WEBVTT 00:00:00.000 --> 00:00:02.439 Welcome back to the Deep Dive. We are here to 00:00:02.439 --> 00:00:05.580 take a massive stack of sources, articles, research 00:00:05.580 --> 00:00:08.580 papers, biographies, and really just distill 00:00:08.580 --> 00:00:10.800 them down into the essential knowledge you need, 00:00:11.000 --> 00:00:13.599 giving you that deep understanding without all 00:00:13.599 --> 00:00:16.980 the information overload. And today we are profiling, 00:00:16.980 --> 00:00:19.750 well... arguably the most prodigious intellect 00:00:19.750 --> 00:00:21.850 in the history of science. We're talking about 00:00:21.850 --> 00:00:25.710 Leonhard Euler, born in Basel in 1707. And if 00:00:25.710 --> 00:00:28.289 you had to pick just one person who completely 00:00:28.289 --> 00:00:31.010 dominated the intellectual landscape of the 18th 00:00:31.010 --> 00:00:33.270 century, it would have to be him. Oh, absolutely. 00:00:33.409 --> 00:00:35.130 He wasn't just a mathematician, though that's 00:00:35.130 --> 00:00:37.329 what he's famous for. The sources all confirm 00:00:37.329 --> 00:00:39.929 he was a polymath in the true sense of the word. 00:00:39.990 --> 00:00:43.429 A physicist, an astronomer. A logician, a geographer. 00:00:43.609 --> 00:00:45.670 And a pioneering engineer. He really did it all. 00:00:45.770 --> 00:00:48.250 And when we use a word like pro... We are not 00:00:48.250 --> 00:00:50.090 exaggerating. I mean, his collected works are 00:00:50.090 --> 00:00:52.070 so massive, they're still not even fully published. 00:00:52.350 --> 00:00:54.850 It's a project that's been ongoing for over a 00:00:54.850 --> 00:00:57.109 century. Right now, it's clocking in at around 00:00:57.109 --> 00:01:01.969 80 volumes. And that that scale is exactly why 00:01:01.969 --> 00:01:04.689 these later titans of mathematics spoke about 00:01:04.689 --> 00:01:08.329 him with this. this almost religious awe. Right. 00:01:08.390 --> 00:01:10.129 You have that famous quote from Pierre Simon 00:01:10.129 --> 00:01:12.469 Laplace who basically commanded everyone, read 00:01:12.469 --> 00:01:15.049 Euler, read Euler. He is the master of us all. 00:01:15.209 --> 00:01:16.650 I mean, that's the kind of endorsement you only 00:01:16.650 --> 00:01:18.849 get from someone who recognizes pure foundational 00:01:18.849 --> 00:01:20.969 genius when they see it. And then there's Carl 00:01:20.969 --> 00:01:24.049 Friedrich Gauss, another absolute heavyweight 00:01:24.049 --> 00:01:26.849 in the history of math. He echoed that same sentiment. 00:01:27.069 --> 00:01:29.890 He did. He said that studying Euler's works will 00:01:29.890 --> 00:01:31.790 remain the best school for the different fields 00:01:31.790 --> 00:01:34.290 of mathematics and nothing else can replace it. 00:01:34.560 --> 00:01:37.560 So our mission today is to give you that best 00:01:37.560 --> 00:01:40.239 school shortcut. We want you to walk away from 00:01:40.239 --> 00:01:42.079 this understanding, not just what Euler discovered, 00:01:42.239 --> 00:01:45.040 but really how he changed the way we all think 00:01:45.040 --> 00:01:47.219 about the physical world and even the language 00:01:47.219 --> 00:01:49.879 we use to describe it. To really set the stage 00:01:49.879 --> 00:01:51.819 for his career, you have to think about this. 00:01:52.400 --> 00:01:55.439 His entire professional life was spent traveling 00:01:55.439 --> 00:01:59.640 between two of the most powerful, most demanding 00:01:59.640 --> 00:02:02.159 intellectual centers in Europe. He served the 00:02:02.159 --> 00:02:04.620 Imperial Russian Academy of Sciences in St. Petersburg. 00:02:04.760 --> 00:02:07.560 And the Berlin Academy in Prussia. He was essentially 00:02:07.560 --> 00:02:09.759 managing the scientific output for competing 00:02:09.759 --> 00:02:12.500 empires. That just, it speaks volumes about how 00:02:12.500 --> 00:02:15.080 indispensable he was. It really does. But here's 00:02:15.080 --> 00:02:18.439 the critical initial aha moment for you. I think 00:02:18.439 --> 00:02:20.919 this is the key piece of knowledge for anyone 00:02:20.919 --> 00:02:23.460 who's ever struggled through algebra or calculus. 00:02:23.860 --> 00:02:26.939 Go on. Euler didn't just solve these high -level 00:02:26.939 --> 00:02:30.349 abstract problems. He created the toolkit. He 00:02:30.349 --> 00:02:33.189 literally invented or popularized the symbols 00:02:33.189 --> 00:02:35.669 that are the bedrock of all modern mathematical 00:02:35.669 --> 00:02:38.009 communication. Let's really linger on that because 00:02:38.009 --> 00:02:40.490 it's so important. When you see fx written on 00:02:40.490 --> 00:02:44.349 a blackboard, the notation for a function, that's 00:02:44.349 --> 00:02:46.990 Euler. When you use the constant e for the base 00:02:46.990 --> 00:02:49.169 of the natural logarithm Euler's number, that's 00:02:49.169 --> 00:02:51.370 him. If you ever encounter the imaginary unit, 00:02:51.550 --> 00:02:54.030 you know, the square root of negative one. That's 00:02:54.030 --> 00:02:56.909 Euler. The capital sigma. Four summations. And 00:02:56.909 --> 00:02:59.090 even the standard use of the Greek letter for 00:02:59.090 --> 00:03:01.969 the circle ratio. All of it. It all owes its 00:03:01.969 --> 00:03:05.030 common, everyday use to Leonhard Euler's incredible 00:03:05.030 --> 00:03:08.050 standardization efforts. He wasn't just a scientist. 00:03:08.310 --> 00:03:10.870 You put it perfectly earlier. He was a linguistic 00:03:10.870 --> 00:03:14.300 architect for science. He took this fractured 00:03:14.300 --> 00:03:17.599 kind of personalized system of notation and gave 00:03:17.599 --> 00:03:20.479 it a unified universal alphabet. And that made 00:03:20.479 --> 00:03:23.500 communication across borders, across disciplines, 00:03:23.539 --> 00:03:27.460 not just easier, but possible. OK, so let's unpack 00:03:27.460 --> 00:03:30.840 the path that led to this monumental contribution. 00:03:30.919 --> 00:03:33.060 We have to start with his early life and education, 00:03:33.419 --> 00:03:36.060 the genesis of this genius. Right. So he was 00:03:36.060 --> 00:03:39.060 born in Basel in 1707. And the first thing that 00:03:39.060 --> 00:03:41.580 really jumps out from the sources is how deep 00:03:41.580 --> 00:03:43.379 his connection to the scientific establishment 00:03:43.379 --> 00:03:45.599 was. I mean, right from birth. It runs right 00:03:45.599 --> 00:03:47.310 through his family tree, doesn't it? It does. 00:03:47.389 --> 00:03:49.969 His father, Paul III Euler, was a reformed church 00:03:49.969 --> 00:03:53.090 pastor. But, and this is the key detail, he was 00:03:53.090 --> 00:03:54.990 also highly educated. He had actually studied 00:03:54.990 --> 00:03:56.949 under Jacob Bernoulli, one of the first pioneers 00:03:56.949 --> 00:03:58.949 of calculus, right there at the University of 00:03:58.949 --> 00:04:01.490 Basel. So that mathematical tradition was literally 00:04:01.490 --> 00:04:04.729 in the household from day one. Exactly. He grew 00:04:04.729 --> 00:04:07.370 up in Rhine, near Basel, and got his initial 00:04:07.370 --> 00:04:10.449 instruction from his father. That was then supplemented 00:04:10.449 --> 00:04:12.610 with some private tutoring from a young theologian 00:04:12.610 --> 00:04:15.509 named Johannes Burkhart. So the path was kind 00:04:15.509 --> 00:04:17.769 of set early, even though his father, like a 00:04:17.769 --> 00:04:19.750 lot of parents, had a very specific career in 00:04:19.750 --> 00:04:23.670 mind for him. But Euler's his innate talent just 00:04:23.670 --> 00:04:26.449 quickly outgrew that initial curriculum. He enrolled 00:04:26.449 --> 00:04:30.060 at the University of Basel in 1720 at the. Well, 00:04:30.100 --> 00:04:32.779 the astonishingly young age of 13. And this wasn't 00:04:32.779 --> 00:04:35.839 just him being precocious. This was raw intellectual 00:04:35.839 --> 00:04:38.680 firepower being applied very, very early. And 00:04:38.680 --> 00:04:40.560 while he was there, he connected with the person 00:04:40.560 --> 00:04:43.889 who would become his most crucial mentor. Johann 00:04:43.889 --> 00:04:46.129 Bernoulli. Jacob's younger brother and himself 00:04:46.129 --> 00:04:48.509 one of the towering mathematical figures of the 00:04:48.509 --> 00:04:50.529 time. And Johann Bernoulli's teaching method 00:04:50.529 --> 00:04:53.410 for Euler is the stuff of legend. It's actually 00:04:53.410 --> 00:04:55.750 something you, the listener, could adopt for 00:04:55.750 --> 00:04:57.649 your own learning, really. Euler wrote about 00:04:57.649 --> 00:05:00.649 it himself. He detailed how Bernoulli was often 00:05:00.649 --> 00:05:03.850 just too busy for formal private lessons. But 00:05:03.850 --> 00:05:06.509 he provided something far more impactful, what 00:05:06.509 --> 00:05:09.790 he called salutary advice. This is where it gets 00:05:09.790 --> 00:05:12.259 really interesting. Bernoulli instructed him 00:05:12.259 --> 00:05:15.300 to go read the most difficult, the densest mathematical 00:05:15.300 --> 00:05:18.139 books he could find, just digest them on his 00:05:18.139 --> 00:05:20.220 own. And then come back every Saturday afternoon 00:05:20.220 --> 00:05:22.720 with all his difficulties and questions. The 00:05:22.720 --> 00:05:25.279 insight here is just crucial, isn't it? Bernoulli 00:05:25.279 --> 00:05:27.720 wasn't just feeding him answers. Not at all. 00:05:27.879 --> 00:05:30.519 By forcing Euler to wrestle with the material 00:05:30.519 --> 00:05:32.959 first, he was teaching him intellectual independence. 00:05:33.699 --> 00:05:36.259 Euler himself noted that when Bernoulli resolved 00:05:36.259 --> 00:05:38.680 one objection he brought, he said, ten others 00:05:38.680 --> 00:05:41.199 at once disappeared. It was a system designed 00:05:41.199 --> 00:05:42.839 to teach him how to resolve these fundamental 00:05:42.839 --> 00:05:45.540 intellectual roadblocks. It turned him into a 00:05:45.540 --> 00:05:48.360 self -learner capable of dominating, well, any 00:05:48.360 --> 00:05:50.420 field he chose. And that guidance fundamentally 00:05:50.420 --> 00:05:53.540 changed his life path. Euler's father had strongly 00:05:53.540 --> 00:05:56.019 intended for him to become a pastor, you know, 00:05:56.040 --> 00:05:57.759 a theologian following the family profession. 00:05:58.199 --> 00:06:00.759 But Bernoulli, seeing the sheer depth of his 00:06:00.759 --> 00:06:04.180 mathematical talent, actively intervened. He 00:06:04.180 --> 00:06:06.220 went and convinced the older Euler to let his 00:06:06.220 --> 00:06:08.899 son pursue mathematics exclusively. I mean... 00:06:09.129 --> 00:06:11.290 Think about that endorsement. That's the 18th 00:06:11.290 --> 00:06:13.949 century's leading mathematician telling a pastor, 00:06:14.089 --> 00:06:16.129 no, your son is meant for something else entirely. 00:06:16.430 --> 00:06:20.569 And he progressed so rapidly. By 1723, at just 00:06:20.569 --> 00:06:23.829 16 years old, he received his Master of Philosophy. 00:06:24.110 --> 00:06:26.589 And his thesis was remarkably ambitious. It was 00:06:26.589 --> 00:06:29.310 a comparison of the philosophical systems of 00:06:29.310 --> 00:06:32.069 René Descartes and Isaac Newton. Even as a teenager, 00:06:32.310 --> 00:06:34.850 he's synthesizing the major intellectual currents 00:06:34.850 --> 00:06:37.569 that were defining modern European thought. But 00:06:37.569 --> 00:06:39.870 genius doesn't always guarantee a job, does it? 00:06:39.949 --> 00:06:42.790 No. Despite his obvious talent, he couldn't immediately 00:06:42.790 --> 00:06:45.709 secure a position at Basel. He wrote his dissertation, 00:06:45.930 --> 00:06:48.949 Dissertatio Physica de Sono, a physical dissertation 00:06:48.949 --> 00:06:52.470 on sound in 1726, but the local institution just 00:06:52.470 --> 00:06:54.410 wasn't ready to hire him. So he looked to the 00:06:54.410 --> 00:06:56.810 wider European stage to prove himself. And this 00:06:56.810 --> 00:06:59.110 is where we see his first real test against his 00:06:59.110 --> 00:07:01.589 contemporaries, the annual Paris Academy Prize 00:07:01.589 --> 00:07:05.310 competition in 1727. And the problem was intensely 00:07:05.310 --> 00:07:07.750 practical. It was determining the best way to 00:07:07.750 --> 00:07:10.550 place the masts on a ship. This wasn't abstract 00:07:10.550 --> 00:07:13.550 algebra. This was applied physics and naval architecture. 00:07:13.689 --> 00:07:16.389 And he took second place, which is a phenomenal 00:07:16.389 --> 00:07:18.970 achievement for a 20 -year -old newcomer. He 00:07:18.970 --> 00:07:21.189 was beaten that year by Pierre Bougier, who was 00:07:21.189 --> 00:07:23.769 known as the father of naval architecture. So 00:07:23.769 --> 00:07:26.600 no shame in that loss. But that initial entry 00:07:26.600 --> 00:07:29.459 just signaled his capabilities. That minor defeat 00:07:29.459 --> 00:07:32.180 only served as motivation, really. Oh, for sure. 00:07:32.339 --> 00:07:34.560 Over the course of his career, Euler would go 00:07:34.560 --> 00:07:36.860 on to dominate that competition. He eventually 00:07:36.860 --> 00:07:40.160 won the coveted Paris Academy Prize. A staggering 00:07:40.160 --> 00:07:42.819 12 times. So this early period really proves 00:07:42.819 --> 00:07:45.439 the dual nature of his genius. He was capable 00:07:45.439 --> 00:07:47.959 of both the highest theoretical abstraction and 00:07:47.959 --> 00:07:50.959 at the same time solving these complex real -world 00:07:50.959 --> 00:07:54.319 engineering and physics problems. So Euler proved 00:07:54.319 --> 00:07:56.600 his worth, but Bessel still wasn't the right 00:07:56.600 --> 00:07:58.639 place for him. And that brings us to section 00:07:58.639 --> 00:08:01.399 two. The transcontinental career that really 00:08:01.399 --> 00:08:04.000 defined his adult working life, starting with 00:08:04.000 --> 00:08:07.680 that first St. Petersburg period in 1727. And 00:08:07.680 --> 00:08:09.779 this Russian connection, thankfully, was secured 00:08:09.779 --> 00:08:12.620 by the Bernoulli family again. Of course. Johann 00:08:12.620 --> 00:08:15.500 Bernoulli's sons, Daniel and Nicholas, were already 00:08:15.500 --> 00:08:18.100 established in St. Petersburg. They were the 00:08:18.100 --> 00:08:20.500 newly founded Imperial Russian Academy of Sciences. 00:08:20.759 --> 00:08:23.699 But unfortunately, Nicholas died tragically of 00:08:23.699 --> 00:08:26.240 appendicitis shortly after arriving. Which created 00:08:26.240 --> 00:08:29.740 a perfect vacancy. It did. And Daniel, recognizing 00:08:29.740 --> 00:08:32.759 Euler's immense talent, immediately secured the 00:08:32.759 --> 00:08:37.100 position for him. So Euler arrived in May 1727. 00:08:37.719 --> 00:08:39.899 And although he was initially placed in the medical 00:08:39.899 --> 00:08:43.179 department, he very quickly transitioned to pure 00:08:43.179 --> 00:08:45.820 mathematics. And he lodged and worked closely 00:08:45.820 --> 00:08:48.179 with Daniel Bernoulli, right? That's right. And 00:08:48.179 --> 00:08:50.139 he didn't just work in the academy. He completely 00:08:50.139 --> 00:08:52.659 immersed himself in the culture. He mastered 00:08:52.659 --> 00:08:55.259 Russian, adapted to the demanding schedules, 00:08:55.559 --> 00:08:58.059 and even took on an additional job as a medic 00:08:58.059 --> 00:08:59.879 in the Russian Navy. It's worth noting, too, 00:08:59.940 --> 00:09:01.659 that he was offered a promotion to lieutenant, 00:09:01.799 --> 00:09:03.940 but he turned it down. He made it very clear 00:09:03.940 --> 00:09:06.740 that his passion was pure scholarship, not command. 00:09:06.990 --> 00:09:09.490 But the political climate in Russia at this time 00:09:09.490 --> 00:09:12.470 was incredibly unstable. The Academy had been 00:09:12.470 --> 00:09:14.230 founded under the progressive vision of Peter 00:09:14.230 --> 00:09:16.789 the Great. But after the death of Catherine I, 00:09:17.070 --> 00:09:19.309 conservative Russian nobility began to regain 00:09:19.309 --> 00:09:21.769 power under the young Peter II. And that meant 00:09:21.769 --> 00:09:25.100 peril for the Academy. Big trouble. The conservative 00:09:25.100 --> 00:09:27.360 faction was deeply suspicious of the foreign 00:09:27.360 --> 00:09:29.919 scientists. Many of them were German or Swiss. 00:09:30.139 --> 00:09:33.320 And this led to significant funding cuts, bureaucratic 00:09:33.320 --> 00:09:36.220 hostility, and just a very uncertain future for 00:09:36.220 --> 00:09:38.919 all the academics. The conditions became so precarious 00:09:38.919 --> 00:09:41.080 that Daniel Bernoulli eventually returned to 00:09:41.080 --> 00:09:43.690 Basel, right? He cited political censorship as 00:09:43.690 --> 00:09:46.570 a major factor. He did. So it really took a shift 00:09:46.570 --> 00:09:49.029 in leadership to stabilize the environment. In 00:09:49.029 --> 00:09:52.590 1730, Peter II died, and Empress Anna, who had 00:09:52.590 --> 00:09:54.909 German ties and was much more sympathetic to 00:09:54.909 --> 00:09:57.929 modern science, took the throne. And this stabilization 00:09:57.929 --> 00:10:00.610 is what allowed Euler to really flourish. He 00:10:00.610 --> 00:10:02.809 ascended so rapidly. He was made a professor 00:10:02.809 --> 00:10:05.830 of physics in 1731. And then when Daniel Bernoulli 00:10:05.830 --> 00:10:09.129 departed in 1733, Euler, who is now just 26, 00:10:09.490 --> 00:10:11.330 succeeded him as the head of the mathematics 00:10:11.330 --> 00:10:14.570 department. And he really cemented his life in 00:10:14.570 --> 00:10:17.870 Russia, both professionally and personally. In 00:10:17.870 --> 00:10:21.690 January 1734, he married Katharina Grissel, who 00:10:21.690 --> 00:10:23.809 was the daughter of a Swiss painter working at 00:10:23.809 --> 00:10:26.990 the Academy. This stability was absolutely vital, 00:10:27.129 --> 00:10:29.350 considering the immense intellectual output he 00:10:29.350 --> 00:10:31.909 was about to unleash. However, despite that stability 00:10:31.909 --> 00:10:34.429 under Empress Anna, the political situation was 00:10:34.429 --> 00:10:36.870 still a long -term concern, especially after 00:10:36.870 --> 00:10:39.049 her death. Right. And when Frederick the Great 00:10:39.049 --> 00:10:40.950 of Prussia began aggressively recruiting him 00:10:40.950 --> 00:10:43.340 for the Berlin Academy... Euler was tempted. 00:10:43.539 --> 00:10:47.679 In 1741, he accepted Frederick's offer. There's 00:10:47.679 --> 00:10:50.179 a small detail that I think illuminates his human 00:10:50.179 --> 00:10:53.100 side. He partly justified the move by claiming 00:10:53.100 --> 00:10:55.740 he needed a milder climate for his deteriorating 00:10:55.740 --> 00:10:58.120 eyesight. But the core motivation was the promise 00:10:58.120 --> 00:11:00.460 of a more stable, more intellectually focused 00:11:00.460 --> 00:11:02.899 environment in Prussia. The Russian Academy, 00:11:03.080 --> 00:11:05.299 by the way, was devastated to lose him. Which 00:11:05.299 --> 00:11:07.740 really demonstrates his immense value to them. 00:11:07.840 --> 00:11:09.740 Absolutely. They actually agreed to keep him 00:11:09.740 --> 00:11:12.220 on the payroll. paying him an annual stipend 00:11:12.220 --> 00:11:14.919 and considering him an active member, even as 00:11:14.919 --> 00:11:17.019 he spent the next 25 years working for their 00:11:17.019 --> 00:11:19.379 direct competition. So this transition marks 00:11:19.379 --> 00:11:24.120 the Berlin period from 1741 to 1766, which is 00:11:24.120 --> 00:11:26.700 often cited as the peak of his sheer analytical 00:11:26.700 --> 00:11:29.679 productivity. He was in an environment perfectly 00:11:29.679 --> 00:11:32.000 tailored for scholarship. And he wasted no time. 00:11:32.399 --> 00:11:34.399 This is where he published the seminal textbooks 00:11:34.399 --> 00:11:38.080 that formalized modern analysis. In 1748, he 00:11:38.080 --> 00:11:40.419 released Introductio in Analysin Infinitorum, 00:11:40.500 --> 00:11:43.139 an absolute foundational text on the theory of 00:11:43.139 --> 00:11:46.580 functions and infinite series. And then in 1755, 00:11:46.960 --> 00:11:50.080 Instituciones Calculae Differentialis, which 00:11:50.080 --> 00:11:52.360 covered differential calculus. You have to understand, 00:11:52.460 --> 00:11:55.519 these weren't just dusty academic monographs. 00:11:55.519 --> 00:11:57.940 They were comprehensive, they were clear, and 00:11:57.940 --> 00:12:00.100 they were standardized teaching tools that rapidly 00:12:00.100 --> 00:12:02.490 circulated all across Europe. And they logged 00:12:02.490 --> 00:12:04.789 in his new notations and methods for generations 00:12:04.789 --> 00:12:07.009 of students to come. But his influence wasn't 00:12:07.009 --> 00:12:09.149 just limited to the mathematical elite. This 00:12:09.149 --> 00:12:11.129 period also produced one of his most beloved 00:12:11.129 --> 00:12:13.450 works. And this one was aimed squarely at the 00:12:13.450 --> 00:12:15.309 lay public. The letters of Euler on different 00:12:15.309 --> 00:12:17.309 subjects in natural philosophy addressed to a 00:12:17.309 --> 00:12:20.409 German princess. He wrote over 200 letters while 00:12:20.409 --> 00:12:22.210 he was tutoring Frederick the Great's niece, 00:12:22.490 --> 00:12:25.049 Princess Friedrich Charlotte, on topics ranging 00:12:25.049 --> 00:12:28.350 from philosophy and optics to magnetism and logic. 00:12:28.830 --> 00:12:31.330 This collection is just a monument to his genius 00:12:31.330 --> 00:12:34.529 for communication. It shows this profound ability, 00:12:34.850 --> 00:12:38.289 which is rare among intellectual giants, to translate 00:12:38.289 --> 00:12:40.570 these incredibly complex scientific principles 00:12:40.570 --> 00:12:45.600 into accessible elegant prose for a general audience. 00:12:45.919 --> 00:12:48.100 And the letters were translated into seven languages, 00:12:48.179 --> 00:12:51.000 published in dozens of editions, and became really 00:12:51.000 --> 00:12:53.620 astonishingly more widely read across Europe 00:12:53.620 --> 00:12:56.179 and the early American states than any of his 00:12:56.179 --> 00:12:58.539 specialized mathematical papers. Yet despite 00:12:58.539 --> 00:13:00.759 running the Berlin Academy's intellectual output 00:13:00.759 --> 00:13:03.179 and its prestige, his relationship with King 00:13:03.179 --> 00:13:06.539 Frederick II was famously strained. The king 00:13:06.539 --> 00:13:09.379 viewed Euler as a brilliant calculator, but intellectually 00:13:09.379 --> 00:13:12.059 unrefined, sort of lacking in the courtly wit 00:13:12.059 --> 00:13:14.100 and general culture that Frederick prized so 00:13:14.100 --> 00:13:16.080 much. And that tension was definitely heightened 00:13:16.080 --> 00:13:18.039 by the presence of Frederick's favorite philosopher, 00:13:18.179 --> 00:13:21.980 Voltaire. Euler was simple, devout, nonpolitical. 00:13:22.019 --> 00:13:24.519 He was the exact opposite of the witty, secular, 00:13:24.720 --> 00:13:27.139 politically connected Voltaire. And Voltaire 00:13:27.139 --> 00:13:30.720 often targeted Euler's lack of polish in intellectual 00:13:30.720 --> 00:13:33.019 debates that were outside of pure mathematics. 00:13:33.460 --> 00:13:35.470 Right. It sounds like Euler was brilliant in 00:13:35.470 --> 00:13:37.669 a structured environment, but maybe a little 00:13:37.669 --> 00:13:39.750 uncomfortable in the cut and thrust of courtly 00:13:39.750 --> 00:13:42.360 debate. That's precisely the dynamic the sources 00:13:42.360 --> 00:13:44.940 suggest. Frederick, who was a connoisseur of 00:13:44.940 --> 00:13:47.399 sophisticated conversation, just found Euler 00:13:47.399 --> 00:13:50.740 boring. He reportedly told d 'Alembert that Euler's 00:13:50.740 --> 00:13:53.899 conversation lacked grace, saying Euler had a 00:13:53.899 --> 00:13:56.320 great genius but just could not sustain a salon 00:13:56.320 --> 00:13:59.460 conversation. And nothing, nothing encapsulates 00:13:59.460 --> 00:14:01.940 Frederick's irritation with Euler's practical 00:14:01.940 --> 00:14:05.019 application quite like the infamous Sanssouci 00:14:05.019 --> 00:14:07.539 fountain incident. This one is always worth dwelling 00:14:07.539 --> 00:14:09.759 on. Frederick wanted this magnificent geometric... 00:14:09.799 --> 00:14:12.500 calculated system of water jets for his garden 00:14:12.500 --> 00:14:15.639 at Sansuti. He tasked Euler with designing the 00:14:15.639 --> 00:14:18.639 optimal system. The result? A spectacular failure. 00:14:18.919 --> 00:14:21.309 The water pressure was never adequate. And the 00:14:21.309 --> 00:14:23.450 fountain never worked as intended. And the King's 00:14:23.450 --> 00:14:26.029 quote on the failure is just brutal. Vanity of 00:14:26.029 --> 00:14:28.809 vanities, vanity of geometry. He pointed the 00:14:28.809 --> 00:14:31.210 finger squarely at Euler and theoretical mathematics. 00:14:31.649 --> 00:14:33.590 But what's fascinating here is the technical 00:14:33.590 --> 00:14:36.750 context. The sources largely agree that the King's 00:14:36.750 --> 00:14:39.350 disappointment was likely unwarranted, at least 00:14:39.350 --> 00:14:42.110 from a purely engineering perspective. So Euler's 00:14:42.110 --> 00:14:44.750 calculations were probably correct. Almost certainly. 00:14:44.950 --> 00:14:47.409 The calculations for the hydraulics, the geometry, 00:14:47.610 --> 00:14:49.419 the fluid movement. They were likely correct. 00:14:49.620 --> 00:14:51.980 The failure probably lay in the poor execution 00:14:51.980 --> 00:14:54.740 of the construction by the builders, or the materials 00:14:54.740 --> 00:14:57.620 used, or the practical limitations of the water 00:14:57.620 --> 00:15:00.360 source. These were issues outside of Euler's 00:15:00.360 --> 00:15:03.019 pure math domain. So the story isn't that Euler 00:15:03.019 --> 00:15:04.779 failed, it's that Frederick preferred to blame 00:15:04.779 --> 00:15:07.440 the theoretical genius he found personally tedious 00:15:07.440 --> 00:15:10.259 rather than, say, the expensive contractors. 00:15:10.700 --> 00:15:13.220 That sharpens the political tension perfectly. 00:15:13.559 --> 00:15:16.220 And despite all this friction, Euler's output 00:15:16.220 --> 00:15:19.580 was relentless. During this 25 -year stint in 00:15:19.580 --> 00:15:23.779 Berlin, he wrote a staggering 380 works, with 00:15:23.779 --> 00:15:26.639 275 of them published. And on top of that, he 00:15:26.639 --> 00:15:28.379 was effectively the chief administrator for the 00:15:28.379 --> 00:15:31.659 Academy. We tend to focus only on the math, but 00:15:31.659 --> 00:15:33.940 his administrative duties were massive. He supervised 00:15:33.940 --> 00:15:36.159 the library, ran the observatory, managed the 00:15:36.159 --> 00:15:39.440 botanical garden, and, this is critical, he oversaw 00:15:39.440 --> 00:15:42.240 the publication of the maps, calendars, and almanacs, 00:15:42.320 --> 00:15:44.659 which were vital sources of income for the academy. 00:15:44.879 --> 00:15:47.259 He was running a scientific conglomerate. This 00:15:47.259 --> 00:15:50.779 dual role just underscores his unbelievable discipline. 00:15:51.059 --> 00:15:53.720 And we have to remember his incredible loyalty 00:15:53.720 --> 00:15:56.559 to St. Petersburg. Even while he was in Berlin, 00:15:56.779 --> 00:15:59.220 he published over 100 memoirs there, he kept 00:15:59.220 --> 00:16:01.639 contact with Russian students, and he even helped 00:16:01.639 --> 00:16:04.059 accommodate them in his own home. That loyalty 00:16:04.059 --> 00:16:06.399 paid off during the tumultuous Seven Years' War. 00:16:06.580 --> 00:16:09.399 In 1760, advancing Russian troops sacked his 00:16:09.399 --> 00:16:11.379 farm in Charlottenburg. But when the Russian 00:16:11.379 --> 00:16:13.720 general, Ivan Petrovich Saltykov, heard about 00:16:13.720 --> 00:16:16.080 this, he immediately paid substantial compensation 00:16:16.080 --> 00:16:18.740 for the damage. And then Empress Elizabeth, upon 00:16:18.740 --> 00:16:21.019 hearing of the incident, added an additional 00:16:21.019 --> 00:16:24.340 4 ,000 rubles, an exorbitant amount at the time. 00:16:24.460 --> 00:16:26.779 He was an international intellectual asset. No 00:16:26.779 --> 00:16:30.159 power could afford to offend him. By 1766, the 00:16:30.159 --> 00:16:32.700 conditions back in St. Petersburg under Catherine 00:16:32.700 --> 00:16:34.919 the Great were far more favorable. They were 00:16:34.919 --> 00:16:37.419 offering better pay and far greater recognition 00:16:37.419 --> 00:16:40.580 than Frederick was ever willing to give. So Euler 00:16:40.580 --> 00:16:43.419 accepted the invitation to return. His demands 00:16:43.419 --> 00:16:45.700 were quite significant, reflecting his recognized 00:16:45.700 --> 00:16:49.769 worth, a 3 ,000 ruble annual salary. a pension 00:16:49.769 --> 00:16:52.049 for his wife, and high -ranking appointments 00:16:52.049 --> 00:16:55.470 for his sons, Catherine met every single demand. 00:16:55.809 --> 00:16:58.230 And this marks the second St. Petersburg period, 00:16:58.309 --> 00:17:02.149 from 1766 to 1783, a time that's really defined 00:17:02.149 --> 00:17:05.109 by physical deterioration, but this unparalleled 00:17:05.109 --> 00:17:07.930 intellectual defiance. Euler's eyesight had been 00:17:07.930 --> 00:17:10.390 failing for decades. Frederick II had already 00:17:10.390 --> 00:17:12.910 dismissively called him Cyclops due to the near 00:17:12.910 --> 00:17:14.970 -total blindness in his right eye, which was 00:17:14.970 --> 00:17:17.670 possibly aggravated by his earlier detailed cartography 00:17:17.670 --> 00:17:20.319 work. Famous quote from Euler when he first lost 00:17:20.319 --> 00:17:22.119 sight in that right eye is just so indicative 00:17:22.119 --> 00:17:24.200 of his character. He said, now I will have fewer 00:17:24.200 --> 00:17:25.980 distractions. I mean, most people would see the 00:17:25.980 --> 00:17:27.980 end of their career. Euler saw an opportunity 00:17:27.980 --> 00:17:31.160 for deeper internal focus. But the tragedy compounded 00:17:31.160 --> 00:17:34.339 in 1766. Complications from cataract surgery 00:17:34.339 --> 00:17:36.859 left him almost totally blind in his remaining 00:17:36.859 --> 00:17:39.740 left eye. Total darkness descended upon him. 00:17:39.940 --> 00:17:41.960 Most academics would have retired immediately. 00:17:42.460 --> 00:17:44.660 But this is where the legend of Euler is truly 00:17:44.660 --> 00:17:48.140 forged. His productivity. It increased. Unbelievable. 00:17:48.519 --> 00:17:51.180 He just adapted his methodology. He hired scribes. 00:17:51.180 --> 00:17:53.480 He relied entirely on his phenomenal memory and 00:17:53.480 --> 00:17:56.119 mental arithmetic. And he dictated his work, 00:17:56.220 --> 00:17:58.799 often to his students like Anders Johan Lexel. 00:17:58.880 --> 00:18:01.039 Let's just quantify that defiance for a minute. 00:18:01.079 --> 00:18:05.180 In 1775, years after losing his sight, he was 00:18:05.180 --> 00:18:07.759 averaging an astonishing one mathematical paper 00:18:07.759 --> 00:18:10.319 per week. He was dictating analysis, formulas, 00:18:10.460 --> 00:18:13.680 and proofs entirely from memory. It is one of 00:18:13.680 --> 00:18:16.039 the most remarkable feats of sustained intellectual 00:18:16.039 --> 00:18:19.039 output in all of human history. It speaks to 00:18:19.039 --> 00:18:20.880 a mind that was just perfectly structured, able 00:18:20.880 --> 00:18:23.400 to manipulate complex, multivariable equations 00:18:23.400 --> 00:18:26.000 purely within the confines of his own skull. 00:18:26.200 --> 00:18:28.039 And on the personal front, this period saw even 00:18:28.039 --> 00:18:30.400 more upheaval. His home was destroyed by a fire 00:18:30.400 --> 00:18:33.519 in 1771, and he lost his first wife, Katharina, 00:18:33.660 --> 00:18:37.170 in 1773. To maintain the stability that was so 00:18:37.170 --> 00:18:39.890 crucial for his ongoing work and for his five 00:18:39.890 --> 00:18:42.750 surviving children, he married Katharina's half 00:18:42.750 --> 00:18:46.829 -sister, Salome Abigail Grissel, in 1776. The 00:18:46.829 --> 00:18:49.029 story of Euler is so often told through his great 00:18:49.029 --> 00:18:51.390 discoveries, but we established earlier that 00:18:51.390 --> 00:18:54.029 his first major contribution was really the language 00:18:54.029 --> 00:18:56.430 he gave to mathematics. Absolutely. So section 00:18:56.430 --> 00:18:59.089 three here is dedicated to this notational revolution 00:18:59.089 --> 00:19:01.750 because he didn't just solve problems. He gave 00:19:01.750 --> 00:19:03.509 us the tools to write them down in the first 00:19:03.509 --> 00:19:06.720 place. Before Euler's textbook standardized everything, 00:19:07.099 --> 00:19:10.559 math was this. This Babylonian mess of competing 00:19:10.559 --> 00:19:13.460 symbols and personalized abbreviations. That's 00:19:13.460 --> 00:19:14.940 a great way to put it. If you wanted to read 00:19:14.940 --> 00:19:17.319 Newton, you had to learn Newton's symbols. If 00:19:17.319 --> 00:19:18.819 you wanted to read Leibniz, you learned his. 00:19:19.660 --> 00:19:22.619 Euler introduced a unified logical language that 00:19:22.619 --> 00:19:24.599 every student today, whether you're in high school 00:19:24.599 --> 00:19:27.380 or graduate school, still uses. Okay, so let's 00:19:27.380 --> 00:19:29.180 walk through the foundational symbols. First, 00:19:29.319 --> 00:19:32.380 the most elementary concept in modern math. The 00:19:32.380 --> 00:19:36.829 function. Euler gave us fx. The use of f of x 00:19:36.829 --> 00:19:39.430 for the value of a function f applied to the 00:19:39.430 --> 00:19:42.450 argument x. Before this, expressing the relationship 00:19:42.450 --> 00:19:44.609 between variables was clunky and descriptive. 00:19:44.970 --> 00:19:47.789 So fx gave us a concise, universally recognized 00:19:47.789 --> 00:19:51.289 shorthand. It revolutionized algebra and calculus 00:19:51.289 --> 00:19:54.109 instantly. It allowed mathematicians to treat 00:19:54.109 --> 00:19:56.490 operations and transformations as objects in 00:19:56.490 --> 00:19:58.569 themselves. Then we have the constants, the letter 00:19:58.569 --> 00:20:01.390 e, Euler's number, the base of the natural logarithm. 00:20:01.819 --> 00:20:04.420 It's central to modeling continuous growth and 00:20:04.420 --> 00:20:07.539 decay everything from population growth to radioactive 00:20:07.539 --> 00:20:11.279 half -life, NYE. The sources suggest it may have 00:20:11.279 --> 00:20:14.079 simply been the next vowel in a series of variables 00:20:14.079 --> 00:20:16.839 he was using. But once he used it in his foundational 00:20:16.839 --> 00:20:21.000 1748 textbook, Introductio in Analysin Infinitorum, 00:20:21.079 --> 00:20:23.640 it stuck instantly because it was clear. It was 00:20:23.640 --> 00:20:26.259 logical. It's the constant that defines exponential 00:20:26.259 --> 00:20:29.160 processes. And crucially, he introduced the letter 00:20:29.160 --> 00:20:31.859 I for the imaginary unit, the square root of 00:20:31.859 --> 00:20:33.940 negative one. This is the absolute foundation 00:20:33.940 --> 00:20:36.400 of complex analysis. And this was a deeply abstract 00:20:36.400 --> 00:20:38.640 concept that had been debated since the 16th 00:20:38.640 --> 00:20:42.119 century. Exactly. By giving it a simple standardized 00:20:42.119 --> 00:20:46.849 symbol, I. Euler normalized and formalized its 00:20:46.849 --> 00:20:50.130 use. You simply cannot do modern physics, electrical 00:20:50.130 --> 00:20:52.329 engineering, signal processing, quantum mechanics 00:20:52.329 --> 00:20:55.309 without the symbol I and the complex number system 00:20:55.309 --> 00:20:57.829 it anchors. He also helped secure the use of 00:20:57.829 --> 00:20:59.809 Bricton. That's right. While he credited the 00:20:59.809 --> 00:21:01.990 Welsh mathematician William Jones with the original 00:21:01.990 --> 00:21:05.670 use back in 1706, it was Euler's extensive use 00:21:05.670 --> 00:21:08.660 of the Greek letter. in his major, widely read 00:21:08.660 --> 00:21:11.039 text that cemented it as the universal standard 00:21:11.039 --> 00:21:13.380 for the ratio of a circle's circumference to 00:21:13.380 --> 00:21:15.319 its diameter. And it wasn't just letters. It 00:21:15.319 --> 00:21:17.619 was operational symbols, too. Right. He introduced 00:21:17.619 --> 00:21:20.490 the Greek letter drat. capital sigma for summations. 00:21:20.630 --> 00:21:22.950 This provided a concise way to represent long 00:21:22.950 --> 00:21:26.029 or infinite series. Before, representing a series 00:21:26.029 --> 00:21:28.549 was just cumbersome prose. Afterwards, it became 00:21:28.549 --> 00:21:30.990 a clean, universally readable equation. And the 00:21:30.990 --> 00:21:33.589 Greek letter A capital delta for finite differences, 00:21:33.710 --> 00:21:35.630 which is essential for numerical methods and 00:21:35.630 --> 00:21:38.460 early calculus work. Even geometry was standardized 00:21:38.460 --> 00:21:41.599 by him. When you look at a textbook today, you 00:21:41.599 --> 00:21:44.200 see triangle sides labeled with lowercase letters 00:21:44.200 --> 00:21:47.259 and the angles opposite them labeled with corresponding 00:21:47.259 --> 00:21:50.380 capital letters. That clear, simple convention, 00:21:50.519 --> 00:21:53.500 that's Euler. He was ensuring that geometry problems 00:21:53.500 --> 00:21:55.900 could be read consistently across continents. 00:21:56.279 --> 00:21:58.559 So the takeaway here is that standardization 00:21:58.559 --> 00:22:01.960 was Euler's first great intellectual act. It 00:22:01.960 --> 00:22:04.779 made math accessible and cumulative. It's so 00:22:04.779 --> 00:22:07.259 rare for one person to have such a profound and 00:22:07.259 --> 00:22:10.099 lasting impact on the global language of an entire 00:22:10.099 --> 00:22:13.339 field. So when you are struggling through a complex 00:22:13.339 --> 00:22:16.339 physics problem today, just remember you are 00:22:16.339 --> 00:22:19.019 using the precise, organized language Euler invented, 00:22:19.339 --> 00:22:22.579 saving centuries of confusion. That's his gift 00:22:22.579 --> 00:22:25.119 to every modern student. And now we transition 00:22:25.119 --> 00:22:27.059 from the language builder to the discoverer. 00:22:27.359 --> 00:22:29.839 Sections 4 and 5 are going to cover the sheer 00:22:29.839 --> 00:22:32.079 breadth of his original theoretical and applied 00:22:32.079 --> 00:22:34.799 research. Let's start with analytical contributions, 00:22:35.059 --> 00:22:36.940 the area where he was truly sovereign. Right. 00:22:37.000 --> 00:22:39.140 The 18th century was dominated by the development 00:22:39.140 --> 00:22:42.240 of infinitesimal calculus, and Euler was the 00:22:42.240 --> 00:22:44.660 most powerful engine in that development. He 00:22:44.660 --> 00:22:46.940 was famous for his frequent use and development 00:22:46.940 --> 00:22:49.519 of power series. Power series expressing functions 00:22:49.519 --> 00:22:53.180 as infinite sums of powers of x. He used them 00:22:53.180 --> 00:22:56.109 like no one before him. If you can express a 00:22:56.109 --> 00:22:58.930 function as an infinite polynomial, you can manipulate 00:22:58.930 --> 00:23:02.289 it, integrate it and differentiate it with comparative 00:23:02.289 --> 00:23:05.000 ease. He showed, for example, the exponential 00:23:05.000 --> 00:23:07.779 function e to the x as that infinite series. 00:23:08.099 --> 00:23:11.000 e to the x equals the sum from n equals zero 00:23:11.000 --> 00:23:14.519 to infinity of x to the n over n factorial. His 00:23:14.519 --> 00:23:16.920 willingness to just work with infinite processes 00:23:16.920 --> 00:23:19.339 allowed him to achieve results that baffled his 00:23:19.339 --> 00:23:21.440 contemporaries. And the most famous example of 00:23:21.440 --> 00:23:23.299 this analytical power has to be his solution 00:23:23.299 --> 00:23:25.839 to the Basel problem. This problem had stood 00:23:25.839 --> 00:23:29.440 open since 1644, posed by Pietro Mangoli, and 00:23:29.440 --> 00:23:31.299 it had frustrated the brightest minds of the 00:23:31.299 --> 00:23:33.619 age, including the elder Bernini. The challenge 00:23:33.619 --> 00:23:35.880 was to find the exact sum of the reciprocals 00:23:35.880 --> 00:23:38.059 of the squares of every natural number. So 1 00:23:38.059 --> 00:23:40.299 over 1 squared plus 1 over 2 squared plus 1 over 00:23:40.299 --> 00:23:42.700 3 squared and so on. It looked like a pure number 00:23:42.700 --> 00:23:45.019 theory problem, and they knew the sum converged 00:23:45.019 --> 00:23:48.140 to some number around 1 .64. But finding the 00:23:48.140 --> 00:23:51.339 exact value proved impossible. Until Euler. In 00:23:51.339 --> 00:23:55.180 1735, he provided the stunning and utterly unexpected 00:23:55.180 --> 00:23:58.839 solution. The sum is exactly squared over 6. 00:23:59.279 --> 00:24:01.539 What made this so groundbreaking was the connection. 00:24:02.380 --> 00:24:04.599 Why should a series involving integers have a 00:24:04.599 --> 00:24:07.640 result tied to the geometric constant of the 00:24:07.640 --> 00:24:10.660 circle? It was a beautiful, inexplicable piece 00:24:10.660 --> 00:24:13.380 of unity, and it relied on a brilliant, though 00:24:13.380 --> 00:24:16.200 initially non -rigorous method, where he treated 00:24:16.200 --> 00:24:18.619 the infinite series like it was a finite polynomial. 00:24:19.079 --> 00:24:21.559 He later provided a more elaborate proof, but 00:24:21.559 --> 00:24:23.500 the result stood. It was a connection between 00:24:23.500 --> 00:24:26.359 two seemingly disparate worlds of math. And his 00:24:26.359 --> 00:24:28.799 work didn't stop with real numbers. He successfully 00:24:28.799 --> 00:24:31.380 defined logarithms for negative and complex numbers, 00:24:31.579 --> 00:24:34.000 which vastly expanded the practical reach of 00:24:34.000 --> 00:24:36.259 the logarithm function. But the absolute pinnacle 00:24:36.259 --> 00:24:38.779 of his achievement in complex analysis is Euler's 00:24:38.779 --> 00:24:41.440 formula. This is truly a high point. He extended 00:24:41.440 --> 00:24:43.140 the exponential function to complex numbers, 00:24:43.259 --> 00:24:45.420 linking it directly to the trigonometric functions. 00:24:45.660 --> 00:24:48.559 The formula states that e to the i equals cos 00:24:48.559 --> 00:24:51.940 plus e sin. This formula provides the fundamental 00:24:51.940 --> 00:24:54.740 link between circular motion, so trigonometry, 00:24:55.019 --> 00:24:57.460 and exponential growth. And if that wasn't beautiful 00:24:57.460 --> 00:25:00.200 enough, if you set the angle 2 .5, you get the 00:25:00.200 --> 00:25:02.500 simplest and most elegant version, which is known 00:25:02.500 --> 00:25:06.000 as Euler's identity. e to the i plus 1 equals 00:25:06.000 --> 00:25:08.660 0. This one short equation is often called the 00:25:08.660 --> 00:25:11.460 greatest formula in mathematics. Richard Feynman 00:25:11.460 --> 00:25:13.500 famously called it that. And think about what 00:25:13.500 --> 00:25:15.960 it connects. You have e, the base of continuous 00:25:15.960 --> 00:25:19.759 growth, i, the imaginary unit. The geometric 00:25:19.759 --> 00:25:23.019 constant, one, the multiplicative identity, and 00:25:23.019 --> 00:25:25.599 zero, the additive identity. Five fundamental 00:25:25.599 --> 00:25:28.180 constants woven into one perfect expression. 00:25:28.640 --> 00:25:31.559 It just epitomizes Euler's unifying vision for 00:25:31.559 --> 00:25:33.539 mathematics. And moving beyond those headline 00:25:33.539 --> 00:25:36.079 results, he also essentially invented the calculus 00:25:36.079 --> 00:25:39.140 of variations. Okay, so what is that, and why 00:25:39.140 --> 00:25:41.279 does it matter? Well, the standard calculus you 00:25:41.279 --> 00:25:43.279 learn in school deals with finding the point, 00:25:43.359 --> 00:25:46.160 a number, where a function is maximized or minimized. 00:25:46.279 --> 00:25:49.109 The calculus of variations is much deeper. It 00:25:49.109 --> 00:25:51.069 finds the function, the actual shape of the path, 00:25:51.250 --> 00:25:54.490 that maximizes or minimizes an integral. So instead 00:25:54.490 --> 00:25:56.670 of finding the quickest single speed, you're 00:25:56.670 --> 00:25:59.289 finding the most efficient path overall. Exactly. 00:25:59.289 --> 00:26:01.869 It's essential for finding things like the shortest 00:26:01.869 --> 00:26:04.069 distance between two points on a curved surface, 00:26:04.269 --> 00:26:06.750 the shape of a soap film stretched between two 00:26:06.750 --> 00:26:09.730 wires, or the optimal trajectory of a satellite. 00:26:10.359 --> 00:26:13.420 He formulated the Euler -Lagrange equation, which 00:26:13.420 --> 00:26:15.559 is the foundational tool for this entire field. 00:26:15.740 --> 00:26:18.279 And it's still used in everything from general 00:26:18.279 --> 00:26:21.680 relativity to classical mechanics. He also contributed 00:26:21.680 --> 00:26:23.640 to the theory of higher transcendental functions 00:26:23.640 --> 00:26:26.680 by introducing the gamma function. That sounds 00:26:26.680 --> 00:26:28.759 a bit intimidating. What purpose does it serve? 00:26:28.940 --> 00:26:31.980 The gamma function, which is denoted az, is essentially 00:26:31.980 --> 00:26:34.500 a way to extend the concept of the factorial, 00:26:34.819 --> 00:26:37.599 which is normally only defined for positive integers, 00:26:37.880 --> 00:26:41.819 like 5, to include non -integers, complex numbers, 00:26:42.000 --> 00:26:43.859 and fractions. So it allows you to calculate 00:26:43.859 --> 00:26:46.349 something like half factorial. Conceptually, 00:26:46.349 --> 00:26:49.529 yes. And this seemingly abstract extension proved 00:26:49.529 --> 00:26:51.650 absolutely crucial for advanced mathematics, 00:26:51.990 --> 00:26:53.970 especially in areas like probability distributions, 00:26:54.410 --> 00:26:57.210 partial differential equations, and complex analysis 00:26:57.210 --> 00:27:00.910 in physics. It generalized a key tool. And we 00:27:00.910 --> 00:27:02.789 should also mention the Euler -McLauren formula, 00:27:02.970 --> 00:27:05.390 another key tool he developed for approximating 00:27:05.390 --> 00:27:08.430 integrals using summation, vital for numerical 00:27:08.430 --> 00:27:11.369 methods. Now let's pivot to number theory, because 00:27:11.369 --> 00:27:14.089 this is where Euler truly created a whole new 00:27:14.089 --> 00:27:16.839 discipline. analytic number theory. This was 00:27:16.839 --> 00:27:19.059 the revolutionary act of uniting number theory, 00:27:19.240 --> 00:27:21.900 which deals with discrete integers and primes 00:27:21.900 --> 00:27:24.640 with continuous analysis. He was using tools 00:27:24.640 --> 00:27:27.759 like infinite series and calculus to solve problems 00:27:27.759 --> 00:27:30.339 about numbers. He demonstrated the power of this 00:27:30.339 --> 00:27:32.960 new approach by proving the infinitude of prime 00:27:32.960 --> 00:27:35.759 numbers. Now, we know Euclid proved this using 00:27:35.759 --> 00:27:38.779 pure number theory by contradiction. How did 00:27:38.779 --> 00:27:41.339 Euler use analysis? He connected the divergence 00:27:41.339 --> 00:27:43.980 of the harmonic series, so 1 plus 1 half plus 00:27:43.980 --> 00:27:46.319 1 third plus 1 fourth and so on, to prime numbers. 00:27:46.759 --> 00:27:48.980 He showed that the sum of the reciprocals of 00:27:48.980 --> 00:27:51.539 the primes, 1 half plus 1 third plus 1 fifth 00:27:51.539 --> 00:27:54.799 plus 1 seventh, also diverges, meaning it approaches 00:27:54.799 --> 00:27:56.900 infinity. And this could only happen if there 00:27:56.900 --> 00:27:58.700 were an infinite number of terms, an infinite 00:27:58.700 --> 00:28:01.319 number of primes. in the series. It's a subtle 00:28:01.319 --> 00:28:03.579 but brilliant argument. It shows that analytic 00:28:03.579 --> 00:28:06.420 methods could yield new insight into these ancient 00:28:06.420 --> 00:28:08.859 integer problems. And that connection led to 00:28:08.859 --> 00:28:11.720 his discovery of the profound Euler product formula 00:28:11.720 --> 00:28:14.579 which linked the sum of reciprocals of integer 00:28:14.579 --> 00:28:17.519 powers, which later became the Riemann zeta function, 00:28:17.839 --> 00:28:21.299 to a product over prime numbers. The implications 00:28:21.299 --> 00:28:24.039 of this formula are still a cornerstone of mathematical 00:28:24.039 --> 00:28:27.319 research today. And in pure number theory, Euler 00:28:27.319 --> 00:28:29.740 was the intellectual heir to Pierre de Fermat. 00:28:29.880 --> 00:28:32.460 He dedicated significant effort to validating 00:28:32.460 --> 00:28:34.740 Fermat's findings, but he also wasn't afraid 00:28:34.740 --> 00:28:37.859 to overturn them. That's right. Fermat had conjectured 00:28:37.859 --> 00:28:40.559 that all Fermat numbers, those of the form 2 00:28:40.559 --> 00:28:44.839 to the n plus 1, were prime. Euler, through brilliant 00:28:44.839 --> 00:28:47.720 calculation, proved that for n equals 5, the 00:28:47.720 --> 00:28:50.559 resulting number, which is over 4 billion, was 00:28:50.559 --> 00:28:53.759 composite. not prime. He showed it was divisible 00:28:53.759 --> 00:28:57.160 by 641, disproving Fermat's conjecture entirely. 00:28:57.599 --> 00:28:59.420 And he also gave us one of the most important 00:28:59.420 --> 00:29:02.059 tools in modern cryptography, the totient function. 00:29:02.519 --> 00:29:05.920 What exactly does the totient function do? The 00:29:05.920 --> 00:29:07.839 totient function, sometimes called Euler's phi 00:29:07.839 --> 00:29:10.319 function, it counts how many positive integers 00:29:10.319 --> 00:29:13.880 less than or equal to n are co -prime to n. And 00:29:13.880 --> 00:29:15.740 two numbers are co -prime if they share no common 00:29:15.740 --> 00:29:18.180 factors other than one. And why is that useful? 00:29:18.440 --> 00:29:20.619 Well, It helps us understand the structure of 00:29:20.619 --> 00:29:23.279 remainders in modular arithmetic, the math of 00:29:23.279 --> 00:29:25.660 clocks and cyclical systems. Euler then used 00:29:25.660 --> 00:29:27.819 this function to generalize Fermat's little theorem, 00:29:27.960 --> 00:29:30.680 resulting in Euler's theorem. And this theorem 00:29:30.680 --> 00:29:32.940 is critical because the principles it establishes 00:29:32.940 --> 00:29:35.119 dealing with exponents and remainders are the 00:29:35.119 --> 00:29:37.839 exact foundational base we use for modern security 00:29:37.839 --> 00:29:41.259 and encryption algorithms like RSA. He also contributed 00:29:41.259 --> 00:29:44.319 to the classical study of perfect numbers. those 00:29:44.319 --> 00:29:47.079 numbers that equal the sum of their proper divisors, 00:29:47.160 --> 00:29:51.019 like 6, where 1 plus 2 plus 3 equals 6. This 00:29:51.019 --> 00:29:54.180 was a fascination since Euclid. It was, and Euler 00:29:54.180 --> 00:29:56.420 solved a long -standing mystery by proving the 00:29:56.420 --> 00:29:58.759 one -to -one correspondence between even perfect 00:29:58.759 --> 00:30:02.460 numbers and Mersenne primes. This result is known 00:30:02.460 --> 00:30:05.400 as the Euclid -Euler theorem, a perfect example 00:30:05.400 --> 00:30:07.980 of his ability to synthesize ancient geometry 00:30:07.980 --> 00:30:11.049 with modern number theory. Okay, moving on to 00:30:11.049 --> 00:30:13.690 section five. If Euler's work in analysis proved 00:30:13.690 --> 00:30:16.309 his theoretical mastery, his contributions to 00:30:16.309 --> 00:30:18.930 geometry, physics, and engineering really demonstrate 00:30:18.930 --> 00:30:22.009 his role as the ultimate applied genius. He wasn't 00:30:22.009 --> 00:30:24.190 just working with abstractions, he was shaping 00:30:24.190 --> 00:30:26.549 the physical world. And the best way to introduce 00:30:26.549 --> 00:30:28.390 this shift of perspective is through the story 00:30:28.390 --> 00:30:30.509 that literally founded an entirely new branch 00:30:30.509 --> 00:30:33.190 of mathematics. Yeah. The Seven Bridges of Konigsberg 00:30:33.190 --> 00:30:36.519 Problem. From 1735. Retell that problem for us 00:30:36.519 --> 00:30:38.220 because it sounds like a simple riddle, but it 00:30:38.220 --> 00:30:40.599 completely changed math forever. The city of 00:30:40.599 --> 00:30:43.180 Königsberg, which is now Kaliningrad, was built 00:30:43.180 --> 00:30:45.799 on the Pregel River, and it contained two large 00:30:45.799 --> 00:30:47.839 islands connected to each other and the mainland 00:30:47.839 --> 00:30:51.000 by seven bridges. And the citizens wondered if 00:30:51.000 --> 00:30:52.980 it was possible to take a walk and cross each 00:30:52.980 --> 00:30:56.240 of the seven bridges exactly once. Euler proved 00:30:56.240 --> 00:30:59.000 it was impossible, but the true genius wasn't 00:30:59.000 --> 00:31:01.779 the no answer. It was the methodology he used 00:31:01.779 --> 00:31:04.920 to reach it. Exactly. Euler realized that the 00:31:04.920 --> 00:31:06.460 solution had nothing to do with the physical 00:31:06.460 --> 00:31:09.200 distances or the size of the islands or the length 00:31:09.200 --> 00:31:11.640 of the bridges. All that mattered was the connectivity. 00:31:11.940 --> 00:31:14.200 He replaced the land masses with points vertices 00:31:14.200 --> 00:31:17.539 and the bridges with lines or edges. Right. And 00:31:17.539 --> 00:31:19.759 he determined that for a path to exist that crosses 00:31:19.759 --> 00:31:23.259 every bridge exactly once, a Eulerian path, the 00:31:23.259 --> 00:31:26.099 graph could have at most two vertices with an 00:31:26.099 --> 00:31:28.400 odd number of edges connected to them. And since 00:31:28.400 --> 00:31:31.059 Konigsberg have four landmasses with an odd number 00:31:31.059 --> 00:31:33.640 of bridges connecting them, such a path was impossible. 00:31:34.039 --> 00:31:37.539 This approach, focusing only on structural relationships 00:31:37.539 --> 00:31:40.440 and ignoring measurement, is recognized as the 00:31:40.440 --> 00:31:43.279 first theorem of graph theory. It's a complete 00:31:43.279 --> 00:31:45.920 shift in mathematical perspective, moving math 00:31:45.920 --> 00:31:48.400 from continuous measurement to discrete connectivity, 00:31:48.819 --> 00:31:51.680 a shift that is crucial for modern computer science 00:31:51.680 --> 00:31:54.779 and networking. He made a similar profound contribution 00:31:54.779 --> 00:31:58.140 to topology, which is the study of spatial properties 00:31:58.140 --> 00:32:00.740 that remain the same even when an object is stretched 00:32:00.740 --> 00:32:03.119 or deformed. That was through the Euler characteristic 00:32:03.119 --> 00:32:06.980 formula. For any convex polyhedron, a 3D shape 00:32:06.980 --> 00:32:09.359 like a cube or a pyramid, the number of vertices, 00:32:09.660 --> 00:32:12.339 V minus the number of edges, E, plus the number 00:32:12.339 --> 00:32:16.039 of faces, F always equals 2. V, E plus F, L equals 00:32:16.039 --> 00:32:18.920 2. It's just beautiful in its simplicity. So 00:32:18.920 --> 00:32:21.359 why is that the origin of topology? Because it's 00:32:21.359 --> 00:32:23.359 a property inherent to the structure of the shape, 00:32:23.500 --> 00:32:26.140 not its size or its curvature. You can distort 00:32:26.140 --> 00:32:28.579 a cube into any shape you want. But as long as 00:32:28.579 --> 00:32:30.859 you don't break or merge edges, the count VE 00:32:30.859 --> 00:32:33.420 plus F will always do too. The study of this 00:32:33.420 --> 00:32:35.799 formula and its generalization to other objects 00:32:35.799 --> 00:32:38.240 is recognized as the starting point for modern 00:32:38.240 --> 00:32:41.240 algebraic topology, a field that seeks structural 00:32:41.240 --> 00:32:43.869 invariance in space. And turning to physics and 00:32:43.869 --> 00:32:46.470 engineering, Euler didn't just apply Newton's 00:32:46.470 --> 00:32:49.369 laws. He refined them to make them more useful 00:32:49.369 --> 00:32:52.789 for the coming industrial age. In his two -volume 00:32:52.789 --> 00:32:56.049 work, Mechanica, he reformulated classical mechanics, 00:32:56.430 --> 00:32:59.470 providing new analytic laws to better describe 00:32:59.470 --> 00:33:02.049 the motion of rigid bodies, which are far more 00:33:02.049 --> 00:33:04.730 complex than the idealized point masses Newton 00:33:04.730 --> 00:33:07.250 often considered. And in structural engineering, 00:33:07.529 --> 00:33:10.569 he gave us two formulas that are absolute cornerstones 00:33:10.569 --> 00:33:13.750 for anyone building things today. First, the 00:33:13.750 --> 00:33:16.490 Euler Bernoulli beam equation. That equation 00:33:16.490 --> 00:33:18.990 is foundational for designing any structure that's 00:33:18.990 --> 00:33:21.190 subject to bending, whether it's a bridge, a 00:33:21.190 --> 00:33:23.670 building frame, or an aircraft wing. It describes 00:33:23.670 --> 00:33:25.670 the relationship between the beam's deflection 00:33:25.670 --> 00:33:28.049 and the loads applied to it, allowing engineers 00:33:28.049 --> 00:33:30.609 to calculate stress with precision. And second, 00:33:30.750 --> 00:33:32.569 and perhaps even more famous in engineering, 00:33:32.789 --> 00:33:35.579 is Euler's critical load formula. This formula 00:33:35.579 --> 00:33:38.700 calculates the exact compressive load at which 00:33:38.700 --> 00:33:41.480 a long, slender column will buckle and fail. 00:33:41.920 --> 00:33:44.940 Before Euler, structural collapse due to buckling 00:33:44.940 --> 00:33:48.019 was often unpredictable. His formula gave engineers 00:33:48.019 --> 00:33:51.359 the definitive boundary, the critical load, telling 00:33:51.359 --> 00:33:53.460 them precisely when the geometry of a column 00:33:53.460 --> 00:33:55.680 will cause catastrophic failure under compression. 00:33:55.980 --> 00:33:58.940 It saved countless lives and billions of dollars 00:33:58.940 --> 00:34:01.539 in future construction. His work in fluid dynamics 00:34:01.539 --> 00:34:04.319 is equally foundational. even if it's often less 00:34:04.319 --> 00:34:08.460 visible. In 1757, he formulated the Euler equations 00:34:08.460 --> 00:34:11.579 for inviscid flow. Inviscid flow being the motion 00:34:11.579 --> 00:34:15.119 of an idealized fluid with no viscosity, no internal 00:34:15.119 --> 00:34:17.960 friction. Right. Which, while theoretical, provides 00:34:17.960 --> 00:34:20.199 the simplest, most powerful starting point for 00:34:20.199 --> 00:34:23.199 understanding how water and air move. These equations 00:34:23.199 --> 00:34:25.219 are still the first tools used in computational 00:34:25.219 --> 00:34:28.000 fluid dynamics. And here's a fact that just shows 00:34:28.000 --> 00:34:31.400 his predictive genius. In 1754, based purely 00:34:31.400 --> 00:34:33.719 on mathematical modeling, Euler was the first 00:34:33.719 --> 00:34:35.940 person to predict the phenomenon of cavitation. 00:34:36.400 --> 00:34:38.940 Cavitation is the formation of vapor bubbles 00:34:38.940 --> 00:34:41.800 in a rapidly flowing liquid when the pressure 00:34:41.800 --> 00:34:44.530 drops below the fluid's vapor pressure. It's 00:34:44.530 --> 00:34:46.849 what causes noise and damage in pumps, propellers, 00:34:46.869 --> 00:34:49.230 and giant turbines. And Euler mathematically 00:34:49.230 --> 00:34:51.730 predicted this phenomenon, these very tiny bubbles 00:34:51.730 --> 00:34:54.670 that challenge giant engineering systems, long 00:34:54.670 --> 00:34:56.409 before it was physically observed in the late 00:34:56.409 --> 00:34:58.909 19th century. That is true scientific prophecy. 00:34:59.190 --> 00:35:01.429 He even weighed in on one of the great historical 00:35:01.429 --> 00:35:04.389 scientific debates in optics, disagreeing with 00:35:04.389 --> 00:35:06.730 Isaac Newton's prevailing theory that light was 00:35:06.730 --> 00:35:09.590 made of particles or corpuscles. Euler was an 00:35:09.590 --> 00:35:12.110 advocate for Christian Huygens' wave theory of 00:35:12.110 --> 00:35:15.530 light. His papers in the 1740s provided strong 00:35:15.530 --> 00:35:18.309 mathematical arguments in favor of light propagating 00:35:18.309 --> 00:35:21.289 as a wave. And while science eventually reconciled 00:35:21.289 --> 00:35:23.710 these two views with quantum theory, Euler's 00:35:23.710 --> 00:35:25.690 work was crucial in pushing the wave theory to 00:35:25.690 --> 00:35:28.250 dominance for a long period, ensuring the scientific 00:35:28.250 --> 00:35:31.030 debate remained dynamic and productive. And beyond 00:35:31.030 --> 00:35:33.730 physics, his influence extended into visualization 00:35:33.730 --> 00:35:38.150 and logic. In 1768, he introduced the Euler diagrams 00:35:38.150 --> 00:35:40.829 to illustrate syllogistic reasoning. These are 00:35:40.829 --> 00:35:43.550 the closed curves we still use today to visually 00:35:43.550 --> 00:35:46.469 represent set relationships, whether they intersect 00:35:46.469 --> 00:35:49.949 or subsets of each other or are entirely disjoint. 00:35:50.170 --> 00:35:52.670 While modern Venn diagrams are a refinement, 00:35:52.889 --> 00:35:55.130 Euler diagrams are the visual starting point 00:35:55.130 --> 00:35:57.050 for thinking about logical relationships and 00:35:57.050 --> 00:36:01.199 set theory. the nascent field of demography, 00:36:01.420 --> 00:36:03.820 he essentially launched modern population modeling 00:36:03.820 --> 00:36:07.579 150 years too early. He did. In the 1760 paper, 00:36:07.760 --> 00:36:10.320 A General Investigation into the Mortality and 00:36:10.320 --> 00:36:13.019 Multiplication of the Human Species, he created 00:36:13.019 --> 00:36:16.139 a comprehensive mathematical model. He showed 00:36:16.139 --> 00:36:18.420 that a population subject to constant rates of 00:36:18.420 --> 00:36:20.659 mortality and fertility would eventually grow 00:36:20.659 --> 00:36:23.159 geometrically, establishing a fixed age structure. 00:36:23.440 --> 00:36:25.480 And that model, the stable population model, 00:36:25.559 --> 00:36:27.820 is fundamental to how demographers study population 00:36:27.820 --> 00:36:30.480 dynamics today. But it wasn't rediscovered and 00:36:30.480 --> 00:36:32.599 widely formalized until Alfred J. Lochte did 00:36:32.599 --> 00:36:35.199 so in the 20th century. He was 150 years ahead 00:36:35.199 --> 00:36:37.599 of the curve, defining the start of formal demographic 00:36:37.599 --> 00:36:39.940 modeling. Finally, we get to one of his most 00:36:39.940 --> 00:36:42.960 unusual and personal interests, music theory. 00:36:43.559 --> 00:36:47.760 In 1739, he wrote Tentamen Novae Theoriae Musicae, 00:36:47.900 --> 00:36:50.179 which was an attempt at a new theory of music, 00:36:50.400 --> 00:36:52.820 trying to ground aesthetic concepts entirely 00:36:52.820 --> 00:36:55.699 in mathematics. And this work was notoriously 00:36:55.699 --> 00:36:58.199 difficult. It was described as too mathematical 00:36:58.199 --> 00:37:00.940 for musicians and too musical for mathematicians. 00:37:01.079 --> 00:37:03.460 It was a beautiful failure, but a failure of 00:37:03.460 --> 00:37:06.400 communication, not insight. Euler was trying 00:37:06.400 --> 00:37:09.039 to quantify harmony. He mathematically derived 00:37:09.039 --> 00:37:11.719 the gratis suavitatis, or degree of agreeableness, 00:37:11.920 --> 00:37:14.400 for musical intervals and chords. He based this 00:37:14.400 --> 00:37:16.880 sweetness on the arithmetic of small prime factors 00:37:16.880 --> 00:37:19.679 found in musical ratios, specifically three and 00:37:19.679 --> 00:37:22.380 five. For instance, a perfect fifth, with a ratio 00:37:22.380 --> 00:37:24.519 of three to two, is harmonically simpler than 00:37:24.519 --> 00:37:26.750 a diminished fifth. His belief was that aesthetic 00:37:26.750 --> 00:37:28.650 pleasure was rooted in the quantitative simplicity 00:37:28.650 --> 00:37:31.949 of ratios. He even devised a graph, the speculum 00:37:31.949 --> 00:37:34.570 musicum, to classify musical genres based on 00:37:34.570 --> 00:37:36.869 their complexity, which has recently drawn renewed 00:37:36.869 --> 00:37:39.530 interest in advanced music theory. It just shows 00:37:39.530 --> 00:37:42.570 his relentless quest to unify knowledge. If mathematics 00:37:42.570 --> 00:37:45.090 governs the stars, mechanics, and light, well, 00:37:45.130 --> 00:37:47.730 why not beauty itself? As we synthesize the life 00:37:47.730 --> 00:37:49.670 of Lee and Mart Euler, we have to just pause 00:37:49.670 --> 00:37:51.889 and consider the sheer magnitude of the mind 00:37:51.889 --> 00:37:54.559 that produced all of this. The volume of his 00:37:54.559 --> 00:37:57.340 output, especially while blind, suggests an organizational 00:37:57.340 --> 00:37:59.880 and computational ability unlike almost anyone 00:37:59.880 --> 00:38:02.860 else in history. The famous tales about his memory 00:38:02.860 --> 00:38:06.699 are just awe -inspiring. By all accounts, he 00:38:06.699 --> 00:38:08.820 possessed a computational memory that was staggering. 00:38:09.619 --> 00:38:12.039 Nicholas Fuss, who was one of Euler's disciples, 00:38:12.440 --> 00:38:14.699 chronicled these abilities. He had memorized 00:38:14.699 --> 00:38:18.039 Virgil's massive epic poem, The Aeneid, in its 00:38:18.039 --> 00:38:21.190 entirety. And in his later years, totally blind, 00:38:21.369 --> 00:38:24.070 he could reportedly recite the poem and instantly 00:38:24.070 --> 00:38:26.590 state the first and last sentence on any given 00:38:26.590 --> 00:38:29.050 page of the exact edition he had learned from 00:38:29.050 --> 00:38:31.269 decades prior. And his mathematical memory was 00:38:31.269 --> 00:38:33.769 equally flawless. He could instantly recall the 00:38:33.769 --> 00:38:35.690 first hundred prime numbers and their powers 00:38:35.690 --> 00:38:38.590 up to the sixth degree, and he used those facts 00:38:38.590 --> 00:38:40.550 as computational scaffolding inside his head. 00:38:40.909 --> 00:38:43.369 This ability to run complex equations solely 00:38:43.369 --> 00:38:45.690 in his memory is why the loss of his sight barely 00:38:45.690 --> 00:38:48.199 slowed him down. And his general knowledge wasn't 00:38:48.199 --> 00:38:50.780 just limited to math and the classics. He had 00:38:50.780 --> 00:38:53.400 comprehensive knowledge of history, Roman writers, 00:38:53.659 --> 00:38:56.800 medicine, botany, chemistry, far more than a 00:38:56.800 --> 00:38:59.519 specialist would be expected to hold. This extraordinary 00:38:59.519 --> 00:39:02.139 mental catalog allowed him to synthesize fields 00:39:02.139 --> 00:39:05.480 and draw connections where others saw only separation. 00:39:05.760 --> 00:39:07.900 It's what makes him a truly universal genius. 00:39:08.460 --> 00:39:11.099 The scale of his legacy sometimes needs a human 00:39:11.099 --> 00:39:14.019 touch, which brings us to a famous, persistent, 00:39:14.179 --> 00:39:17.360 but ultimately false anecdote involving the French 00:39:17.360 --> 00:39:19.739 philosopher Denis Diderot. Right. This legend 00:39:19.739 --> 00:39:21.840 supposedly takes place during Euler's second 00:39:21.840 --> 00:39:24.300 tenure in St. Petersburg under Catherine the 00:39:24.300 --> 00:39:26.960 Great. Diderot, who was known for his atheistic 00:39:26.960 --> 00:39:29.159 leanings, was influencing the Russian court, 00:39:29.260 --> 00:39:32.039 and the empress allegedly asked Euler to publicly 00:39:32.039 --> 00:39:34.849 silence him. The story goes that Euler approached 00:39:34.849 --> 00:39:37.130 Diderot and, with perfect conviction, announced 00:39:37.130 --> 00:39:40.190 a mathematical proof of God's existence. He states 00:39:40.190 --> 00:39:43.090 the non -sequitur. Sir, a plus b to the n divided 00:39:43.090 --> 00:39:47.170 by n equals x, hence God exists, reply. And Diderot, 00:39:47.289 --> 00:39:49.329 allegedly knowing nothing of mathematics, was 00:39:49.329 --> 00:39:51.929 left speechless, mocked by the court, and soon 00:39:51.929 --> 00:39:54.320 asked to leave Russia. It's a wonderfully dramatic, 00:39:54.559 --> 00:39:58.260 if perhaps a bit anti -intellectual story. But 00:39:58.260 --> 00:40:01.019 as historians confirm, it's a legend popularized 00:40:01.019 --> 00:40:04.019 later, and it simply isn't true. Not at all. 00:40:04.599 --> 00:40:06.639 Diderot was a capable mathematician himself, 00:40:06.880 --> 00:40:09.420 and the incident as described never occurred. 00:40:09.769 --> 00:40:12.610 The story is perhaps best appreciated as a cautionary 00:40:12.610 --> 00:40:15.150 tale about the intimidation factor of using specialized 00:40:15.150 --> 00:40:18.230 jargon rather than a factual account of Euler's 00:40:18.230 --> 00:40:21.070 rhetorical style. His real rhetorical power was 00:40:21.070 --> 00:40:23.550 in the elegance of his written proofs, not in 00:40:23.550 --> 00:40:26.269 courtly verbal trickery. His career came to an 00:40:26.269 --> 00:40:29.670 abrupt yet strangely fitting end. On September 00:40:29.670 --> 00:40:33.309 18, 1783, Euler was in St. Petersburg discussing 00:40:33.309 --> 00:40:36.110 the newly discovered planet Uranus and calculating 00:40:36.110 --> 00:40:38.789 its orbit with his student, Anders Johan Luxell. 00:40:39.130 --> 00:40:41.150 He suffered a brain hemorrhage and died instantly. 00:40:41.409 --> 00:40:43.809 The famous eulogy, written by the French mathematician 00:40:43.809 --> 00:40:46.349 and philosopher Marquis de Condorcet, provides 00:40:46.349 --> 00:40:48.769 the perfect epitaph for this immense intellectual 00:40:48.769 --> 00:40:51.789 engine. He wrote, Il cessa de calculer de vivre. 00:40:52.050 --> 00:40:54.829 He ceased to calculate and to live. His legacy 00:40:54.829 --> 00:40:57.170 is undisputed. John von Neumann called him the 00:40:57.170 --> 00:41:00.309 greatest virtuoso of the period. And Henri Poincaré 00:41:00.309 --> 00:41:02.750 went even further, simply calling him the god 00:41:02.750 --> 00:41:05.099 of mathematics. And for those who want to grasp 00:41:05.099 --> 00:41:07.519 the sheer scope one final time, the publishing 00:41:07.519 --> 00:41:10.179 effort to collate his opera omnia, his complete 00:41:10.179 --> 00:41:13.280 works, is still ongoing. The current total is 00:41:13.280 --> 00:41:16.099 80 volumes, cataloged systematically by the Innistrum 00:41:16.099 --> 00:41:19.780 Index from E1 to E866, covering every aspect 00:41:19.780 --> 00:41:22.019 of analysis, mechanics, physics, and correspondence. 00:41:22.659 --> 00:41:26.179 This body of work is the best school Gauss spoke 00:41:26.179 --> 00:41:28.460 of. He appears on the Swiss 10 -franc banknote 00:41:28.460 --> 00:41:30.880 and numerous stamps. His influence isn't just 00:41:30.880 --> 00:41:33.039 academic, it's embedded in the foundational... 00:41:33.230 --> 00:41:35.690 we all use to describe the universe. So what 00:41:35.690 --> 00:41:37.989 does this all mean for you? Euler's life shows 00:41:37.989 --> 00:41:40.190 the triumph of disciplined intellect, overcoming 00:41:40.190 --> 00:41:42.489 political instability and even total physical 00:41:42.489 --> 00:41:44.829 blindness through the power of a perfectly structured 00:41:44.829 --> 00:41:47.349 mind. But let's connect back to that unusual 00:41:47.349 --> 00:41:49.769 passion we discussed earlier, his attempt to 00:41:49.769 --> 00:41:53.369 quantify harmony, the grotus suavitatis. He believed 00:41:53.369 --> 00:41:55.230 musical sweetness could be derived precisely 00:41:55.230 --> 00:41:57.889 from the arithmetic of small prime numbers, the 00:41:57.889 --> 00:42:00.469 simplicity of the ratios three and five. It was 00:42:00.469 --> 00:42:02.389 a mathematical definition of aesthetic beauty. 00:42:02.780 --> 00:42:04.940 So here is the final provocative thought for 00:42:04.940 --> 00:42:08.280 you to mull over. If aesthetic concepts as subjective 00:42:08.280 --> 00:42:11.079 as musical harmony can be precisely derived from 00:42:11.079 --> 00:42:14.980 elegant, underlying quantitative laws, if beauty 00:42:14.980 --> 00:42:18.119 has a numerical root, then consider whether all 00:42:18.119 --> 00:42:20.559 forms of aesthetic pleasure or natural beauty, 00:42:20.699 --> 00:42:22.840 whether it's the structure of a flower, the flow 00:42:22.840 --> 00:42:25.199 of a river, or the shape of a seashell, might 00:42:25.199 --> 00:42:27.860 also be rooted in similarly elegant, universal 00:42:27.860 --> 00:42:30.000 quantitative laws that are just waiting to be 00:42:30.000 --> 00:42:31.960 discovered. Euler's work suggests that knowledge 00:42:31.960 --> 00:42:33.960 is ultimately not a collection of separate subjects, 00:42:34.039 --> 00:42:41.639 but one universe. Thank you for joining us on 00:42:41.639 --> 00:42:43.500 this deep dive into Leonhard Euler. We'll see 00:42:43.500 --> 00:42:43.940 you next time.