WEBVTT 00:00:00.000 --> 00:00:03.520 Imagine being the lead architect of like a 100 00:00:03.520 --> 00:00:07.519 story modern soaring skyscraper Okay, or trying 00:00:07.519 --> 00:00:10.259 to engineer the aerodynamics of a massive cargo 00:00:10.259 --> 00:00:12.880 ship or even you know Just sitting down to write 00:00:12.880 --> 00:00:17.079 a basic piece of software But here's the catch 00:00:17.079 --> 00:00:20.160 you are not allowed to use numbers Wow good luck 00:00:20.160 --> 00:00:22.899 with that right you have no alphabet You have 00:00:22.899 --> 00:00:25.800 no standardized symbols of any kind and the crazy 00:00:25.800 --> 00:00:28.960 thing is 300 years ago That was basically the 00:00:28.960 --> 00:00:30.980 state of mathematics. It really was. It was just 00:00:30.980 --> 00:00:34.460 a chaotic mess of competing languages until one 00:00:34.460 --> 00:00:36.600 man sat down and essentially wrote the source 00:00:36.600 --> 00:00:41.460 code for reality. And the wildest part. He did 00:00:41.460 --> 00:00:43.600 his absolute best work after going completely 00:00:43.600 --> 00:00:46.000 blind. It sounds like an impossible premise for 00:00:46.000 --> 00:00:48.159 a life, doesn't it? I mean, you can't build complex 00:00:48.159 --> 00:00:50.119 structures, whether they're physical buildings, 00:00:50.460 --> 00:00:53.579 digital networks, or abstract theoretical frameworks. 00:00:53.719 --> 00:00:56.679 If you don't first have a reliable way to communicate 00:00:56.679 --> 00:00:59.240 the foundational concepts, you need a shared 00:00:59.240 --> 00:01:00.799 grammar. And the man we are talking about today 00:01:00.799 --> 00:01:03.359 didn't just learn the grammar, he invented it. 00:01:03.579 --> 00:01:05.420 Which brings us perfectly to our mission for 00:01:05.420 --> 00:01:08.359 today's deep dive. we are looking at a massive, 00:01:08.700 --> 00:01:11.719 comprehensive, encyclopedic entry on Leonhard 00:01:11.719 --> 00:01:14.180 Euler. A giant of history. Born in Switzerland 00:01:14.180 --> 00:01:17.680 in 1707. And Euler is the man who standardized 00:01:17.680 --> 00:01:20.239 the language of math and science. As I was going 00:01:20.239 --> 00:01:22.019 through these sources, I was trying to wrap my 00:01:22.019 --> 00:01:24.599 head around the, well, the sheer volume of his 00:01:24.599 --> 00:01:26.680 output. It's hard to even comprehend. Right. 00:01:26.900 --> 00:01:32.239 Between 1725 and 1783, Euler averaged 800 pages 00:01:32.239 --> 00:01:36.370 of published work a year. 800 pages. Every year. 00:01:36.530 --> 00:01:39.209 It is staggering. He offered roughly a quarter 00:01:39.209 --> 00:01:41.849 of all the combined output in mathematics, physics, 00:01:42.090 --> 00:01:44.489 mechanics, astronomy, and navigation in the entire 00:01:44.489 --> 00:01:47.209 18th century. A quarter of everything. Yes. But 00:01:47.209 --> 00:01:49.370 what makes Euler so captivating isn't just the 00:01:49.370 --> 00:01:52.310 page count. It is the contrast of his life. Oh, 00:01:52.310 --> 00:01:54.829 so. Well, on one hand, you have his infinite, 00:01:54.969 --> 00:01:57.250 perfectly ordered, brilliant abstract mind. And 00:01:57.250 --> 00:01:59.349 then on the other, you have the incredibly messy, 00:01:59.549 --> 00:02:01.810 physical, and frankly, highly volatile political 00:02:01.810 --> 00:02:03.950 world he was forced to navigate every day. OK, 00:02:03.950 --> 00:02:06.640 let's unpack this. Because to understand what 00:02:06.640 --> 00:02:09.860 Euler built, we really have to look at this alphabet 00:02:09.860 --> 00:02:13.340 of modern math that he gave us. Right. If you 00:02:13.340 --> 00:02:15.819 listening to this right now have ever taken a 00:02:15.819 --> 00:02:18.680 high school algebra or calculus class, you're 00:02:18.680 --> 00:02:23.439 speaking Euler's language. He popularized or, 00:02:23.439 --> 00:02:26.479 you know, outright invented notations that we 00:02:26.479 --> 00:02:28.520 just take for absolute granted today. We don't 00:02:28.520 --> 00:02:30.639 even think about them. Exactly. Like, he was 00:02:30.639 --> 00:02:33.780 the first to use the notation f of x to denote 00:02:33.780 --> 00:02:36.039 the value of a function. Yes, and it didn't stop 00:02:36.039 --> 00:02:38.759 there. He gave us the letter e for the base of 00:02:38.759 --> 00:02:41.199 the natural logarithm. He started using the letter 00:02:41.199 --> 00:02:44.759 i to express imaginary numbers. Wow. And he introduced 00:02:44.759 --> 00:02:47.620 the Greek letter sigma for mathematical summations. 00:02:47.780 --> 00:02:50.990 And perhaps most famously, He popularized using 00:02:50.990 --> 00:02:53.870 the Greek letter pi to denote the ratio of a 00:02:53.870 --> 00:02:55.830 circle's circumference to its diameter. Right, 00:02:55.870 --> 00:02:58.090 the classic pi. Now, the sources do note that 00:02:58.090 --> 00:03:00.849 a mathematician named William Jones technically 00:03:00.849 --> 00:03:03.909 used pi in that context first. Yeah. But it was 00:03:03.909 --> 00:03:06.110 Euler who put it in his widely circulated textbooks 00:03:06.110 --> 00:03:07.990 and really cemented it in the global consciousness. 00:03:08.129 --> 00:03:10.770 He made it the standard. Right. And the analogy 00:03:10.770 --> 00:03:13.030 that kept hitting me while reading this is that 00:03:13.030 --> 00:03:15.710 Euler is basically a foundational software developer. 00:03:15.770 --> 00:03:17.810 Oh, that's interesting. How do you mean? Well, 00:03:17.870 --> 00:03:20.409 he didn't just write a really great app, like 00:03:20.409 --> 00:03:23.449 a single brilliant theorem that solved one specific 00:03:23.449 --> 00:03:26.629 problem. He invented the coding language itself. 00:03:26.689 --> 00:03:30.669 I see. He wrote the C++ or the Python of mathematics. 00:03:31.389 --> 00:03:33.840 Every other developer, every... physicist, every 00:03:33.840 --> 00:03:36.599 structural engineer for the next 300 years, was 00:03:36.599 --> 00:03:39.599 essentially forced to use Euler's coding language 00:03:39.599 --> 00:03:41.719 to build their own apps. What's fascinating here 00:03:41.719 --> 00:03:43.879 is the collaborative compounding effect of that 00:03:43.879 --> 00:03:46.819 standardization, because before Euler, mathematicians 00:03:46.819 --> 00:03:49.840 in France might be using entirely different notation 00:03:49.840 --> 00:03:52.580 than mathematicians in England or Germany. It 00:03:52.580 --> 00:03:55.840 was just completely fragmented. Exactly. By standardizing 00:03:55.840 --> 00:03:58.659 this mathematical grammar, Euler allowed humanity 00:03:58.659 --> 00:04:02.139 to finally collaborate across borders and, crucially, 00:04:01.930 --> 00:04:04.430 across centuries. Yeah, that's a huge shift. 00:04:04.789 --> 00:04:07.949 You simply cannot have a global scientific revolution 00:04:07.949 --> 00:04:10.229 if everyone is speaking a different theoretical 00:04:10.229 --> 00:04:13.069 language. So he builds this alphabet. But an 00:04:13.069 --> 00:04:15.650 alphabet is just a tool, right? How did he actually 00:04:15.650 --> 00:04:18.689 use it to change mathematics? That's where things 00:04:18.689 --> 00:04:21.230 get really incredible. Because the sources talk 00:04:21.230 --> 00:04:24.470 a lot about Euler's identity, or Euler's formula, 00:04:25.149 --> 00:04:28.449 which uses his own standardized notations to 00:04:28.449 --> 00:04:32.589 establish that E to the power of i times pi plus 00:04:32.589 --> 00:04:35.829 one equals zero. That is correct. e to the i 00:04:35.829 --> 00:04:39.209 pi plus one equals zero. The physicist Richard 00:04:39.209 --> 00:04:42.089 Feynman famously called it the most remarkable 00:04:42.089 --> 00:04:44.529 formula in mathematics. Hold on. I need to visualize 00:04:44.529 --> 00:04:46.269 this because for a lot of people that just sounds 00:04:46.269 --> 00:04:48.389 like a random string of math words. Fair enough. 00:04:48.490 --> 00:04:51.529 You're taking in a rational number. e raising 00:04:51.529 --> 00:04:53.730 it to the power of an imaginary number multiplied 00:04:53.730 --> 00:04:56.269 by another irrational number pi, and somehow 00:04:56.269 --> 00:04:58.509 that perfectly equals negative one. So when you 00:04:58.509 --> 00:05:01.009 add one, you get zero. It sounds like magic, 00:05:01.189 --> 00:05:03.329 doesn't it? How is that even mathematically possible? 00:05:03.529 --> 00:05:06.589 It is profoundly elegant once you see the mechanism 00:05:06.589 --> 00:05:08.430 behind it. Let's break it down simply. Okay, 00:05:08.529 --> 00:05:11.949 please do. So the number E deals with continuous 00:05:11.949 --> 00:05:15.370 compounding growth, and the letter I represents 00:05:15.370 --> 00:05:18.250 imaginary numbers. Now, instead of thinking of 00:05:18.250 --> 00:05:20.850 I as just some weird, impossible number, think 00:05:20.850 --> 00:05:23.529 of it spatially. Spatially, like on a graph. 00:05:23.649 --> 00:05:25.529 Exactly. If you have a normal number line going 00:05:25.529 --> 00:05:28.589 left to right, multiplying by I is like rotating 00:05:28.589 --> 00:05:31.329 90 degrees upward into a new vertical dimension. 00:05:31.889 --> 00:05:34.410 This creates what we call the complex plane. 00:05:34.750 --> 00:05:37.290 OK, so I is basically like a rotational instruction. 00:05:37.449 --> 00:05:40.170 Yes. Exactly. And then you have pi, which of 00:05:40.170 --> 00:05:42.290 course relates to circles. Right. So when you 00:05:42.290 --> 00:05:45.829 take compounding growth, E, and apply this rotational 00:05:45.829 --> 00:05:48.589 instruction, I, for exactly pi amount of time, 00:05:48.629 --> 00:05:50.990 which is halfway around a circle. Oh, wow. You 00:05:50.990 --> 00:05:53.810 start at positive 1 on the number line, rotate 00:05:53.810 --> 00:05:56.209 180 degrees through this alternate dimension, 00:05:56.550 --> 00:05:59.370 and land precisely on negative 1. Add 1, and 00:05:59.370 --> 00:06:02.480 you get 0. That is wild. It bridges exponential 00:06:02.480 --> 00:06:05.279 growth, circular geometry, and the complex plane 00:06:05.279 --> 00:06:08.399 all in one impossibly simple equation. It's just 00:06:08.399 --> 00:06:11.000 perfectly neat. It's as if you took the five 00:06:11.000 --> 00:06:14.360 fundamental constants of reality zero, one, pi, 00:06:14.399 --> 00:06:16.959 e, and i, and discovered they all fit together 00:06:16.959 --> 00:06:19.819 into a single flawless puzzle piece. So he has 00:06:19.819 --> 00:06:22.639 this unique ability to strip away the noise and 00:06:22.639 --> 00:06:25.379 find these deep underlying connections, which 00:06:25.379 --> 00:06:27.639 makes me want to look at how he applied that 00:06:27.639 --> 00:06:30.839 same vision to problems that had stumped everyone 00:06:30.839 --> 00:06:35.079 else. He had this habit of taking real -world 00:06:35.079 --> 00:06:37.720 dead ends and turning them into abstract games. 00:06:38.000 --> 00:06:40.819 Yes, his ability to look at an unsolvable problem 00:06:40.819 --> 00:06:43.560 and completely reframe it was unparalleled. Let's 00:06:43.560 --> 00:06:45.399 talk about the Basel problem first because this 00:06:45.399 --> 00:06:48.199 one drove mathematicians absolutely crazy. Hawaii 00:06:48.199 --> 00:06:50.879 was the bane of the mathematical community for 00:06:50.879 --> 00:06:53.279 a long time. For almost a century, the greatest 00:06:53.279 --> 00:06:55.500 minds in Europe had been totally stumped by it. 00:06:55.939 --> 00:06:59.040 The challenge was to find the exact sum of the 00:06:59.040 --> 00:07:01.850 reciprocals of the squares of every natural number. 00:07:01.870 --> 00:07:04.529 Which is a mouthful. Yeah. So 1 over 1 squared 00:07:04.529 --> 00:07:07.410 plus 1 over 2 squared plus 1 over 3 squared onto 00:07:07.410 --> 00:07:10.250 infinity. No one could figure out exactly what 00:07:10.250 --> 00:07:12.709 that infinite series converged to. Nobody. And 00:07:12.709 --> 00:07:16.050 then Euler comes along in 1735 and proves it 00:07:16.050 --> 00:07:19.189 equals pi squared divided by 6, which is an amazing 00:07:19.189 --> 00:07:22.649 flex. But where does pi come from in a list of 00:07:22.649 --> 00:07:25.149 squares? It is a brilliant question, and the 00:07:25.149 --> 00:07:27.990 answer shows exactly how Euler's mind worked. 00:07:28.730 --> 00:07:31.290 Euler looked at this infinite sum of fractions, 00:07:31.689 --> 00:07:34.779 and he essentially asked... What if I treat this 00:07:34.779 --> 00:07:38.839 endless addition problem like a giant algebraic 00:07:38.839 --> 00:07:41.040 polynomial? A polynomial. Now in algebra, if 00:07:41.040 --> 00:07:42.980 you have a polynomial, you can break it down 00:07:42.980 --> 00:07:45.319 into its roots, the points where the equation 00:07:45.319 --> 00:07:48.579 crosses zero. Right, like finding x when y is 00:07:48.579 --> 00:07:51.879 zero. Exactly. Then Euler looked around for something 00:07:51.879 --> 00:07:55.120 in nature that crosses zero infinitely many times. 00:07:55.199 --> 00:07:57.540 Like a wave. And he found it in the sine wave. 00:07:57.600 --> 00:08:00.579 A sine wave oscillates up and down forever, crossing 00:08:00.579 --> 00:08:03.389 zero exactly at multiples of pi. one pi, two 00:08:03.389 --> 00:08:06.350 pi, three pi, and so on. Oh, because sine deals 00:08:06.350 --> 00:08:09.050 with circles and angles. Precisely. Euler set 00:08:09.050 --> 00:08:11.750 the infinite polynomial of those fractions equal 00:08:11.750 --> 00:08:14.009 to the infinite series that describes a sine 00:08:14.009 --> 00:08:16.449 wave. That is so clever. By matching the mathematical 00:08:16.449 --> 00:08:18.329 parts of the waves to the parts of the fractions, 00:08:18.750 --> 00:08:21.290 the roots of the sine wave, which are built entirely 00:08:21.290 --> 00:08:23.810 out of pi, suddenly dropped right into the equation. 00:08:24.069 --> 00:08:26.730 Wow. He proved that an infinite sum of squares 00:08:26.730 --> 00:08:29.209 is secretly describing the geometry of a circle. 00:08:29.589 --> 00:08:32.799 That is genuinely beautiful. He finds the hidden 00:08:32.799 --> 00:08:37.460 architecture, but his knack for solving the unsolvable 00:08:37.460 --> 00:08:39.860 wasn't just limited to pure number theory. Not 00:08:39.860 --> 00:08:41.740 at all. He applied this to the physical world 00:08:41.740 --> 00:08:44.419 too, which brings us to the seven bridges of 00:08:44.419 --> 00:08:47.379 Konigsberg. Yes, a very famous problem. Konigsberg 00:08:47.379 --> 00:08:50.840 was a real city in Prussia. It was set on a river 00:08:50.840 --> 00:08:54.740 and it had two large islands. These islands were 00:08:54.740 --> 00:08:56.600 connected to each other and to the mainland. 00:08:57.000 --> 00:09:00.360 by exactly seven bridges. Right. And the citizens 00:09:00.360 --> 00:09:03.379 had this local riddle. Could you walk a continuous 00:09:03.379 --> 00:09:06.299 path through the city that crossed each of the 00:09:06.299 --> 00:09:09.179 seven bridges exactly once? A seemingly innocent 00:09:09.179 --> 00:09:11.159 puzzle. But I have to push back here, or at least 00:09:11.159 --> 00:09:12.779 ask the question that I think anyone listening 00:09:12.779 --> 00:09:15.080 would have. Sure. The Konigsberg Bridge problem 00:09:15.080 --> 00:09:18.159 sounds like a simple, you know, distraction puzzle 00:09:18.159 --> 00:09:19.759 you'd find on the back of a kid's restaurant 00:09:19.759 --> 00:09:22.879 menu. Like a maze. Yeah. Like, can you draw a 00:09:22.879 --> 00:09:25.059 line through this maze without picking up your 00:09:25.059 --> 00:09:28.340 crown? Why is the greatest mathematician of the 00:09:28.340 --> 00:09:30.980 18th century wasting his precious time on a local 00:09:30.980 --> 00:09:33.840 walking riddle? It is a completely fair question. 00:09:34.299 --> 00:09:36.539 And the answer is that Euler didn't just solve 00:09:36.539 --> 00:09:39.879 the puzzle. He proved the puzzle was fundamentally 00:09:39.879 --> 00:09:42.620 rigged by looking at the invisible rules governing 00:09:42.620 --> 00:09:46.000 it. Rigged how? Well, if we connect this to the 00:09:46.000 --> 00:09:49.139 bigger picture, Euler realized a very simple, 00:09:49.419 --> 00:09:52.100 undeniable rule of movement. If you walk onto 00:09:52.100 --> 00:09:55.639 a landmass by crossing one bridge, You have to 00:09:55.639 --> 00:09:58.240 leave that landmass by crossing a different bridge. 00:09:58.539 --> 00:10:01.039 Right. One bridge in, one bridge out. Exactly. 00:10:01.379 --> 00:10:04.039 That means every single landmass in the city 00:10:04.039 --> 00:10:07.000 must have an even number of bridges attached 00:10:07.000 --> 00:10:10.220 to it, pairs of ins and outs, unless it is the 00:10:10.220 --> 00:10:12.460 absolute start of your journey or the absolute 00:10:12.460 --> 00:10:15.590 end of your journey. Oh, wow. Let me guess, Konigsberg 00:10:15.590 --> 00:10:18.129 didn't have even numbers. Not a single one. All 00:10:18.129 --> 00:10:20.649 four land masses in Konigsberg had an odd number 00:10:20.649 --> 00:10:23.029 of bridges connecting to them. So it was impossible 00:10:23.029 --> 00:10:25.330 from the start? Exactly. By identifying this 00:10:25.330 --> 00:10:27.610 invisible law, Euler proved that no such path 00:10:27.610 --> 00:10:30.370 could ever exist. The physical distances between 00:10:30.370 --> 00:10:32.389 the bridges didn't matter. The size of the islands 00:10:32.389 --> 00:10:34.809 didn't matter. He just ignored all the real world 00:10:34.809 --> 00:10:37.350 stuff? He stripped away the actual geography 00:10:37.350 --> 00:10:40.309 of the city, reducing the land masses to simple 00:10:40.309 --> 00:10:43.759 points. what we now call vertices, and the bridges 00:10:43.759 --> 00:10:46.580 to simple lines connecting them, which we call 00:10:46.580 --> 00:10:49.559 edges. He threw out the map entirely. He did. 00:10:49.860 --> 00:10:52.379 And in doing so, he didn't just solve a riddle, 00:10:52.480 --> 00:10:55.460 he invented graph theory. Which is the foundation 00:10:55.460 --> 00:10:58.120 of so much of our modern world. I mean, it's 00:10:58.120 --> 00:11:00.379 modern computer science, it's network analysis, 00:11:00.879 --> 00:11:03.460 it's literally how your GPS routes you through 00:11:03.460 --> 00:11:06.580 traffic. by treating intersections as vertices 00:11:06.580 --> 00:11:09.740 and roads as edges. Exactly. By stripping away 00:11:09.740 --> 00:11:12.340 the physical distractions, Euler also laid the 00:11:12.340 --> 00:11:15.019 groundwork for topology. What exactly is that? 00:11:15.179 --> 00:11:17.059 That is the mathematical study of properties 00:11:17.059 --> 00:11:20.100 that are preserved through deformation. In topology, 00:11:20.120 --> 00:11:23.419 you can stretch or twist or bend an object like 00:11:23.419 --> 00:11:26.240 turning a coffee mug into a donut shape without 00:11:26.240 --> 00:11:28.700 breaking its fundamental mathematical properties. 00:11:28.840 --> 00:11:31.240 It requires an incredible amount of mental visualization 00:11:31.240 --> 00:11:34.029 to be able to look at a bustling physical city 00:11:34.029 --> 00:11:37.029 and abstract it into pure floating vertices and 00:11:37.029 --> 00:11:39.549 edges in your mind. It really does. But the profound 00:11:39.549 --> 00:11:43.090 almost cruel irony of Euler's life is that as 00:11:43.090 --> 00:11:46.470 his internal mental vision was expanding, his 00:11:46.470 --> 00:11:49.460 actual physical vision was failing. It is one 00:11:49.460 --> 00:11:51.600 of the most poignant aspects of his biography. 00:11:52.120 --> 00:11:55.340 His eyesight deteriorated relentlessly throughout 00:11:55.340 --> 00:11:58.679 his career. By 1738, he was almost completely 00:11:58.679 --> 00:12:01.879 blind in his right eye. He blamed it on a severe 00:12:01.879 --> 00:12:04.940 fever he'd had and, you know, the incredibly 00:12:04.940 --> 00:12:07.460 straining cartography work he was doing, making 00:12:07.460 --> 00:12:09.980 maps for the St. Petersburg Academy. A very demanding 00:12:09.980 --> 00:12:12.200 job. The king of Prussia, Frederick the Great, 00:12:12.340 --> 00:12:14.700 used to actually mock him and call him Cyclops. 00:12:14.889 --> 00:12:17.830 Which is just awful. Yeah. And then years later, 00:12:18.169 --> 00:12:20.950 he develops a cataract in his left eye, a botched 00:12:20.950 --> 00:12:23.330 surgery follows, and he is left almost totally 00:12:23.330 --> 00:12:25.669 blind. For a mathematician who works with highly 00:12:25.669 --> 00:12:28.750 complex equations, massive polynomials, and geometric 00:12:28.750 --> 00:12:31.250 proofs, you would expect total blindness to be 00:12:31.250 --> 00:12:33.370 a career -ending tragedy. Right. But here's where 00:12:33.370 --> 00:12:35.429 it gets really interesting. When most people 00:12:35.429 --> 00:12:37.990 lose a vital sense, their output drops. They 00:12:37.990 --> 00:12:40.509 retreat. Naturally. But when Euler lost his sight, 00:12:40.669 --> 00:12:43.470 he famously said, now I will have fewer distractions. 00:12:43.769 --> 00:12:46.649 And the sources show that his productivity didn't 00:12:46.649 --> 00:12:50.809 just hold steady. It skyrocketed. By 1775, he 00:12:50.809 --> 00:12:53.389 was entirely blind, and he was dictating roughly 00:12:53.389 --> 00:12:56.370 one mathematical paper a week to his scribes. 00:12:56.649 --> 00:12:59.830 One paper a week at the highest possible level 00:12:59.830 --> 00:13:02.350 of theoretical mathematics. I found myself comparing 00:13:02.350 --> 00:13:05.529 his mind to an unparalleled offline supercomputer. 00:13:05.870 --> 00:13:08.090 He couldn't look anything up anymore, so he just 00:13:08.090 --> 00:13:10.429 stored reality in his head. A perfect way to 00:13:10.429 --> 00:13:13.090 describe it. And how he organized it is fascinating. 00:13:13.399 --> 00:13:16.080 The sources note that he had memorized the entirety 00:13:16.080 --> 00:13:19.039 of Virgil's epic poem, the Aeneid, but not just 00:13:19.039 --> 00:13:21.620 the words. He could recite the first and last 00:13:21.620 --> 00:13:24.440 sentence of any page from the exact edition he 00:13:24.440 --> 00:13:27.179 read as a kid. That is photographic memory on 00:13:27.179 --> 00:13:29.360 another level. He had a perfectly constructed 00:13:29.360 --> 00:13:32.600 mental desktop. He knew the first 100 prime numbers 00:13:32.600 --> 00:13:34.899 perfectly, and he could calculate their powers 00:13:34.899 --> 00:13:37.100 up to the sixth degree instantly in his head. 00:13:37.259 --> 00:13:39.419 What's truly remarkable here isn't just the memory, 00:13:39.580 --> 00:13:41.820 it's the psychological marvel of it. What do 00:13:41.820 --> 00:13:44.190 you mean? We have this pervasive cultural trope 00:13:44.190 --> 00:13:47.669 of the tortured neurotic genius. The person whose 00:13:47.669 --> 00:13:50.110 brilliant mind is a burden that isolates them 00:13:50.110 --> 00:13:52.440 and makes them bitter. Right, the mad scientist 00:13:52.440 --> 00:13:55.519 stereoscience. Exactly. But the historical accounts 00:13:55.519 --> 00:13:58.860 describe Euler as maintaining a remarkably kind, 00:13:59.240 --> 00:14:02.220 generous, and good -natured disposition, even 00:14:02.220 --> 00:14:04.440 in total darkness. He wasn't lashing out at the 00:14:04.440 --> 00:14:06.820 world because of his condition. Not at all. His 00:14:06.820 --> 00:14:09.980 mind was so rigorously organized that blindness 00:14:09.980 --> 00:14:13.080 didn't trap him in the dark. In a strange way, 00:14:13.120 --> 00:14:15.419 it freed him. Freed him from the distractions. 00:14:15.480 --> 00:14:18.960 Yes. It allowed him to live entirely within the 00:14:18.960 --> 00:14:21.360 pure, perfect architecture of his mathematics, 00:14:22.080 --> 00:14:24.340 completely untethered from the physical messiness 00:14:24.340 --> 00:14:27.059 of light and ink. But while Euler's internal 00:14:27.059 --> 00:14:30.379 world was perfectly ordered, the external 18th 00:14:30.379 --> 00:14:33.080 century world he lived in was anything but. It 00:14:33.080 --> 00:14:35.279 was chaotic, it was highly political, and it 00:14:35.279 --> 00:14:38.240 was deeply flawed. Yes, his career was constantly 00:14:38.240 --> 00:14:40.620 buffeted by the shifting political winds of Europe. 00:14:40.830 --> 00:14:43.230 He spent his professional life bouncing back 00:14:43.230 --> 00:14:45.269 and forth between the Imperial Russian Academy 00:14:45.269 --> 00:14:47.990 in St. Petersburg and Frederick the Great's Berlin 00:14:47.990 --> 00:14:50.350 Academy in Prussia. And these weren't just peaceful 00:14:50.350 --> 00:14:52.710 academic sabbaticals. These moves were driven 00:14:52.710 --> 00:14:55.950 by regime changes, strict censorship, and outright 00:14:55.950 --> 00:14:58.710 war. Very dangerous times. At one point during 00:14:58.710 --> 00:15:01.529 the Seven Years' War, Euler's farm in Charlottenburg 00:15:01.529 --> 00:15:04.610 was literally sacked by advancing Russian troops. 00:15:04.840 --> 00:15:06.679 Though it is worth noting the Russian general 00:15:06.679 --> 00:15:09.179 was so horrified that his troops had looted the 00:15:09.179 --> 00:15:11.740 great oiler's estate that he personally paid 00:15:11.740 --> 00:15:14.600 compensation immediately. Oh, wow. And the Empress 00:15:14.600 --> 00:15:17.620 of Russia added an exorbitant sum on top of it. 00:15:17.940 --> 00:15:21.320 He was viewed as a highly valued, almost untouchable 00:15:21.320 --> 00:15:23.759 commodity. But his time in the Prussian court 00:15:23.759 --> 00:15:26.769 wasn't exactly easy either. Euler was devoutly 00:15:26.769 --> 00:15:29.129 religious, he was a simple man, and by all accounts, 00:15:29.389 --> 00:15:32.529 he was not a skilled witty debater. No, he was 00:15:32.529 --> 00:15:34.509 quite straightforward. Which made him an easy 00:15:34.509 --> 00:15:37.769 target for the sophisticated, cynical intellectuals 00:15:37.769 --> 00:15:40.700 at Fredericks Court, specifically Voltaire. who 00:15:40.700 --> 00:15:43.259 just loved to verbally spar with him and humiliate 00:15:43.259 --> 00:15:45.500 him in public. Voltaire thrived in the realm 00:15:45.500 --> 00:15:48.700 of philosophy, ambiguity, and sharp rhetoric, 00:15:48.980 --> 00:15:51.940 whereas Euler operated in absolute truths. They 00:15:51.940 --> 00:15:54.960 were oil and water. So what does this all mean? 00:15:55.019 --> 00:15:56.840 As I read these accounts, I couldn't help but 00:15:56.840 --> 00:15:59.480 wonder, was Euler basically just the ultimate 00:15:59.480 --> 00:16:01.860 math nerd getting shoved into lockers by the 00:16:01.860 --> 00:16:04.179 cool kids at the royal court? It can certainly 00:16:04.179 --> 00:16:06.200 seem that way from a distance. I mean, Frederick 00:16:06.200 --> 00:16:09.179 the Great famously mocked Euler when the water 00:16:09.179 --> 00:16:11.700 fountains at his summer palace on Susi failed 00:16:11.700 --> 00:16:14.759 to work. Ah, yes, the fountain incident. The 00:16:14.759 --> 00:16:17.720 king wrote this scathing letter complaining that 00:16:17.720 --> 00:16:20.419 Euler had miscalculated the force needed to raise 00:16:20.419 --> 00:16:24.240 the water, famously scoffing, vanity of vanities, 00:16:24.860 --> 00:16:27.279 vanity of geometry. This raises an important 00:16:27.279 --> 00:16:29.600 question about the friction between theoretical 00:16:29.600 --> 00:16:32.419 science and practical execution. And it is a 00:16:32.419 --> 00:16:35.419 criticism that often gets leveled at purely theoretical 00:16:35.419 --> 00:16:39.009 thinkers. But we have to defend Euler here. Okay, 00:16:39.250 --> 00:16:41.870 defend him? Modern engineering analysis has looked 00:16:41.870 --> 00:16:44.990 back at those specific fountain plants. Euler's 00:16:44.990 --> 00:16:47.330 calculations for the fluid dynamics were perfectly 00:16:47.330 --> 00:16:49.769 sound. So the math was actually right. The math 00:16:49.769 --> 00:16:51.769 was completely right. The problem was that the 00:16:51.769 --> 00:16:54.389 18th century builders simply didn't have the 00:16:54.389 --> 00:16:57.110 materials or the execution skills to construct 00:16:57.110 --> 00:16:59.549 pipes that could handle the pressure Euler had 00:16:59.549 --> 00:17:01.809 accurately calculated. Oh, the pipes couldn't 00:17:01.809 --> 00:17:04.480 take it. Yeah. The wooden and lead pipes burst 00:17:04.480 --> 00:17:06.460 because of practical engineering limitations, 00:17:06.599 --> 00:17:09.680 not theoretical failures. And furthermore, it 00:17:09.680 --> 00:17:13.819 is a huge mistake to view Euler as just an abstract 00:17:13.819 --> 00:17:16.420 theorist disconnected from the real physical 00:17:16.420 --> 00:17:18.900 world. Because his practical applications were 00:17:18.900 --> 00:17:21.319 everywhere. Exactly. He wasn't just doing math 00:17:21.319 --> 00:17:24.539 for math's sake. In fluid dynamics, he mathematically 00:17:24.539 --> 00:17:27.740 predicted the phenomenon of cavitation in 1754, 00:17:28.059 --> 00:17:30.480 long before it could even be observed physically. 00:17:30.599 --> 00:17:32.839 Wait, what actually is cavitation? for those 00:17:32.839 --> 00:17:35.480 who aren't engineers. It is what happens when 00:17:35.480 --> 00:17:39.119 the pressure in a flowing liquid drops so abruptly 00:17:39.119 --> 00:17:42.279 that the liquid actually boils and forms vapor 00:17:42.279 --> 00:17:45.240 bubbles, like around a rapidly spinning ship 00:17:45.240 --> 00:17:47.579 propeller. When those bubbles collapse, they 00:17:47.579 --> 00:17:49.940 create massive shock waves that can tear metal 00:17:49.940 --> 00:17:52.400 apart. Euler predicted that purely through math. 00:17:52.559 --> 00:17:54.920 That is incredible. He also helped develop the 00:17:54.920 --> 00:17:57.500 Euler -Bernoulli beam equation, and he figured 00:17:57.500 --> 00:18:00.519 out the formula for a structural column's critical 00:18:00.519 --> 00:18:02.500 buckling load. Which means what, practically 00:18:02.500 --> 00:18:05.200 speaking? Imagine pressing down on the top of 00:18:05.200 --> 00:18:07.440 a dry piece of spaghetti. At a certain point 00:18:07.440 --> 00:18:10.720 of pressure, it bows outward and snaps. Euler 00:18:10.720 --> 00:18:13.680 created the exact mathematical formula to predict 00:18:13.680 --> 00:18:16.339 exactly how much weight a column of any material 00:18:16.339 --> 00:18:19.390 can take before it bends and buckles. These aren't 00:18:19.390 --> 00:18:22.549 abstract concepts. These are the literal structural 00:18:22.549 --> 00:18:24.930 engineering blueprints that allow us to build 00:18:24.930 --> 00:18:28.069 skyscrapers safely today. It is amazing how his 00:18:28.069 --> 00:18:31.609 reputation swung between being mocked by kings 00:18:31.609 --> 00:18:34.970 for a fountain to being viewed as this untouchable, 00:18:35.069 --> 00:18:38.049 almost divine mathematical deity. He was a paradox 00:18:38.049 --> 00:18:40.289 to them. Which brings up my absolute favorite 00:18:40.289 --> 00:18:42.849 anecdote from the sources. We have to clearly 00:18:42.849 --> 00:18:45.869 state upfront, historians widely agree this story 00:18:45.869 --> 00:18:49.119 is likely a legend, an apocryphal myth. but it 00:18:49.119 --> 00:18:51.500 so perfectly illustrates how the rest of the 00:18:51.500 --> 00:18:54.420 world viewed his intellect. Ah, the Diderot story. 00:18:54.599 --> 00:18:56.640 Yes, so the legend goes that the famous French 00:18:56.640 --> 00:18:59.099 philosopher, Ditas Diderot, who was a prominent 00:18:59.099 --> 00:19:01.799 atheist, was visiting the Russian court. And 00:19:01.799 --> 00:19:03.740 causing trouble, as usual. Right, and he's going 00:19:03.740 --> 00:19:05.839 around making all these eloquent philosophical 00:19:05.839 --> 00:19:07.920 arguments against the existence of God, and he's 00:19:07.920 --> 00:19:10.079 completely flustering everyone. So the Empress 00:19:10.079 --> 00:19:12.880 asks Euler to step in and confront him. The ultimate 00:19:12.880 --> 00:19:15.240 showdown. Euler walks right up to Diderot in 00:19:15.240 --> 00:19:17.230 the middle of the court. looks him dead in the 00:19:17.230 --> 00:19:20.390 eye, and says, in a tone of absolute conviction, 00:19:20.809 --> 00:19:24.809 sir, a plus b to the nth power divided by n equals 00:19:24.809 --> 00:19:29.309 x. Hence, God exists. Reply. And according to 00:19:29.309 --> 00:19:31.769 the legend, Diderot knowing absolutely nothing 00:19:31.769 --> 00:19:34.690 about advanced mathematics is completely dumbstruck 00:19:34.690 --> 00:19:37.230 by this algebraic nonsense. He has no idea what 00:19:37.230 --> 00:19:39.539 to say. The whole court erupts into laughter 00:19:39.539 --> 00:19:41.980 and Diderot is so embarrassed he immediately 00:19:41.980 --> 00:19:44.440 asks for safe passage out of Russia. It's just 00:19:44.440 --> 00:19:47.839 a brilliant image slaying a philosophical opponent 00:19:47.839 --> 00:19:51.680 with literally meaningless algebra just because 00:19:51.680 --> 00:19:53.859 your reputation is so massive that no one dares 00:19:53.859 --> 00:19:55.779 question your math. They just assumed he was 00:19:55.779 --> 00:19:57.829 right. Even if it's totally fake, it tells you 00:19:57.829 --> 00:19:59.369 everything you need to know about Euler's aura. 00:19:59.529 --> 00:20:01.890 It does. It highlights the reverence people held 00:20:01.890 --> 00:20:04.450 for his intellect. He was the man who could quantify 00:20:04.450 --> 00:20:07.109 the divine or at least quantify the universe 00:20:07.109 --> 00:20:09.509 in a way that felt divine to the rest of humanity. 00:20:09.650 --> 00:20:11.710 And the reality is every single one of you listening 00:20:11.710 --> 00:20:14.710 to this deep dive right now is living inside 00:20:14.710 --> 00:20:17.349 Leonhard Euler's calculations. Every single day. 00:20:17.670 --> 00:20:19.589 Every time you drive over a suspension bridge 00:20:19.589 --> 00:20:22.089 and trust that the structural columns won't buckle 00:20:22.089 --> 00:20:24.759 under the weight of traffic. Every time you open 00:20:24.759 --> 00:20:28.160 a GPS map app on your phone to find the fastest 00:20:28.160 --> 00:20:30.759 route using network edges and vertices. Yep. 00:20:31.460 --> 00:20:33.339 Even when you go to a concert and appreciate 00:20:33.339 --> 00:20:35.720 the acoustics of the hall or just listen to a 00:20:35.720 --> 00:20:39.039 piano. Yes, his work in music theory shouldn't 00:20:39.039 --> 00:20:41.859 be overlooked. He spent immense time applying 00:20:41.859 --> 00:20:44.400 mathematical concepts to music, developing ways 00:20:44.400 --> 00:20:47.299 to numerically describe the subdivision of octaves 00:20:47.299 --> 00:20:50.339 into fractional parts using prime numbers. He 00:20:50.339 --> 00:20:53.539 saw math everywhere. He viewed harmony and pitched 00:20:53.539 --> 00:20:56.140 through the exact same lens of pure mathematics 00:20:56.140 --> 00:20:58.680 as he viewed fluid dynamics. He saw the code 00:20:58.680 --> 00:21:01.480 behind the chords. It's just incredible. Euler 00:21:01.480 --> 00:21:03.940 worked relentlessly, entirely blind until the 00:21:03.940 --> 00:21:05.880 very last day of his life. He never stopped. 00:21:06.039 --> 00:21:08.539 The sources say that in September 1783, at the 00:21:08.539 --> 00:21:10.980 age of 76, he was having lunch with his family, 00:21:11.460 --> 00:21:13.359 discussing the newly discovered planet Uranus, 00:21:13.740 --> 00:21:15.799 and actively calculating its orbit in his head. 00:21:16.240 --> 00:21:18.720 Over lunch. Over lunch. He collapsed from a brain 00:21:18.720 --> 00:21:21.069 hemorrhage and passed away. The Marquis de Condorcet 00:21:21.069 --> 00:21:23.250 wrote eulogy for the French Academy that I think 00:21:23.250 --> 00:21:25.849 serves as the most fitting epitaph possible for 00:21:25.849 --> 00:21:28.589 a man of his nature. What did he write? He simply 00:21:28.589 --> 00:21:32.549 wrote, he ceased to calculate and to live. For 00:21:32.549 --> 00:21:34.869 Euler, those two things were entirely synonymous. 00:21:35.250 --> 00:21:37.950 He ceased to calculate and to live. It's beautiful. 00:21:38.190 --> 00:21:41.130 It really is. So as we wrap up our dive into 00:21:41.130 --> 00:21:43.289 the life of the man who wrote the operating system 00:21:43.289 --> 00:21:45.630 of the modern world, I want to leave you with 00:21:45.630 --> 00:21:49.349 one final thought to mull over. We started by 00:21:49.349 --> 00:21:52.569 imagining trying to build a massive skyscraper 00:21:52.569 --> 00:21:55.569 without an alphabet. A daunting task. Euler was 00:21:55.569 --> 00:21:58.410 a blind man in the 18th century who used pure 00:21:58.410 --> 00:22:00.670 mathematics to abstract the physical world of 00:22:00.670 --> 00:22:03.269 bridges, fluid dynamics, and celestial orbits 00:22:03.269 --> 00:22:06.369 into immortal perfect equations. He mapped the 00:22:06.369 --> 00:22:08.390 invisible. If you could see all of those hidden 00:22:08.390 --> 00:22:11.269 structures in the dark, what unseen patterns 00:22:11.269 --> 00:22:14.130 governing our own chaotic modern lives are sitting 00:22:14.130 --> 00:22:15.950 right in front of us right now, just waiting 00:22:15.950 --> 00:22:17.509 for new language to describe them.