:15] In ancient times, the Egyptians, Greeks, and Romans all managed without a proper zero.
[00:07:22] They sometimes left blank spaces or used placeholder symbols to mark gaps in their number systems, but this left plenty of room for confusion.
[00:07:34] Other civilisations, such as the Babylonians, used placeholders more systematically, and the Maya independently came up with zero.
[00:07:44] Still, the real breakthrough—treating zero not merely as a placeholder but as a fully-fledged number—happened in India.
[00:07:54] By the 7th century CE, the Indian mathematician Brahmagupta had formally defined zero, describing how it worked with addition, subtraction, multiplication, and division.
[00:08:08] This was revolutionary.
[00:08:10] Once you accept that zero is a number, you can write much bigger numbers more easily, and you can carry out more complex calculations—which was a key step toward developing advanced mathematics.
[00:08:25] To explain why this is so important, let’s take the example of the number 10.
[00:08:32] In the Roman numeral system, this was denominated by the letter X, whereas in the Hindu–Arabic numeral system, the number 10 is represented by two numbers: 1 and 0.
[00:08:47] The 1 represents one ten, and the 0 represents zero units.
[00:08:53] In the case of a small number like 10, it isn’t such a massive issue.
[00:08:58] But take a larger number. 1,906, for example.
[00:09:05] Writing that in Roman numerals is a real pain.
[00:09:08] It is MCMVI, M represents 1,000, C represents 100, M represents 1,000 again, and because it has a C in front of it, you take 100 away from 1,000 to give 900, then the V represents five, and the I represents 1, which gives you 1,906.
[00:09:31] Then, contrast this with writing it in decimals. 1906.
[00:09:38] No doubt, even a Roman would admit that the decimal system is significantly superior.
[00:09:44] And this is thanks to the invention of zero.
[00:09:48] But zero didn’t just help us to write numbers.
[00:09:52] It allowed much more efficient calculations, including algebra and calculus, and the ability to work with much larger numbers.
[00:10:02] And the binary system, the system of 1s and 0s used by almost all computers and electronic devices, well this would be impossible without zero.
[00:10:14] So, while zero might seem like nothing, nothing important at least, the reality is anything but.
[00:10:23] Now, moving on to our next mathematical development, let’s talk about angles and perspective.
[00:10:32] As anyone can observe, if you look at a road disappearing into the horizon, it seems to narrow and get smaller the farther it goes.
[00:10:43] Or, when you see a row of trees in a field, the ones in the distance appear much smaller than the ones closer to you, even though, in reality, they’re all the same size.
[00:10:57] These are things that we all observe with our eyes, we know they are true, but for centuries, there was no mathematical framework to explain why this happens or how to represent it accurately.
[00:11:13] This changed during the Italian Renaissance.
[00:11:17] Wealthy city-states like Florence, Venice, and Milan were flourishing, and their patronage of artists, architects, and scientists encouraged innovation and exploration of new ideas.
[00:11:31] At the same time, Italian merchants were heavily involved in trade with the Islamic world, where advanced mathematical concepts had been preserved and expanded upon after originating in ancient Greece and India.
[00:11:46] One key area of focus was geometry, the mathematical study of shapes and space.
[00:11:55] Renaissance thinkers built on the work of ancient Greek mathematicians like Euclid, whose Elements laid out the foundations of geometry.
[00:12:05] But instead of simply studying these concepts in the abstract, they sought to apply them to real-world problems, including art and architecture.
[00:12:18] It was the architect and engineer Filippo Brunelleschi who first formalised the rules of linear perspective, a mathematical method for creating the illusion of depth on a flat surface.
[00:12:33] Using geometry, Brunelleschi demonstrated that parallel lines appear to converge as they recede into the distance, meeting at a single vanishing point on the horizon.
[00:12:46] This principle explained why objects appear smaller the farther away they are, and for the first time, it allowed artists to depict realistic three-dimensional scenes on two-dimensional canvases.
[00:13:02] That’s why, sometime in the early 15th century, you start to see frescoes–wall paintings–that accurately depict perspective. Of course, artists had tried to do this before, they had tried to accurately represent what they could clearly see in front of them, but it wasn’t until Brunelleschi and his mathematical rules of perspective that artists had a guidebook for how to do it.
[00:13:32] But the importance of perspective went beyond art.
[00:13:36] It influenced architecture, urban planning, and even the way we think about space and representation. By combining observation with mathematical precision, linear perspective symbolised the Renaissance spirit of connecting science, art, and the human experience.
[00:13:57] Now, we will stick in Italy and stick to the theme of geometry for development number three, however, this development was not cultural or artistic but military.
[00:14:11] If I ask you to picture a castle, perhaps you might think about a large square building with high walls and a tower in the middle, often strategically built on top of a hill or on high ground.
[00:14:26] For a long time, this was a very effective design.
[00:14:31] Invading enemy forces had to approach the castle and either put ladders against the walls or smash down the door. The higher the castle walls, the greater the advantage the defenders would have; archers would be able to fire greater distances, you could pour all sorts of horrible stuff onto people trying to climb up, and climbing up a ladder and getting over the walls just took a lot longer to do.
[00:15:00] But gunpowder and the invention of cannons changed all of this.
[00:15:06] A big high wall was an easy target to hit, and if a castle wall got a direct hit from a cannonball, well, that wasn’t good news for your castle.
[00:15:18] So, how do mathematics and geometry come in?
[00:15:23] Well, mathematics and geometry completely redesigned how fortresses were built, transforming them into structures that could better withstand the power of cannons.
[00:15:36] This new design was called the Star fort, or the trace italienne, and it was a revolutionary approach to military defence.
[00:15:47] Instead of tall, vertical walls that were easy targets for cannon fire, star forts had lower, thicker walls that were much harder to damage.
[00:15:59] These walls were also angled in such a way that cannonballs would hit them at an angle, skimming the side and reducing the force of the impact.
[00:16:11] But the most striking feature was the star-like shape of the fortress.
[00:16:16] The star shape wasn’t just decorative—it was highly strategic.
[00:16:21] The walls of the fort stuck out in triangular points, creating what were called bastions. These bastions allowed defenders to fire at approaching enemies from multiple angles, covering areas that would have been blind spots in a traditional square castle.
[00:16:41] If an enemy tried to approach one part of the fort, they would be exposed to fire from two or three different directions at once.
[00:16:51] Surrounding the fort was often something called a glacis, a gently sloping hill that made it even harder for attackers to get close. This slope forced cannonballs to lose much of their energy before they even reached the walls. Between the glacis and the main walls were moats or open spaces, further slowing down attackers and making them easier to target.
[00:17:18] This new design meant that an attacking army couldn’t simply march up and bombard the walls—they had to deal with a network of defences that made their job far more complicated.
[00:17:31] Sieges became longer and much more difficult, which gave defenders more time to prepare and resist.
[00:17:38] And, given their superiority over traditional castle design, the geometry of star forts spread across Europe, and influenced fortress design for centuries.
[00:17:50] Now, for our fourth mathematical development, let’s cast our gaze skywards.
[00:17:57] As long as humans have had functioning eyes, we’ve observed the sun, moon, stars, and wandering planets and asked ourselves why they move the way they do.
[00:18:09] To come back to our “calculating paradigm”, it begins with observation and measurement, then moves on to mathematical manipulation, and finally, real-world application.
[00:18:22] But the drive to understand the heavens wasn’t just about cosmic curiosity; it also had a vital practical purpose: figuring out where on Earth we are.
[00:18:34] And nowhere was this harder than at sea.
[00:18:38] Without land-based markers, sailors had to rely on celestial objects, and while the idea of latitude was understood, accurately calculating it proved difficult.
[00:18:52] Early mariners used primitive star charts and tools to measure angles between the stars and the horizon, but their calculations could be cumbersome and they were prone to error.
[00:19:06] In the late 17th and early 18th centuries, however, two breakthroughs changed everything.
[00:19:14] First, mathematicians refined trigonometry and spherical geometry, building on centuries of work from Islamic and European scholars.
[00:19:24] For the first time, navigators had detailed tables and a sound theoretical framework to convert an angle measured on a spherical Earth into a precise location.
[00:19:39] Second, better printing methods allowed the widespread distribution of astronomical almanacs—carefully compiled lists of the sun’s, moon’s, and key stars’ positions for every day of the year.
[00:19:54] With one of these almanacs, and a sextant, the instrument sailors use to measure angles, a sailor could measure the angle between, let's say, the midday sun and the horizon, then compare it with the expected angle for any given date and latitude.
[00:20:13] By applying trigonometric formulas, they solved the geometry of the triangle formed by the horizon, the celestial object, and the observer’s position.
[00:20:24] With these calculations, it finally became possible to determine latitude with remarkable precision, even in the middle of the open ocean, surrounded by nothing but open water.
[00:20:38] This marriage of precise measurement and advanced mathematics was the reason 18th-century mariners gained far greater navigational confidence than their predecessors.
[00:20:49] Voyages became more predictable, trade routes multiplied, and the world became more interconnected than ever before.
[00:20:58] Of course, the separate and equally challenging problem of calculating longitude remained, but that’s a story for another day.
[00:21:07] Right then, mathematical development number five, the last one we will talk about today.
[00:21:13] It is something that probably sounds decidedly unsexy, but it was incredibly important. And that was the development of double-entry bookkeeping, which came about in the 15th century.
[00:21:28] The idea is simple, but it revolutionised how businesses kept track of money.
[00:21:36] It works like this. Every financial transaction is recorded in at least two accounts—once as a credit and once as a debit.
[00:21:47] For example, if a merchant sells goods, they record both the income from the sale and the reduction in inventory.
[00:21:57] This method provided businesses with a clear and reliable way to track profits, losses, and overall financial health.
[00:22:06] It's not known who exactly “invented it”, but it was formalised by an Italian monk named Luca Pacioli, who is often called the “father of accounting.”
[00:22:18] Again, unless you have a deep-seated passion for accounting, it might sound deadly boring, but it is hard to overstate its importance.
[00:22:28] It provided merchants with mathematical precision, giving them the tools to expand their operations, secure loans, and make long-term investments. This Italian monk arguably did more for capitalism and global economic growth than anyone has since.
[00:22:47] So, there we have it, levers, zeroes, linear perspective, star forts, and even double-entry bookkeeping.
[00:22:55] These are but a few of the many ways in which mathematics changed the world.
[00:23:03] OK then, that is it for today's episode on mathematics, and with that comes the end of this three-part mini-series on the theme of “scientific discovery”.
[00:23:13] In case you missed them, in part one, we had the science of leap years, and in part two, it was the discovery of DNA.
[00:23:21] And I should also say that this was a member request from a fascinating guy called Lorenzo, who asked if I would dedicate this episode to his brother, Francesco.
[00:23:32] So, Lorenzo, or indeed, Francesco, here you go, this one is for you.
[00:23:38] You've been listening to English Learning for Curious Minds by Leonardo English.
[00:23:43] I'm Alastair Budge, you stay safe, and I'll catch you in the next episode.
[00:00:05] Hello, hello hello, and welcome to English Learning for Curious Minds, by Leonardo English.
[00:00:12] The show where you can listen to fascinating stories and learn weird and wonderful things about the world at the same time as improving your English.
[00:00:21] I'm Alastair Budge, and today is part three of our three-part mini-series on the theme of important scientific discovery.
[00:00:30] As a reminder, in case you missed them, part one was on the science of leap years and how humans found an elegant solution to the problem of the inexact orbit of the Earth around the sun.
[00:00:43] Part two was on the amazing story of the discovery of the structure of DNA, and how two men in particular tend to get all the credit, but the story is a little more complicated than that.
[00:00:57] And in part three, today’s episode, we are going to talk about how mathematics changed the world.
[00:01:04] So much of the world around us, arguably almost everything, can be explained through some form of mathematical discovery, and today we are going to talk about some of the most weird and wonderful.
[00:01:17] From Italian Renaissance forts to why the number zero was so revolutionary, we have a lot to get through, so let’s get right into it.
[00:01:28] The Greek mathematician Archimedes is once reported to have said, “Give me a place to stand on, and I will move the Earth”.
[00:01:39] It wasn’t a boast about his physical strength—there is no evidence to suggest he was any stronger or weaker than any other man of his time.
[00:01:49] Instead, it was a statement about a profound principle he had mathematically proven: leverage.
[00:01:58] Leverage is the idea that, by using a long enough lever and the right fulcrum [the point where the lever pivots], you can exert a small amount of force to move a much heavier object.
[00:02:15] Archimedes wasn’t the first person to “discover” this; anyone with a long piece of wood could see that they could put it under something heavy, and if they pressed down on the other end of it, the longer the distance between them and the thing they were trying to lift, the less force they needed to lift the object.
[00:02:36] Anyone could see that this was true, but Archimedes explained why it was true.
[00:02:45] He was able to prove how this works using the principles of geometry and balance, showing that the mechanics of levers were not just an observation but a predictable, universal truth governed by mathematics.
[00:03:03] Once he understood how it worked, there were practical implications.
[00:03:09] For example, he designed systems that allowed sailors to use ropes and pulley systems to lift up heavy objects more easily, thereby decreasing the time required to load and unload ships.
[00:03:24] It is just one example, but shows us the various steps that typically take place when mathematics interacts with the real world, or as a father and son pair of professors from the University of Stockholm and the University of Minnesota put it, “the calculating paradigm”.
[00:03:44] To start, there is a real-world problem to be analysed.
[00:03:49] In the case of the lever, anyone can observe the natural phenomenon; the longer the distance between the thing you are trying to lift and the force you are exerting on it, the less force you need.
[00:04:03] The first step is to measure this.
[00:04:07] What is the impact of changing the length of the lever, or adjusting the position of the fulcrum, the point on which the lever is balanced?
[00:04:16] When you change things, what happens?
[00:04:20] The second step is what these professors, the Malmbergs, called “manipulation”.
[00:04:27] In other words, you figure out the mathematical reason for what you observe.
[00:04:34] In the case of Archimedes and the lever, he observed that the relationship between the lengths of the lever arms and the force required was proportional.
[00:04:46] The longer the arm where force was applied compared to the shorter arm lifting the weight, the greater the mechanical advantage.
[00:04:54] This wasn’t just a rough guess—it was a precise mathematical relationship that could be calculated and predicted.
[00:05:04] This was what led Archimedes to famously quote that, with a lever long enough, he could move the Earth.
[00:05:12] It’s true, mathematically, at least, it is possible; he had proven it.
[00:05:18] And now the final step of this “calculating paradigm” is what the Malmbergs called “application”.
[00:05:27] In other words, how can this mathematical understanding be applied to the real world?
[00:05:34] In the case of Archimedes, it was by designing things like pulley systems, which worked on the principle of levers and allowed sailors to lift much heavier objects than they would have been able to had they just been trying to pull them up with ropes, without pulley systems.
[00:05:52] This is, of course, just one theoretical model to look at mathematical developments, but it is a useful one when it comes to thinking about the real-world application of mathematics, how this comes from real-world observations, and how mathematics is the science of transforming and explaining what we see so that we improve the world around us.
[00:06:18] So, with that in mind, let’s talk about some other ways in which mathematics changed the world.
[00:06:26] The first is a seemingly simple one. The invention of zero.
[00:06:31] Now, if you didn’t know about this before, you could be forgiven for thinking that it is a strange addition to this episode. Zero means nothing, any child can observe that you can have one apple, you can have two apples, you can have no apples, or “zero” apples.
[00:06:50] But while the concept of having “nothing” may seem obvious, the mathematical representation of nothing wasn’t always so clear.
[00:07:01] In fact, the invention of zero as a number was one of the most revolutionary moments in mathematical history.
[00:07:09] For much of human history, you see, people counted without zero.
[00:07:15] In ancient times, the Egyptians, Greeks, and Romans all managed without a proper zero.
[00:07:22] They sometimes left blank spaces or used placeholder symbols to mark gaps in their number systems, but this left plenty of room for confusion.
[00:07:34] Other civilisations, such as the Babylonians, used placeholders more systematically, and the Maya independently came up with zero.
[00:07:44] Still, the real breakthrough—treating zero not merely as a placeholder but as a fully-fledged number—happened in India.
[00:07:54] By the 7th century CE, the Indian mathematician Brahmagupta had formally defined zero, describing how it worked with addition, subtraction, multiplication, and division.
[00:08:08] This was revolutionary.
[00:08:10] Once you accept that zero is a number, you can write much bigger numbers more easily, and you can carry out more complex calculations—which was a key step toward developing advanced mathematics.
[00:08:25] To explain why this is so important, let’s take the example of the number 10.
[00:08:32] In the Roman numeral system, this was denominated by the letter X, whereas in the Hindu–Arabic numeral system, the number 10 is represented by two numbers: 1 and 0.
[00:08:47] The 1 represents one ten, and the 0 represents zero units.
[00:08:53] In the case of a small number like 10, it isn’t such a massive issue.
[00:08:58] But take a larger number. 1,906, for example.
[00:09:05] Writing that in Roman numerals is a real pain.
[00:09:08] It is MCMVI, M represents 1,000, C represents 100, M represents 1,000 again, and because it has a C in front of it, you take 100 away from 1,000 to give 900, then the V represents five, and the I represents 1, which gives you 1,906.
[00:09:31] Then, contrast this with writing it in decimals. 1906.
[00:09:38] No doubt, even a Roman would admit that the decimal system is significantly superior.
[00:09:44] And this is thanks to the invention of zero.
[00:09:48] But zero didn’t just help us to write numbers.
[00:09:52] It allowed much more efficient calculations, including algebra and calculus, and the ability to work with much larger numbers.
[00:10:02] And the binary system, the system of 1s and 0s used by almost all computers and electronic devices, well this would be impossible without zero.
[00:10:14] So, while zero might seem like nothing, nothing important at least, the reality is anything but.
[00:10:23] Now, moving on to our next mathematical development, let’s talk about angles and perspective.
[00:10:32] As anyone can observe, if you look at a road disappearing into the horizon, it seems to narrow and get smaller the farther it goes.
[00:10:43] Or, when you see a row of trees in a field, the ones in the distance appear much smaller than the ones closer to you, even though, in reality, they’re all the same size.
[00:10:57] These are things that we all observe with our eyes, we know they are true, but for centuries, there was no mathematical framework to explain why this happens or how to represent it accurately.
[00:11:13] This changed during the Italian Renaissance.
[00:11:17] Wealthy city-states like Florence, Venice, and Milan were flourishing, and their patronage of artists, architects, and scientists encouraged innovation and exploration of new ideas.
[00:11:31] At the same time, Italian merchants were heavily involved in trade with the Islamic world, where advanced mathematical concepts had been preserved and expanded upon after originating in ancient Greece and India.
[00:11:46] One key area of focus was geometry, the mathematical study of shapes and space.
[00:11:55] Renaissance thinkers built on the work of ancient Greek mathematicians like Euclid, whose Elements laid out the foundations of geometry.
[00:12:05] But instead of simply studying these concepts in the abstract, they sought to apply them to real-world problems, including art and architecture.
[00:12:18] It was the architect and engineer Filippo Brunelleschi who first formalised the rules of linear perspective, a mathematical method for creating the illusion of depth on a flat surface.
[00:12:33] Using geometry, Brunelleschi demonstrated that parallel lines appear to converge as they recede into the distance, meeting at a single vanishing point on the horizon.
[00:12:46] This principle explained why objects appear smaller the farther away they are, and for the first time, it allowed artists to depict realistic three-dimensional scenes on two-dimensional canvases.
[00:13:02] That’s why, sometime in the early 15th century, you start to see frescoes–wall paintings–that accurately depict perspective. Of course, artists had tried to do this before, they had tried to accurately represent what they could clearly see in front of them, but it wasn’t until Brunelleschi and his mathematical rules of perspective that artists had a guidebook for how to do it.
[00:13:32] But the importance of perspective went beyond art.
[00:13:36] It influenced architecture, urban planning, and even the way we think about space and representation. By combining observation with mathematical precision, linear perspective symbolised the Renaissance spirit of connecting science, art, and the human experience.
[00:13:57] Now, we will stick in Italy and stick to the theme of geometry for development number three, however, this development was not cultural or artistic but military.
[00:14:11] If I ask you to picture a castle, perhaps you might think about a large square building with high walls and a tower in the middle, often strategically built on top of a hill or on high ground.
[00:14:26] For a long time, this was a very effective design.
[00:14:31] Invading enemy forces had to approach the castle and either put ladders against the walls or smash down the door. The higher the castle walls, the greater the advantage the defenders would have; archers would be able to fire greater distances, you could pour all sorts of horrible stuff onto people trying to climb up, and climbing up a ladder and getting over the walls just took a lot longer to do.
[00:15:00] But gunpowder and the invention of cannons changed all of this.
[00:15:06] A big high wall was an easy target to hit, and if a castle wall got a direct hit from a cannonball, well, that wasn’t good news for your castle.
[00:15:18] So, how do mathematics and geometry come in?
[00:15:23] Well, mathematics and geometry completely redesigned how fortresses were built, transforming them into structures that could better withstand the power of cannons.
[00:15:36] This new design was called the Star fort, or the trace italienne, and it was a revolutionary approach to military defence.
[00:15:47] Instead of tall, vertical walls that were easy targets for cannon fire, star forts had lower, thicker walls that were much harder to damage.
[00:15:59] These walls were also angled in such a way that cannonballs would hit them at an angle, skimming the side and reducing the force of the impact.
[00:16:11] But the most striking feature was the star-like shape of the fortress.
[00:16:16] The star shape wasn’t just decorative—it was highly strategic.
[00:16:21] The walls of the fort stuck out in triangular points, creating what were called bastions. These bastions allowed defenders to fire at approaching enemies from multiple angles, covering areas that would have been blind spots in a traditional square castle.
[00:16:41] If an enemy tried to approach one part of the fort, they would be exposed to fire from two or three different directions at once.
[00:16:51] Surrounding the fort was often something called a glacis, a gently sloping hill that made it even harder for attackers to get close. This slope forced cannonballs to lose much of their energy before they even reached the walls. Between the glacis and the main walls were moats or open spaces, further slowing down attackers and making them easier to target.
[00:17:18] This new design meant that an attacking army couldn’t simply march up and bombard the walls—they had to deal with a network of defences that made their job far more complicated.
[00:17:31] Sieges became longer and much more difficult, which gave defenders more time to prepare and resist.
[00:17:38] And, given their superiority over traditional castle design, the geometry of star forts spread across Europe, and influenced fortress design for centuries.
[00:17:50] Now, for our fourth mathematical development, let’s cast our gaze skywards.
[00:17:57] As long as humans have had functioning eyes, we’ve observed the sun, moon, stars, and wandering planets and asked ourselves why they move the way they do.
[00:18:09] To come back to our “calculating paradigm”, it begins with observation and measurement, then moves on to mathematical manipulation, and finally, real-world application.
[00:18:22] But the drive to understand the heavens wasn’t just about cosmic curiosity; it also had a vital practical purpose: figuring out where on Earth we are.
[00:18:34] And nowhere was this harder than at sea.
[00:18:38] Without land-based markers, sailors had to rely on celestial objects, and while the idea of latitude was understood, accurately calculating it proved difficult.
[00:18:52] Early mariners used primitive star charts and tools to measure angles between the stars and the horizon, but their calculations could be cumbersome and they were prone to error.
[00:19:06] In the late 17th and early 18th centuries, however, two breakthroughs changed everything.
[00:19:14] First, mathematicians refined trigonometry and spherical geometry, building on centuries of work from Islamic and European scholars.
[00:19:24] For the first time, navigators had detailed tables and a sound theoretical framework to convert an angle measured on a spherical Earth into a precise location.
[00:19:39] Second, better printing methods allowed the widespread distribution of astronomical almanacs—carefully compiled lists of the sun’s, moon’s, and key stars’ positions for every day of the year.
[00:19:54] With one of these almanacs, and a sextant, the instrument sailors use to measure angles, a sailor could measure the angle between, let's say, the midday sun and the horizon, then compare it with the expected angle for any given date and latitude.
[00:20:13] By applying trigonometric formulas, they solved the geometry of the triangle formed by the horizon, the celestial object, and the observer’s position.
[00:20:24] With these calculations, it finally became possible to determine latitude with remarkable precision, even in the middle of the open ocean, surrounded by nothing but open water.
[00:20:38] This marriage of precise measurement and advanced mathematics was the reason 18th-century mariners gained far greater navigational confidence than their predecessors.
[00:20:49] Voyages became more predictable, trade routes multiplied, and the world became more interconnected than ever before.
[00:20:58] Of course, the separate and equally challenging problem of calculating longitude remained, but that’s a story for another day.
[00:21:07] Right then, mathematical development number five, the last one we will talk about today.
[00:21:13] It is something that probably sounds decidedly unsexy, but it was incredibly important. And that was the development of double-entry bookkeeping, which came about in the 15th century.
[00:21:28] The idea is simple, but it revolutionised how businesses kept track of money.
[00:21:36] It works like this. Every financial transaction is recorded in at least two accounts—once as a credit and once as a debit.
[00:21:47] For example, if a merchant sells goods, they record both the income from the sale and the reduction in inventory.
[00:21:57] This method provided businesses with a clear and reliable way to track profits, losses, and overall financial health.
[00:22:06] It's not known who exactly “invented it”, but it was formalised by an Italian monk named Luca Pacioli, who is often called the “father of accounting.”
[00:22:18] Again, unless you have a deep-seated passion for accounting, it might sound deadly boring, but it is hard to overstate its importance.
[00:22:28] It provided merchants with mathematical precision, giving them the tools to expand their operations, secure loans, and make long-term investments. This Italian monk arguably did more for capitalism and global economic growth than anyone has since.
[00:22:47] So, there we have it, levers, zeroes, linear perspective, star forts, and even double-entry bookkeeping.
[00:22:55] These are but a few of the many ways in which mathematics changed the world.
[00:23:03] OK then, that is it for today's episode on mathematics, and with that comes the end of this three-part mini-series on the theme of “scientific discovery”.
[00:23:13] In case you missed them, in part one, we had the science of leap years, and in part two, it was the discovery of DNA.
[00:23:21] And I should also say that this was a member request from a fascinating guy called Lorenzo, who asked if I would dedicate this episode to his brother, Francesco.
[00:23:32] So, Lorenzo, or indeed, Francesco, here you go, this one is for you.
[00:23:38] You've been listening to English Learning for Curious Minds by Leonardo English.
[00:23:43] I'm Alastair Budge, you stay safe, and I'll catch you in the next episode.
[00:00:05] Hello, hello hello, and welcome to English Learning for Curious Minds, by Leonardo English.
[00:00:12] The show where you can listen to fascinating stories and learn weird and wonderful things about the world at the same time as improving your English.
[00:00:21] I'm Alastair Budge, and today is part three of our three-part mini-series on the theme of important scientific discovery.
[00:00:30] As a reminder, in case you missed them, part one was on the science of leap years and how humans found an elegant solution to the problem of the inexact orbit of the Earth around the sun.
[00:00:43] Part two was on the amazing story of the discovery of the structure of DNA, and how two men in particular tend to get all the credit, but the story is a little more complicated than that.
[00:00:57] And in part three, today’s episode, we are going to talk about how mathematics changed the world.
[00:01:04] So much of the world around us, arguably almost everything, can be explained through some form of mathematical discovery, and today we are going to talk about some of the most weird and wonderful.
[00:01:17] From Italian Renaissance forts to why the number zero was so revolutionary, we have a lot to get through, so let’s get right into it.
[00:01:28] The Greek mathematician Archimedes is once reported to have said, “Give me a place to stand on, and I will move the Earth”.
[00:01:39] It wasn’t a boast about his physical strength—there is no evidence to suggest he was any stronger or weaker than any other man of his time.
[00:01:49] Instead, it was a statement about a profound principle he had mathematically proven: leverage.
[00:01:58] Leverage is the idea that, by using a long enough lever and the right fulcrum [the point where the lever pivots], you can exert a small amount of force to move a much heavier object.
[00:02:15] Archimedes wasn’t the first person to “discover” this; anyone with a long piece of wood could see that they could put it under something heavy, and if they pressed down on the other end of it, the longer the distance between them and the thing they were trying to lift, the less force they needed to lift the object.
[00:02:36] Anyone could see that this was true, but Archimedes explained why it was true.
[00:02:45] He was able to prove how this works using the principles of geometry and balance, showing that the mechanics of levers were not just an observation but a predictable, universal truth governed by mathematics.
[00:03:03] Once he understood how it worked, there were practical implications.
[00:03:09] For example, he designed systems that allowed sailors to use ropes and pulley systems to lift up heavy objects more easily, thereby decreasing the time required to load and unload ships.
[00:03:24] It is just one example, but shows us the various steps that typically take place when mathematics interacts with the real world, or as a father and son pair of professors from the University of Stockholm and the University of Minnesota put it, “the calculating paradigm”.
[00:03:44] To start, there is a real-world problem to be analysed.
[00:03:49] In the case of the lever, anyone can observe the natural phenomenon; the longer the distance between the thing you are trying to lift and the force you are exerting on it, the less force you need.
[00:04:03] The first step is to measure this.
[00:04:07] What is the impact of changing the length of the lever, or adjusting the position of the fulcrum, the point on which the lever is balanced?
[00:04:16] When you change things, what happens?
[00:04:20] The second step is what these professors, the Malmbergs, called “manipulation”.
[00:04:27] In other words, you figure out the mathematical reason for what you observe.
[00:04:34] In the case of Archimedes and the lever, he observed that the relationship between the lengths of the lever arms and the force required was proportional.
[00:04:46] The longer the arm where force was applied compared to the shorter arm lifting the weight, the greater the mechanical advantage.
[00:04:54] This wasn’t just a rough guess—it was a precise mathematical relationship that could be calculated and predicted.
[00:05:04] This was what led Archimedes to famously quote that, with a lever long enough, he could move the Earth.
[00:05:12] It’s true, mathematically, at least, it is possible; he had proven it.
[00:05:18] And now the final step of this “calculating paradigm” is what the Malmbergs called “application”.
[00:05:27] In other words, how can this mathematical understanding be applied to the real world?
[00:05:34] In the case of Archimedes, it was by designing things like pulley systems, which worked on the principle of levers and allowed sailors to lift much heavier objects than they would have been able to had they just been trying to pull them up with ropes, without pulley systems.
[00:05:52] This is, of course, just one theoretical model to look at mathematical developments, but it is a useful one when it comes to thinking about the real-world application of mathematics, how this comes from real-world observations, and how mathematics is the science of transforming and explaining what we see so that we improve the world around us.
[00:06:18] So, with that in mind, let’s talk about some other ways in which mathematics changed the world.
[00:06:26] The first is a seemingly simple one. The invention of zero.
[00:06:31] Now, if you didn’t know about this before, you could be forgiven for thinking that it is a strange addition to this episode. Zero means nothing, any child can observe that you can have one apple, you can have two apples, you can have no apples, or “zero” apples.
[00:06:50] But while the concept of having “nothing” may seem obvious, the mathematical representation of nothing wasn’t always so clear.
[00:07:01] In fact, the invention of zero as a number was one of the most revolutionary moments in mathematical history.
[00:07:09] For much of human history, you see, people counted without zero.
[00:07:15] In ancient times, the Egyptians, Greeks, and Romans all managed without a proper zero.
[00:07:22] They sometimes left blank spaces or used placeholder symbols to mark gaps in their number systems, but this left plenty of room for confusion.
[00:07:34] Other civilisations, such as the Babylonians, used placeholders more systematically, and the Maya independently came up with zero.
[00:07:44] Still, the real breakthrough—treating zero not merely as a placeholder but as a fully-fledged number—happened in India.
[00:07:54] By the 7th century CE, the Indian mathematician Brahmagupta had formally defined zero, describing how it worked with addition, subtraction, multiplication, and division.
[00:08:08] This was revolutionary.
[00:08:10] Once you accept that zero is a number, you can write much bigger numbers more easily, and you can carry out more complex calculations—which was a key step toward developing advanced mathematics.
[00:08:25] To explain why this is so important, let’s take the example of the number 10.
[00:08:32] In the Roman numeral system, this was denominated by the letter X, whereas in the Hindu–Arabic numeral system, the number 10 is represented by two numbers: 1 and 0.
[00:08:47] The 1 represents one ten, and the 0 represents zero units.
[00:08:53] In the case of a small number like 10, it isn’t such a massive issue.
[00:08:58] But take a larger number. 1,906, for example.
[00:09:05] Writing that in Roman numerals is a real pain.
[00:09:08] It is MCMVI, M represents 1,000, C represents 100, M represents 1,000 again, and because it has a C in front of it, you take 100 away from 1,000 to give 900, then the V represents five, and the I represents 1, which gives you 1,906.
[00:09:31] Then, contrast this with writing it in decimals. 1906.
[00:09:38] No doubt, even a Roman would admit that the decimal system is significantly superior.
[00:09:44] And this is thanks to the invention of zero.
[00:09:48] But zero didn’t just help us to write numbers.
[00:09:52] It allowed much more efficient calculations, including algebra and calculus, and the ability to work with much larger numbers.
[00:10:02] And the binary system, the system of 1s and 0s used by almost all computers and electronic devices, well this would be impossible without zero.
[00:10:14] So, while zero might seem like nothing, nothing important at least, the reality is anything but.
[00:10:23] Now, moving on to our next mathematical development, let’s talk about angles and perspective.
[00:10:32] As anyone can observe, if you look at a road disappearing into the horizon, it seems to narrow and get smaller the farther it goes.
[00:10:43] Or, when you see a row of trees in a field, the ones in the distance appear much smaller than the ones closer to you, even though, in reality, they’re all the same size.
[00:10:57] These are things that we all observe with our eyes, we know they are true, but for centuries, there was no mathematical framework to explain why this happens or how to represent it accurately.
[00:11:13] This changed during the Italian Renaissance.
[00:11:17] Wealthy city-states like Florence, Venice, and Milan were flourishing, and their patronage of artists, architects, and scientists encouraged innovation and exploration of new ideas.
[00:11:31] At the same time, Italian merchants were heavily involved in trade with the Islamic world, where advanced mathematical concepts had been preserved and expanded upon after originating in ancient Greece and India.
[00:11:46] One key area of focus was geometry, the mathematical study of shapes and space.
[00:11:55] Renaissance thinkers built on the work of ancient Greek mathematicians like Euclid, whose Elements laid out the foundations of geometry.
[00:12:05] But instead of simply studying these concepts in the abstract, they sought to apply them to real-world problems, including art and architecture.
[00:12:18] It was the architect and engineer Filippo Brunelleschi who first formalised the rules of linear perspective, a mathematical method for creating the illusion of depth on a flat surface.
[00:12:33] Using geometry, Brunelleschi demonstrated that parallel lines appear to converge as they recede into the distance, meeting at a single vanishing point on the horizon.
[00:12:46] This principle explained why objects appear smaller the farther away they are, and for the first time, it allowed artists to depict realistic three-dimensional scenes on two-dimensional canvases.
[00:13:02] That’s why, sometime in the early 15th century, you start to see frescoes–wall paintings–that accurately depict perspective. Of course, artists had tried to do this before, they had tried to accurately represent what they could clearly see in front of them, but it wasn’t until Brunelleschi and his mathematical rules of perspective that artists had a guidebook for how to do it.
[00:13:32] But the importance of perspective went beyond art.
[00:13:36] It influenced architecture, urban planning, and even the way we think about space and representation. By combining observation with mathematical precision, linear perspective symbolised the Renaissance spirit of connecting science, art, and the human experience.
[00:13:57] Now, we will stick in Italy and stick to the theme of geometry for development number three, however, this development was not cultural or artistic but military.
[00:14:11] If I ask you to picture a castle, perhaps you might think about a large square building with high walls and a tower in the middle, often strategically built on top of a hill or on high ground.
[00:14:26] For a long time, this was a very effective design.
[00:14:31] Invading enemy forces had to approach the castle and either put ladders against the walls or smash down the door. The higher the castle walls, the greater the advantage the defenders would have; archers would be able to fire greater distances, you could pour all sorts of horrible stuff onto people trying to climb up, and climbing up a ladder and getting over the walls just took a lot longer to do.
[00:15:00] But gunpowder and the invention of cannons changed all of this.
[00:15:06] A big high wall was an easy target to hit, and if a castle wall got a direct hit from a cannonball, well, that wasn’t good news for your castle.
[00:15:18] So, how do mathematics and geometry come in?
[00:15:23] Well, mathematics and geometry completely redesigned how fortresses were built, transforming them into structures that could better withstand the power of cannons.
[00:15:36] This new design was called the Star fort, or the trace italienne, and it was a revolutionary approach to military defence.
[00:15:47] Instead of tall, vertical walls that were easy targets for cannon fire, star forts had lower, thicker walls that were much harder to damage.
[00:15:59] These walls were also angled in such a way that cannonballs would hit them at an angle, skimming the side and reducing the force of the impact.
[00:16:11] But the most striking feature was the star-like shape of the fortress.
[00:16:16] The star shape wasn’t just decorative—it was highly strategic.
[00:16:21] The walls of the fort stuck out in triangular points, creating what were called bastions. These bastions allowed defenders to fire at approaching enemies from multiple angles, covering areas that would have been blind spots in a traditional square castle.
[00:16:41] If an enemy tried to approach one part of the fort, they would be exposed to fire from two or three different directions at once.
[00:16:51] Surrounding the fort was often something called a glacis, a gently sloping hill that made it even harder for attackers to get close. This slope forced cannonballs to lose much of their energy before they even reached the walls. Between the glacis and the main walls were moats or open spaces, further slowing down attackers and making them easier to target.
[00:17:18] This new design meant that an attacking army couldn’t simply march up and bombard the walls—they had to deal with a network of defences that made their job far more complicated.
[00:17:31] Sieges became longer and much more difficult, which gave defenders more time to prepare and resist.
[00:17:38] And, given their superiority over traditional castle design, the geometry of star forts spread across Europe, and influenced fortress design for centuries.
[00:17:50] Now, for our fourth mathematical development, let’s cast our gaze skywards.
[00:17:57] As long as humans have had functioning eyes, we’ve observed the sun, moon, stars, and wandering planets and asked ourselves why they move the way they do.
[00:18:09] To come back to our “calculating paradigm”, it begins with observation and measurement, then moves on to mathematical manipulation, and finally, real-world application.
[00:18:22] But the drive to understand the heavens wasn’t just about cosmic curiosity; it also had a vital practical purpose: figuring out where on Earth we are.
[00:18:34] And nowhere was this harder than at sea.
[00:18:38] Without land-based markers, sailors had to rely on celestial objects, and while the idea of latitude was understood, accurately calculating it proved difficult.
[00:18:52] Early mariners used primitive star charts and tools to measure angles between the stars and the horizon, but their calculations could be cumbersome and they were prone to error.
[00:19:06] In the late 17th and early 18th centuries, however, two breakthroughs changed everything.
[00:19:14] First, mathematicians refined trigonometry and spherical geometry, building on centuries of work from Islamic and European scholars.
[00:19:24] For the first time, navigators had detailed tables and a sound theoretical framework to convert an angle measured on a spherical Earth into a precise location.
[00:19:39] Second, better printing methods allowed the widespread distribution of astronomical almanacs—carefully compiled lists of the sun’s, moon’s, and key stars’ positions for every day of the year.
[00:19:54] With one of these almanacs, and a sextant, the instrument sailors use to measure angles, a sailor could measure the angle between, let's say, the midday sun and the horizon, then compare it with the expected angle for any given date and latitude.
[00:20:13] By applying trigonometric formulas, they solved the geometry of the triangle formed by the horizon, the celestial object, and the observer’s position.
[00:20:24] With these calculations, it finally became possible to determine latitude with remarkable precision, even in the middle of the open ocean, surrounded by nothing but open water.
[00:20:38] This marriage of precise measurement and advanced mathematics was the reason 18th-century mariners gained far greater navigational confidence than their predecessors.
[00:20:49] Voyages became more predictable, trade routes multiplied, and the world became more interconnected than ever before.
[00:20:58] Of course, the separate and equally challenging problem of calculating longitude remained, but that’s a story for another day.
[00:21:07] Right then, mathematical development number five, the last one we will talk about today.
[00:21:13] It is something that probably sounds decidedly unsexy, but it was incredibly important. And that was the development of double-entry bookkeeping, which came about in the 15th century.
[00:21:28] The idea is simple, but it revolutionised how businesses kept track of money.
[00:21:36] It works like this. Every financial transaction is recorded in at least two accounts—once as a credit and once as a debit.
[00:21:47] For example, if a merchant sells goods, they record both the income from the sale and the reduction in inventory.
[00:21:57] This method provided businesses with a clear and reliable way to track profits, losses, and overall financial health.
[00:22:06] It's not known who exactly “invented it”, but it was formalised by an Italian monk named Luca Pacioli, who is often called the “father of accounting.”
[00:22:18] Again, unless you have a deep-seated passion for accounting, it might sound deadly boring, but it is hard to overstate its importance.
[00:22:28] It provided merchants with mathematical precision, giving them the tools to expand their operations, secure loans, and make long-term investments. This Italian monk arguably did more for capitalism and global economic growth than anyone has since.
[00:22:47] So, there we have it, levers, zeroes, linear perspective, star forts, and even double-entry bookkeeping.
[00:22:55] These are but a few of the many ways in which mathematics changed the world.
[00:23:03] OK then, that is it for today's episode on mathematics, and with that comes the end of this three-part mini-series on the theme of “scientific discovery”.
[00:23:13] In case you missed them, in part one, we had the science of leap years, and in part two, it was the discovery of DNA.
[00:23:21] And I should also say that this was a member request from a fascinating guy called Lorenzo, who asked if I would dedicate this episode to his brother, Francesco.
[00:23:32] So, Lorenzo, or indeed, Francesco, here you go, this one is for you.
[00:23:38] You've been listening to English Learning for Curious Minds by Leonardo English.
[00:23:43] I'm Alastair Budge, you stay safe, and I'll catch you in the next episode.
