Imagine being the lead architect of like a 100
story modern soaring skyscraper Okay, or trying
to engineer the aerodynamics of a massive cargo
ship or even you know Just sitting down to write
a basic piece of software But here's the catch
you are not allowed to use numbers Wow good luck
with that right you have no alphabet You have
no standardized symbols of any kind and the crazy
thing is 300 years ago That was basically the
state of mathematics. It really was. It was just
a chaotic mess of competing languages until one
man sat down and essentially wrote the source
code for reality. And the wildest part. He did
his absolute best work after going completely
blind. It sounds like an impossible premise for
a life, doesn't it? I mean, you can't build complex
structures, whether they're physical buildings,
digital networks, or abstract theoretical frameworks.
If you don't first have a reliable way to communicate
the foundational concepts, you need a shared
grammar. And the man we are talking about today
didn't just learn the grammar, he invented it.
Which brings us perfectly to our mission for
today's deep dive. we are looking at a massive,
comprehensive, encyclopedic entry on Leonhard
Euler. A giant of history. Born in Switzerland
in 1707. And Euler is the man who standardized
the language of math and science. As I was going
through these sources, I was trying to wrap my
head around the, well, the sheer volume of his
output. It's hard to even comprehend. Right.
Between 1725 and 1783, Euler averaged 800 pages
of published work a year. 800 pages. Every year.
It is staggering. He offered roughly a quarter
of all the combined output in mathematics, physics,
mechanics, astronomy, and navigation in the entire
18th century. A quarter of everything. Yes. But
what makes Euler so captivating isn't just the
page count. It is the contrast of his life. Oh,
so. Well, on one hand, you have his infinite,
perfectly ordered, brilliant abstract mind. And
then on the other, you have the incredibly messy,
physical, and frankly, highly volatile political
world he was forced to navigate every day. OK,
let's unpack this. Because to understand what
Euler built, we really have to look at this alphabet
of modern math that he gave us. Right. If you
listening to this right now have ever taken a
high school algebra or calculus class, you're
speaking Euler's language. He popularized or,
you know, outright invented notations that we
just take for absolute granted today. We don't
even think about them. Exactly. Like, he was
the first to use the notation f of x to denote
the value of a function. Yes, and it didn't stop
there. He gave us the letter e for the base of
the natural logarithm. He started using the letter
i to express imaginary numbers. Wow. And he introduced
the Greek letter sigma for mathematical summations.
And perhaps most famously, He popularized using
the Greek letter pi to denote the ratio of a
circle's circumference to its diameter. Right,
the classic pi. Now, the sources do note that
a mathematician named William Jones technically
used pi in that context first. Yeah. But it was
Euler who put it in his widely circulated textbooks
and really cemented it in the global consciousness.
He made it the standard. Right. And the analogy
that kept hitting me while reading this is that
Euler is basically a foundational software developer.
Oh, that's interesting. How do you mean? Well,
he didn't just write a really great app, like
a single brilliant theorem that solved one specific
problem. He invented the coding language itself.
I see. He wrote the C++ or the Python of mathematics.
Every other developer, every... physicist, every
structural engineer for the next 300 years, was
essentially forced to use Euler's coding language
to build their own apps. What's fascinating here
is the collaborative compounding effect of that
standardization, because before Euler, mathematicians
in France might be using entirely different notation
than mathematicians in England or Germany. It
was just completely fragmented. Exactly. By standardizing
this mathematical grammar, Euler allowed humanity
to finally collaborate across borders and, crucially,
across centuries. Yeah, that's a huge shift.
You simply cannot have a global scientific revolution
if everyone is speaking a different theoretical
language. So he builds this alphabet. But an
alphabet is just a tool, right? How did he actually
use it to change mathematics? That's where things
get really incredible. Because the sources talk
a lot about Euler's identity, or Euler's formula,
which uses his own standardized notations to
establish that E to the power of i times pi plus
one equals zero. That is correct. e to the i
pi plus one equals zero. The physicist Richard
Feynman famously called it the most remarkable
formula in mathematics. Hold on. I need to visualize
this because for a lot of people that just sounds
like a random string of math words. Fair enough.
You're taking in a rational number. e raising
it to the power of an imaginary number multiplied
by another irrational number pi, and somehow
that perfectly equals negative one. So when you
add one, you get zero. It sounds like magic,
doesn't it? How is that even mathematically possible?
It is profoundly elegant once you see the mechanism
behind it. Let's break it down simply. Okay,
please do. So the number E deals with continuous
compounding growth, and the letter I represents
imaginary numbers. Now, instead of thinking of
I as just some weird, impossible number, think
of it spatially. Spatially, like on a graph.
Exactly. If you have a normal number line going
left to right, multiplying by I is like rotating
90 degrees upward into a new vertical dimension.
This creates what we call the complex plane.
OK, so I is basically like a rotational instruction.
Yes. Exactly. And then you have pi, which of
course relates to circles. Right. So when you
take compounding growth, E, and apply this rotational
instruction, I, for exactly pi amount of time,
which is halfway around a circle. Oh, wow. You
start at positive 1 on the number line, rotate
180 degrees through this alternate dimension,
and land precisely on negative 1. Add 1, and
you get 0. That is wild. It bridges exponential
growth, circular geometry, and the complex plane
all in one impossibly simple equation. It's just
perfectly neat. It's as if you took the five
fundamental constants of reality zero, one, pi,
e, and i, and discovered they all fit together
into a single flawless puzzle piece. So he has
this unique ability to strip away the noise and
find these deep underlying connections, which
makes me want to look at how he applied that
same vision to problems that had stumped everyone
else. He had this habit of taking real -world
dead ends and turning them into abstract games.
Yes, his ability to look at an unsolvable problem
and completely reframe it was unparalleled. Let's
talk about the Basel problem first because this
one drove mathematicians absolutely crazy. Hawaii
was the bane of the mathematical community for
a long time. For almost a century, the greatest
minds in Europe had been totally stumped by it.
The challenge was to find the exact sum of the
reciprocals of the squares of every natural number.
Which is a mouthful. Yeah. So 1 over 1 squared
plus 1 over 2 squared plus 1 over 3 squared onto
infinity. No one could figure out exactly what
that infinite series converged to. Nobody. And
then Euler comes along in 1735 and proves it
equals pi squared divided by 6, which is an amazing
flex. But where does pi come from in a list of
squares? It is a brilliant question, and the
answer shows exactly how Euler's mind worked.
Euler looked at this infinite sum of fractions,
and he essentially asked... What if I treat this
endless addition problem like a giant algebraic
polynomial? A polynomial. Now in algebra, if
you have a polynomial, you can break it down
into its roots, the points where the equation
crosses zero. Right, like finding x when y is
zero. Exactly. Then Euler looked around for something
in nature that crosses zero infinitely many times.
Like a wave. And he found it in the sine wave.
A sine wave oscillates up and down forever, crossing
zero exactly at multiples of pi. one pi, two
pi, three pi, and so on. Oh, because sine deals
with circles and angles. Precisely. Euler set
the infinite polynomial of those fractions equal
to the infinite series that describes a sine
wave. That is so clever. By matching the mathematical
parts of the waves to the parts of the fractions,
the roots of the sine wave, which are built entirely
out of pi, suddenly dropped right into the equation.
Wow. He proved that an infinite sum of squares
is secretly describing the geometry of a circle.
That is genuinely beautiful. He finds the hidden
architecture, but his knack for solving the unsolvable
wasn't just limited to pure number theory. Not
at all. He applied this to the physical world
too, which brings us to the seven bridges of
Konigsberg. Yes, a very famous problem. Konigsberg
was a real city in Prussia. It was set on a river
and it had two large islands. These islands were
connected to each other and to the mainland.
by exactly seven bridges. Right. And the citizens
had this local riddle. Could you walk a continuous
path through the city that crossed each of the
seven bridges exactly once? A seemingly innocent
puzzle. But I have to push back here, or at least
ask the question that I think anyone listening
would have. Sure. The Konigsberg Bridge problem
sounds like a simple, you know, distraction puzzle
you'd find on the back of a kid's restaurant
menu. Like a maze. Yeah. Like, can you draw a
line through this maze without picking up your
crown? Why is the greatest mathematician of the
18th century wasting his precious time on a local
walking riddle? It is a completely fair question.
And the answer is that Euler didn't just solve
the puzzle. He proved the puzzle was fundamentally
rigged by looking at the invisible rules governing
it. Rigged how? Well, if we connect this to the
bigger picture, Euler realized a very simple,
undeniable rule of movement. If you walk onto
a landmass by crossing one bridge, You have to
leave that landmass by crossing a different bridge.
Right. One bridge in, one bridge out. Exactly.
That means every single landmass in the city
must have an even number of bridges attached
to it, pairs of ins and outs, unless it is the
absolute start of your journey or the absolute
end of your journey. Oh, wow. Let me guess, Konigsberg
didn't have even numbers. Not a single one. All
four land masses in Konigsberg had an odd number
of bridges connecting to them. So it was impossible
from the start? Exactly. By identifying this
invisible law, Euler proved that no such path
could ever exist. The physical distances between
the bridges didn't matter. The size of the islands
didn't matter. He just ignored all the real world
stuff? He stripped away the actual geography
of the city, reducing the land masses to simple
points. what we now call vertices, and the bridges
to simple lines connecting them, which we call
edges. He threw out the map entirely. He did.
And in doing so, he didn't just solve a riddle,
he invented graph theory. Which is the foundation
of so much of our modern world. I mean, it's
modern computer science, it's network analysis,
it's literally how your GPS routes you through
traffic. by treating intersections as vertices
and roads as edges. Exactly. By stripping away
the physical distractions, Euler also laid the
groundwork for topology. What exactly is that?
That is the mathematical study of properties
that are preserved through deformation. In topology,
you can stretch or twist or bend an object like
turning a coffee mug into a donut shape without
breaking its fundamental mathematical properties.
It requires an incredible amount of mental visualization
to be able to look at a bustling physical city
and abstract it into pure floating vertices and
edges in your mind. It really does. But the profound
almost cruel irony of Euler's life is that as
his internal mental vision was expanding, his
actual physical vision was failing. It is one
of the most poignant aspects of his biography.
His eyesight deteriorated relentlessly throughout
his career. By 1738, he was almost completely
blind in his right eye. He blamed it on a severe
fever he'd had and, you know, the incredibly
straining cartography work he was doing, making
maps for the St. Petersburg Academy. A very demanding
job. The king of Prussia, Frederick the Great,
used to actually mock him and call him Cyclops.
Which is just awful. Yeah. And then years later,
he develops a cataract in his left eye, a botched
surgery follows, and he is left almost totally
blind. For a mathematician who works with highly
complex equations, massive polynomials, and geometric
proofs, you would expect total blindness to be
a career -ending tragedy. Right. But here's where
it gets really interesting. When most people
lose a vital sense, their output drops. They
retreat. Naturally. But when Euler lost his sight,
he famously said, now I will have fewer distractions.
And the sources show that his productivity didn't
just hold steady. It skyrocketed. By 1775, he
was entirely blind, and he was dictating roughly
one mathematical paper a week to his scribes.
One paper a week at the highest possible level
of theoretical mathematics. I found myself comparing
his mind to an unparalleled offline supercomputer.
He couldn't look anything up anymore, so he just
stored reality in his head. A perfect way to
describe it. And how he organized it is fascinating.
The sources note that he had memorized the entirety
of Virgil's epic poem, the Aeneid, but not just
the words. He could recite the first and last
sentence of any page from the exact edition he
read as a kid. That is photographic memory on
another level. He had a perfectly constructed
mental desktop. He knew the first 100 prime numbers
perfectly, and he could calculate their powers
up to the sixth degree instantly in his head.
What's truly remarkable here isn't just the memory,
it's the psychological marvel of it. What do
you mean? We have this pervasive cultural trope
of the tortured neurotic genius. The person whose
brilliant mind is a burden that isolates them
and makes them bitter. Right, the mad scientist
stereoscience. Exactly. But the historical accounts
describe Euler as maintaining a remarkably kind,
generous, and good -natured disposition, even
in total darkness. He wasn't lashing out at the
world because of his condition. Not at all. His
mind was so rigorously organized that blindness
didn't trap him in the dark. In a strange way,
it freed him. Freed him from the distractions.
Yes. It allowed him to live entirely within the
pure, perfect architecture of his mathematics,
completely untethered from the physical messiness
of light and ink. But while Euler's internal
world was perfectly ordered, the external 18th
century world he lived in was anything but. It
was chaotic, it was highly political, and it
was deeply flawed. Yes, his career was constantly
buffeted by the shifting political winds of Europe.
He spent his professional life bouncing back
and forth between the Imperial Russian Academy
in St. Petersburg and Frederick the Great's Berlin
Academy in Prussia. And these weren't just peaceful
academic sabbaticals. These moves were driven
by regime changes, strict censorship, and outright
war. Very dangerous times. At one point during
the Seven Years' War, Euler's farm in Charlottenburg
was literally sacked by advancing Russian troops.
Though it is worth noting the Russian general
was so horrified that his troops had looted the
great oiler's estate that he personally paid
compensation immediately. Oh, wow. And the Empress
of Russia added an exorbitant sum on top of it.
He was viewed as a highly valued, almost untouchable
commodity. But his time in the Prussian court
wasn't exactly easy either. Euler was devoutly
religious, he was a simple man, and by all accounts,
he was not a skilled witty debater. No, he was
quite straightforward. Which made him an easy
target for the sophisticated, cynical intellectuals
at Fredericks Court, specifically Voltaire. who
just loved to verbally spar with him and humiliate
him in public. Voltaire thrived in the realm
of philosophy, ambiguity, and sharp rhetoric,
whereas Euler operated in absolute truths. They
were oil and water. So what does this all mean?
As I read these accounts, I couldn't help but
wonder, was Euler basically just the ultimate
math nerd getting shoved into lockers by the
cool kids at the royal court? It can certainly
seem that way from a distance. I mean, Frederick
the Great famously mocked Euler when the water
fountains at his summer palace on Susi failed
to work. Ah, yes, the fountain incident. The
king wrote this scathing letter complaining that
Euler had miscalculated the force needed to raise
the water, famously scoffing, vanity of vanities,
vanity of geometry. This raises an important
question about the friction between theoretical
science and practical execution. And it is a
criticism that often gets leveled at purely theoretical
thinkers. But we have to defend Euler here. Okay,
defend him? Modern engineering analysis has looked
back at those specific fountain plants. Euler's
calculations for the fluid dynamics were perfectly
sound. So the math was actually right. The math
was completely right. The problem was that the
18th century builders simply didn't have the
materials or the execution skills to construct
pipes that could handle the pressure Euler had
accurately calculated. Oh, the pipes couldn't
take it. Yeah. The wooden and lead pipes burst
because of practical engineering limitations,
not theoretical failures. And furthermore, it
is a huge mistake to view Euler as just an abstract
theorist disconnected from the real physical
world. Because his practical applications were
everywhere. Exactly. He wasn't just doing math
for math's sake. In fluid dynamics, he mathematically
predicted the phenomenon of cavitation in 1754,
long before it could even be observed physically.
Wait, what actually is cavitation? for those
who aren't engineers. It is what happens when
the pressure in a flowing liquid drops so abruptly
that the liquid actually boils and forms vapor
bubbles, like around a rapidly spinning ship
propeller. When those bubbles collapse, they
create massive shock waves that can tear metal
apart. Euler predicted that purely through math.
That is incredible. He also helped develop the
Euler -Bernoulli beam equation, and he figured
out the formula for a structural column's critical
buckling load. Which means what, practically
speaking? Imagine pressing down on the top of
a dry piece of spaghetti. At a certain point
of pressure, it bows outward and snaps. Euler
created the exact mathematical formula to predict
exactly how much weight a column of any material
can take before it bends and buckles. These aren't
abstract concepts. These are the literal structural
engineering blueprints that allow us to build
skyscrapers safely today. It is amazing how his
reputation swung between being mocked by kings
for a fountain to being viewed as this untouchable,
almost divine mathematical deity. He was a paradox
to them. Which brings up my absolute favorite
anecdote from the sources. We have to clearly
state upfront, historians widely agree this story
is likely a legend, an apocryphal myth. but it
so perfectly illustrates how the rest of the
world viewed his intellect. Ah, the Diderot story.
Yes, so the legend goes that the famous French
philosopher, Ditas Diderot, who was a prominent
atheist, was visiting the Russian court. And
causing trouble, as usual. Right, and he's going
around making all these eloquent philosophical
arguments against the existence of God, and he's
completely flustering everyone. So the Empress
asks Euler to step in and confront him. The ultimate
showdown. Euler walks right up to Diderot in
the middle of the court. looks him dead in the
eye, and says, in a tone of absolute conviction,
sir, a plus b to the nth power divided by n equals
x. Hence, God exists. Reply. And according to
the legend, Diderot knowing absolutely nothing
about advanced mathematics is completely dumbstruck
by this algebraic nonsense. He has no idea what
to say. The whole court erupts into laughter
and Diderot is so embarrassed he immediately
asks for safe passage out of Russia. It's just
a brilliant image slaying a philosophical opponent
with literally meaningless algebra just because
your reputation is so massive that no one dares
question your math. They just assumed he was
right. Even if it's totally fake, it tells you
everything you need to know about Euler's aura.
It does. It highlights the reverence people held
for his intellect. He was the man who could quantify
the divine or at least quantify the universe
in a way that felt divine to the rest of humanity.
And the reality is every single one of you listening
to this deep dive right now is living inside
Leonhard Euler's calculations. Every single day.
Every time you drive over a suspension bridge
and trust that the structural columns won't buckle
under the weight of traffic. Every time you open
a GPS map app on your phone to find the fastest
route using network edges and vertices. Yep.
Even when you go to a concert and appreciate
the acoustics of the hall or just listen to a
piano. Yes, his work in music theory shouldn't
be overlooked. He spent immense time applying
mathematical concepts to music, developing ways
to numerically describe the subdivision of octaves
into fractional parts using prime numbers. He
saw math everywhere. He viewed harmony and pitched
through the exact same lens of pure mathematics
as he viewed fluid dynamics. He saw the code
behind the chords. It's just incredible. Euler
worked relentlessly, entirely blind until the
very last day of his life. He never stopped.
The sources say that in September 1783, at the
age of 76, he was having lunch with his family,
discussing the newly discovered planet Uranus,
and actively calculating its orbit in his head.
Over lunch. Over lunch. He collapsed from a brain
hemorrhage and passed away. The Marquis de Condorcet
wrote eulogy for the French Academy that I think
serves as the most fitting epitaph possible for
a man of his nature. What did he write? He simply
wrote, he ceased to calculate and to live. For
Euler, those two things were entirely synonymous.
He ceased to calculate and to live. It's beautiful.
It really is. So as we wrap up our dive into
the life of the man who wrote the operating system
of the modern world, I want to leave you with
one final thought to mull over. We started by
imagining trying to build a massive skyscraper
without an alphabet. A daunting task. Euler was
a blind man in the 18th century who used pure
mathematics to abstract the physical world of
bridges, fluid dynamics, and celestial orbits
into immortal perfect equations. He mapped the
invisible. If you could see all of those hidden
structures in the dark, what unseen patterns
governing our own chaotic modern lives are sitting
right in front of us right now, just waiting
for new language to describe them.