Welcome back to the Deep Dive. We are here to
take a massive stack of sources, articles, research
papers, biographies, and really just distill
them down into the essential knowledge you need,
giving you that deep understanding without all
the information overload. And today we are profiling,
well... arguably the most prodigious intellect
in the history of science. We're talking about
Leonhard Euler, born in Basel in 1707. And if
you had to pick just one person who completely
dominated the intellectual landscape of the 18th
century, it would have to be him. Oh, absolutely.
He wasn't just a mathematician, though that's
what he's famous for. The sources all confirm
he was a polymath in the true sense of the word.
A physicist, an astronomer. A logician, a geographer.
And a pioneering engineer. He really did it all.
And when we use a word like pro... We are not
exaggerating. I mean, his collected works are
so massive, they're still not even fully published.
It's a project that's been ongoing for over a
century. Right now, it's clocking in at around
80 volumes. And that that scale is exactly why
these later titans of mathematics spoke about
him with this. this almost religious awe. Right.
You have that famous quote from Pierre Simon
Laplace who basically commanded everyone, read
Euler, read Euler. He is the master of us all.
I mean, that's the kind of endorsement you only
get from someone who recognizes pure foundational
genius when they see it. And then there's Carl
Friedrich Gauss, another absolute heavyweight
in the history of math. He echoed that same sentiment.
He did. He said that studying Euler's works will
remain the best school for the different fields
of mathematics and nothing else can replace it.
So our mission today is to give you that best
school shortcut. We want you to walk away from
this understanding, not just what Euler discovered,
but really how he changed the way we all think
about the physical world and even the language
we use to describe it. To really set the stage
for his career, you have to think about this.
His entire professional life was spent traveling
between two of the most powerful, most demanding
intellectual centers in Europe. He served the
Imperial Russian Academy of Sciences in St. Petersburg.
And the Berlin Academy in Prussia. He was essentially
managing the scientific output for competing
empires. That just, it speaks volumes about how
indispensable he was. It really does. But here's
the critical initial aha moment for you. I think
this is the key piece of knowledge for anyone
who's ever struggled through algebra or calculus.
Go on. Euler didn't just solve these high -level
abstract problems. He created the toolkit. He
literally invented or popularized the symbols
that are the bedrock of all modern mathematical
communication. Let's really linger on that because
it's so important. When you see fx written on
a blackboard, the notation for a function, that's
Euler. When you use the constant e for the base
of the natural logarithm Euler's number, that's
him. If you ever encounter the imaginary unit,
you know, the square root of negative one. That's
Euler. The capital sigma. Four summations. And
even the standard use of the Greek letter for
the circle ratio. All of it. It all owes its
common, everyday use to Leonhard Euler's incredible
standardization efforts. He wasn't just a scientist.
You put it perfectly earlier. He was a linguistic
architect for science. He took this fractured
kind of personalized system of notation and gave
it a unified universal alphabet. And that made
communication across borders, across disciplines,
not just easier, but possible. OK, so let's unpack
the path that led to this monumental contribution.
We have to start with his early life and education,
the genesis of this genius. Right. So he was
born in Basel in 1707. And the first thing that
really jumps out from the sources is how deep
his connection to the scientific establishment
was. I mean, right from birth. It runs right
through his family tree, doesn't it? It does.
His father, Paul III Euler, was a reformed church
pastor. But, and this is the key detail, he was
also highly educated. He had actually studied
under Jacob Bernoulli, one of the first pioneers
of calculus, right there at the University of
Basel. So that mathematical tradition was literally
in the household from day one. Exactly. He grew
up in Rhine, near Basel, and got his initial
instruction from his father. That was then supplemented
with some private tutoring from a young theologian
named Johannes Burkhart. So the path was kind
of set early, even though his father, like a
lot of parents, had a very specific career in
mind for him. But Euler's his innate talent just
quickly outgrew that initial curriculum. He enrolled
at the University of Basel in 1720 at the. Well,
the astonishingly young age of 13. And this wasn't
just him being precocious. This was raw intellectual
firepower being applied very, very early. And
while he was there, he connected with the person
who would become his most crucial mentor. Johann
Bernoulli. Jacob's younger brother and himself
one of the towering mathematical figures of the
time. And Johann Bernoulli's teaching method
for Euler is the stuff of legend. It's actually
something you, the listener, could adopt for
your own learning, really. Euler wrote about
it himself. He detailed how Bernoulli was often
just too busy for formal private lessons. But
he provided something far more impactful, what
he called salutary advice. This is where it gets
really interesting. Bernoulli instructed him
to go read the most difficult, the densest mathematical
books he could find, just digest them on his
own. And then come back every Saturday afternoon
with all his difficulties and questions. The
insight here is just crucial, isn't it? Bernoulli
wasn't just feeding him answers. Not at all.
By forcing Euler to wrestle with the material
first, he was teaching him intellectual independence.
Euler himself noted that when Bernoulli resolved
one objection he brought, he said, ten others
at once disappeared. It was a system designed
to teach him how to resolve these fundamental
intellectual roadblocks. It turned him into a
self -learner capable of dominating, well, any
field he chose. And that guidance fundamentally
changed his life path. Euler's father had strongly
intended for him to become a pastor, you know,
a theologian following the family profession.
But Bernoulli, seeing the sheer depth of his
mathematical talent, actively intervened. He
went and convinced the older Euler to let his
son pursue mathematics exclusively. I mean...
Think about that endorsement. That's the 18th
century's leading mathematician telling a pastor,
no, your son is meant for something else entirely.
And he progressed so rapidly. By 1723, at just
16 years old, he received his Master of Philosophy.
And his thesis was remarkably ambitious. It was
a comparison of the philosophical systems of
René Descartes and Isaac Newton. Even as a teenager,
he's synthesizing the major intellectual currents
that were defining modern European thought. But
genius doesn't always guarantee a job, does it?
No. Despite his obvious talent, he couldn't immediately
secure a position at Basel. He wrote his dissertation,
Dissertatio Physica de Sono, a physical dissertation
on sound in 1726, but the local institution just
wasn't ready to hire him. So he looked to the
wider European stage to prove himself. And this
is where we see his first real test against his
contemporaries, the annual Paris Academy Prize
competition in 1727. And the problem was intensely
practical. It was determining the best way to
place the masts on a ship. This wasn't abstract
algebra. This was applied physics and naval architecture.
And he took second place, which is a phenomenal
achievement for a 20 -year -old newcomer. He
was beaten that year by Pierre Bougier, who was
known as the father of naval architecture. So
no shame in that loss. But that initial entry
just signaled his capabilities. That minor defeat
only served as motivation, really. Oh, for sure.
Over the course of his career, Euler would go
on to dominate that competition. He eventually
won the coveted Paris Academy Prize. A staggering
12 times. So this early period really proves
the dual nature of his genius. He was capable
of both the highest theoretical abstraction and
at the same time solving these complex real -world
engineering and physics problems. So Euler proved
his worth, but Bessel still wasn't the right
place for him. And that brings us to section
two. The transcontinental career that really
defined his adult working life, starting with
that first St. Petersburg period in 1727. And
this Russian connection, thankfully, was secured
by the Bernoulli family again. Of course. Johann
Bernoulli's sons, Daniel and Nicholas, were already
established in St. Petersburg. They were the
newly founded Imperial Russian Academy of Sciences.
But unfortunately, Nicholas died tragically of
appendicitis shortly after arriving. Which created
a perfect vacancy. It did. And Daniel, recognizing
Euler's immense talent, immediately secured the
position for him. So Euler arrived in May 1727.
And although he was initially placed in the medical
department, he very quickly transitioned to pure
mathematics. And he lodged and worked closely
with Daniel Bernoulli, right? That's right. And
he didn't just work in the academy. He completely
immersed himself in the culture. He mastered
Russian, adapted to the demanding schedules,
and even took on an additional job as a medic
in the Russian Navy. It's worth noting, too,
that he was offered a promotion to lieutenant,
but he turned it down. He made it very clear
that his passion was pure scholarship, not command.
But the political climate in Russia at this time
was incredibly unstable. The Academy had been
founded under the progressive vision of Peter
the Great. But after the death of Catherine I,
conservative Russian nobility began to regain
power under the young Peter II. And that meant
peril for the Academy. Big trouble. The conservative
faction was deeply suspicious of the foreign
scientists. Many of them were German or Swiss.
And this led to significant funding cuts, bureaucratic
hostility, and just a very uncertain future for
all the academics. The conditions became so precarious
that Daniel Bernoulli eventually returned to
Basel, right? He cited political censorship as
a major factor. He did. So it really took a shift
in leadership to stabilize the environment. In
1730, Peter II died, and Empress Anna, who had
German ties and was much more sympathetic to
modern science, took the throne. And this stabilization
is what allowed Euler to really flourish. He
ascended so rapidly. He was made a professor
of physics in 1731. And then when Daniel Bernoulli
departed in 1733, Euler, who is now just 26,
succeeded him as the head of the mathematics
department. And he really cemented his life in
Russia, both professionally and personally. In
January 1734, he married Katharina Grissel, who
was the daughter of a Swiss painter working at
the Academy. This stability was absolutely vital,
considering the immense intellectual output he
was about to unleash. However, despite that stability
under Empress Anna, the political situation was
still a long -term concern, especially after
her death. Right. And when Frederick the Great
of Prussia began aggressively recruiting him
for the Berlin Academy... Euler was tempted.
In 1741, he accepted Frederick's offer. There's
a small detail that I think illuminates his human
side. He partly justified the move by claiming
he needed a milder climate for his deteriorating
eyesight. But the core motivation was the promise
of a more stable, more intellectually focused
environment in Prussia. The Russian Academy,
by the way, was devastated to lose him. Which
really demonstrates his immense value to them.
Absolutely. They actually agreed to keep him
on the payroll. paying him an annual stipend
and considering him an active member, even as
he spent the next 25 years working for their
direct competition. So this transition marks
the Berlin period from 1741 to 1766, which is
often cited as the peak of his sheer analytical
productivity. He was in an environment perfectly
tailored for scholarship. And he wasted no time.
This is where he published the seminal textbooks
that formalized modern analysis. In 1748, he
released Introductio in Analysin Infinitorum,
an absolute foundational text on the theory of
functions and infinite series. And then in 1755,
Instituciones Calculae Differentialis, which
covered differential calculus. You have to understand,
these weren't just dusty academic monographs.
They were comprehensive, they were clear, and
they were standardized teaching tools that rapidly
circulated all across Europe. And they logged
in his new notations and methods for generations
of students to come. But his influence wasn't
just limited to the mathematical elite. This
period also produced one of his most beloved
works. And this one was aimed squarely at the
lay public. The letters of Euler on different
subjects in natural philosophy addressed to a
German princess. He wrote over 200 letters while
he was tutoring Frederick the Great's niece,
Princess Friedrich Charlotte, on topics ranging
from philosophy and optics to magnetism and logic.
This collection is just a monument to his genius
for communication. It shows this profound ability,
which is rare among intellectual giants, to translate
these incredibly complex scientific principles
into accessible elegant prose for a general audience.
And the letters were translated into seven languages,
published in dozens of editions, and became really
astonishingly more widely read across Europe
and the early American states than any of his
specialized mathematical papers. Yet despite
running the Berlin Academy's intellectual output
and its prestige, his relationship with King
Frederick II was famously strained. The king
viewed Euler as a brilliant calculator, but intellectually
unrefined, sort of lacking in the courtly wit
and general culture that Frederick prized so
much. And that tension was definitely heightened
by the presence of Frederick's favorite philosopher,
Voltaire. Euler was simple, devout, nonpolitical.
He was the exact opposite of the witty, secular,
politically connected Voltaire. And Voltaire
often targeted Euler's lack of polish in intellectual
debates that were outside of pure mathematics.
Right. It sounds like Euler was brilliant in
a structured environment, but maybe a little
uncomfortable in the cut and thrust of courtly
debate. That's precisely the dynamic the sources
suggest. Frederick, who was a connoisseur of
sophisticated conversation, just found Euler
boring. He reportedly told d 'Alembert that Euler's
conversation lacked grace, saying Euler had a
great genius but just could not sustain a salon
conversation. And nothing, nothing encapsulates
Frederick's irritation with Euler's practical
application quite like the infamous Sanssouci
fountain incident. This one is always worth dwelling
on. Frederick wanted this magnificent geometric...
calculated system of water jets for his garden
at Sansuti. He tasked Euler with designing the
optimal system. The result? A spectacular failure.
The water pressure was never adequate. And the
fountain never worked as intended. And the King's
quote on the failure is just brutal. Vanity of
vanities, vanity of geometry. He pointed the
finger squarely at Euler and theoretical mathematics.
But what's fascinating here is the technical
context. The sources largely agree that the King's
disappointment was likely unwarranted, at least
from a purely engineering perspective. So Euler's
calculations were probably correct. Almost certainly.
The calculations for the hydraulics, the geometry,
the fluid movement. They were likely correct.
The failure probably lay in the poor execution
of the construction by the builders, or the materials
used, or the practical limitations of the water
source. These were issues outside of Euler's
pure math domain. So the story isn't that Euler
failed, it's that Frederick preferred to blame
the theoretical genius he found personally tedious
rather than, say, the expensive contractors.
That sharpens the political tension perfectly.
And despite all this friction, Euler's output
was relentless. During this 25 -year stint in
Berlin, he wrote a staggering 380 works, with
275 of them published. And on top of that, he
was effectively the chief administrator for the
Academy. We tend to focus only on the math, but
his administrative duties were massive. He supervised
the library, ran the observatory, managed the
botanical garden, and, this is critical, he oversaw
the publication of the maps, calendars, and almanacs,
which were vital sources of income for the academy.
He was running a scientific conglomerate. This
dual role just underscores his unbelievable discipline.
And we have to remember his incredible loyalty
to St. Petersburg. Even while he was in Berlin,
he published over 100 memoirs there, he kept
contact with Russian students, and he even helped
accommodate them in his own home. That loyalty
paid off during the tumultuous Seven Years' War.
In 1760, advancing Russian troops sacked his
farm in Charlottenburg. But when the Russian
general, Ivan Petrovich Saltykov, heard about
this, he immediately paid substantial compensation
for the damage. And then Empress Elizabeth, upon
hearing of the incident, added an additional
4 ,000 rubles, an exorbitant amount at the time.
He was an international intellectual asset. No
power could afford to offend him. By 1766, the
conditions back in St. Petersburg under Catherine
the Great were far more favorable. They were
offering better pay and far greater recognition
than Frederick was ever willing to give. So Euler
accepted the invitation to return. His demands
were quite significant, reflecting his recognized
worth, a 3 ,000 ruble annual salary. a pension
for his wife, and high -ranking appointments
for his sons, Catherine met every single demand.
And this marks the second St. Petersburg period,
from 1766 to 1783, a time that's really defined
by physical deterioration, but this unparalleled
intellectual defiance. Euler's eyesight had been
failing for decades. Frederick II had already
dismissively called him Cyclops due to the near
-total blindness in his right eye, which was
possibly aggravated by his earlier detailed cartography
work. Famous quote from Euler when he first lost
sight in that right eye is just so indicative
of his character. He said, now I will have fewer
distractions. I mean, most people would see the
end of their career. Euler saw an opportunity
for deeper internal focus. But the tragedy compounded
in 1766. Complications from cataract surgery
left him almost totally blind in his remaining
left eye. Total darkness descended upon him.
Most academics would have retired immediately.
But this is where the legend of Euler is truly
forged. His productivity. It increased. Unbelievable.
He just adapted his methodology. He hired scribes.
He relied entirely on his phenomenal memory and
mental arithmetic. And he dictated his work,
often to his students like Anders Johan Lexel.
Let's just quantify that defiance for a minute.
In 1775, years after losing his sight, he was
averaging an astonishing one mathematical paper
per week. He was dictating analysis, formulas,
and proofs entirely from memory. It is one of
the most remarkable feats of sustained intellectual
output in all of human history. It speaks to
a mind that was just perfectly structured, able
to manipulate complex, multivariable equations
purely within the confines of his own skull.
And on the personal front, this period saw even
more upheaval. His home was destroyed by a fire
in 1771, and he lost his first wife, Katharina,
in 1773. To maintain the stability that was so
crucial for his ongoing work and for his five
surviving children, he married Katharina's half
-sister, Salome Abigail Grissel, in 1776. The
story of Euler is so often told through his great
discoveries, but we established earlier that
his first major contribution was really the language
he gave to mathematics. Absolutely. So section
three here is dedicated to this notational revolution
because he didn't just solve problems. He gave
us the tools to write them down in the first
place. Before Euler's textbook standardized everything,
math was this. This Babylonian mess of competing
symbols and personalized abbreviations. That's
a great way to put it. If you wanted to read
Newton, you had to learn Newton's symbols. If
you wanted to read Leibniz, you learned his.
Euler introduced a unified logical language that
every student today, whether you're in high school
or graduate school, still uses. Okay, so let's
walk through the foundational symbols. First,
the most elementary concept in modern math. The
function. Euler gave us fx. The use of f of x
for the value of a function f applied to the
argument x. Before this, expressing the relationship
between variables was clunky and descriptive.
So fx gave us a concise, universally recognized
shorthand. It revolutionized algebra and calculus
instantly. It allowed mathematicians to treat
operations and transformations as objects in
themselves. Then we have the constants, the letter
e, Euler's number, the base of the natural logarithm.
It's central to modeling continuous growth and
decay everything from population growth to radioactive
half -life, NYE. The sources suggest it may have
simply been the next vowel in a series of variables
he was using. But once he used it in his foundational
1748 textbook, Introductio in Analysin Infinitorum,
it stuck instantly because it was clear. It was
logical. It's the constant that defines exponential
processes. And crucially, he introduced the letter
I for the imaginary unit, the square root of
negative one. This is the absolute foundation
of complex analysis. And this was a deeply abstract
concept that had been debated since the 16th
century. Exactly. By giving it a simple standardized
symbol, I. Euler normalized and formalized its
use. You simply cannot do modern physics, electrical
engineering, signal processing, quantum mechanics
without the symbol I and the complex number system
it anchors. He also helped secure the use of
Bricton. That's right. While he credited the
Welsh mathematician William Jones with the original
use back in 1706, it was Euler's extensive use
of the Greek letter. in his major, widely read
text that cemented it as the universal standard
for the ratio of a circle's circumference to
its diameter. And it wasn't just letters. It
was operational symbols, too. Right. He introduced
the Greek letter drat. capital sigma for summations.
This provided a concise way to represent long
or infinite series. Before, representing a series
was just cumbersome prose. Afterwards, it became
a clean, universally readable equation. And the
Greek letter A capital delta for finite differences,
which is essential for numerical methods and
early calculus work. Even geometry was standardized
by him. When you look at a textbook today, you
see triangle sides labeled with lowercase letters
and the angles opposite them labeled with corresponding
capital letters. That clear, simple convention,
that's Euler. He was ensuring that geometry problems
could be read consistently across continents.
So the takeaway here is that standardization
was Euler's first great intellectual act. It
made math accessible and cumulative. It's so
rare for one person to have such a profound and
lasting impact on the global language of an entire
field. So when you are struggling through a complex
physics problem today, just remember you are
using the precise, organized language Euler invented,
saving centuries of confusion. That's his gift
to every modern student. And now we transition
from the language builder to the discoverer.
Sections 4 and 5 are going to cover the sheer
breadth of his original theoretical and applied
research. Let's start with analytical contributions,
the area where he was truly sovereign. Right.
The 18th century was dominated by the development
of infinitesimal calculus, and Euler was the
most powerful engine in that development. He
was famous for his frequent use and development
of power series. Power series expressing functions
as infinite sums of powers of x. He used them
like no one before him. If you can express a
function as an infinite polynomial, you can manipulate
it, integrate it and differentiate it with comparative
ease. He showed, for example, the exponential
function e to the x as that infinite series.
e to the x equals the sum from n equals zero
to infinity of x to the n over n factorial. His
willingness to just work with infinite processes
allowed him to achieve results that baffled his
contemporaries. And the most famous example of
this analytical power has to be his solution
to the Basel problem. This problem had stood
open since 1644, posed by Pietro Mangoli, and
it had frustrated the brightest minds of the
age, including the elder Bernini. The challenge
was to find the exact sum of the reciprocals
of the squares of every natural number. So 1
over 1 squared plus 1 over 2 squared plus 1 over
3 squared and so on. It looked like a pure number
theory problem, and they knew the sum converged
to some number around 1 .64. But finding the
exact value proved impossible. Until Euler. In
1735, he provided the stunning and utterly unexpected
solution. The sum is exactly squared over 6.
What made this so groundbreaking was the connection.
Why should a series involving integers have a
result tied to the geometric constant of the
circle? It was a beautiful, inexplicable piece
of unity, and it relied on a brilliant, though
initially non -rigorous method, where he treated
the infinite series like it was a finite polynomial.
He later provided a more elaborate proof, but
the result stood. It was a connection between
two seemingly disparate worlds of math. And his
work didn't stop with real numbers. He successfully
defined logarithms for negative and complex numbers,
which vastly expanded the practical reach of
the logarithm function. But the absolute pinnacle
of his achievement in complex analysis is Euler's
formula. This is truly a high point. He extended
the exponential function to complex numbers,
linking it directly to the trigonometric functions.
The formula states that e to the i equals cos
plus e sin. This formula provides the fundamental
link between circular motion, so trigonometry,
and exponential growth. And if that wasn't beautiful
enough, if you set the angle 2 .5, you get the
simplest and most elegant version, which is known
as Euler's identity. e to the i plus 1 equals
0. This one short equation is often called the
greatest formula in mathematics. Richard Feynman
famously called it that. And think about what
it connects. You have e, the base of continuous
growth, i, the imaginary unit. The geometric
constant, one, the multiplicative identity, and
zero, the additive identity. Five fundamental
constants woven into one perfect expression.
It just epitomizes Euler's unifying vision for
mathematics. And moving beyond those headline
results, he also essentially invented the calculus
of variations. Okay, so what is that, and why
does it matter? Well, the standard calculus you
learn in school deals with finding the point,
a number, where a function is maximized or minimized.
The calculus of variations is much deeper. It
finds the function, the actual shape of the path,
that maximizes or minimizes an integral. So instead
of finding the quickest single speed, you're
finding the most efficient path overall. Exactly.
It's essential for finding things like the shortest
distance between two points on a curved surface,
the shape of a soap film stretched between two
wires, or the optimal trajectory of a satellite.
He formulated the Euler -Lagrange equation, which
is the foundational tool for this entire field.
And it's still used in everything from general
relativity to classical mechanics. He also contributed
to the theory of higher transcendental functions
by introducing the gamma function. That sounds
a bit intimidating. What purpose does it serve?
The gamma function, which is denoted az, is essentially
a way to extend the concept of the factorial,
which is normally only defined for positive integers,
like 5, to include non -integers, complex numbers,
and fractions. So it allows you to calculate
something like half factorial. Conceptually,
yes. And this seemingly abstract extension proved
absolutely crucial for advanced mathematics,
especially in areas like probability distributions,
partial differential equations, and complex analysis
in physics. It generalized a key tool. And we
should also mention the Euler -McLauren formula,
another key tool he developed for approximating
integrals using summation, vital for numerical
methods. Now let's pivot to number theory, because
this is where Euler truly created a whole new
discipline. analytic number theory. This was
the revolutionary act of uniting number theory,
which deals with discrete integers and primes
with continuous analysis. He was using tools
like infinite series and calculus to solve problems
about numbers. He demonstrated the power of this
new approach by proving the infinitude of prime
numbers. Now, we know Euclid proved this using
pure number theory by contradiction. How did
Euler use analysis? He connected the divergence
of the harmonic series, so 1 plus 1 half plus
1 third plus 1 fourth and so on, to prime numbers.
He showed that the sum of the reciprocals of
the primes, 1 half plus 1 third plus 1 fifth
plus 1 seventh, also diverges, meaning it approaches
infinity. And this could only happen if there
were an infinite number of terms, an infinite
number of primes. in the series. It's a subtle
but brilliant argument. It shows that analytic
methods could yield new insight into these ancient
integer problems. And that connection led to
his discovery of the profound Euler product formula
which linked the sum of reciprocals of integer
powers, which later became the Riemann zeta function,
to a product over prime numbers. The implications
of this formula are still a cornerstone of mathematical
research today. And in pure number theory, Euler
was the intellectual heir to Pierre de Fermat.
He dedicated significant effort to validating
Fermat's findings, but he also wasn't afraid
to overturn them. That's right. Fermat had conjectured
that all Fermat numbers, those of the form 2
to the n plus 1, were prime. Euler, through brilliant
calculation, proved that for n equals 5, the
resulting number, which is over 4 billion, was
composite. not prime. He showed it was divisible
by 641, disproving Fermat's conjecture entirely.
And he also gave us one of the most important
tools in modern cryptography, the totient function.
What exactly does the totient function do? The
totient function, sometimes called Euler's phi
function, it counts how many positive integers
less than or equal to n are co -prime to n. And
two numbers are co -prime if they share no common
factors other than one. And why is that useful?
Well, It helps us understand the structure of
remainders in modular arithmetic, the math of
clocks and cyclical systems. Euler then used
this function to generalize Fermat's little theorem,
resulting in Euler's theorem. And this theorem
is critical because the principles it establishes
dealing with exponents and remainders are the
exact foundational base we use for modern security
and encryption algorithms like RSA. He also contributed
to the classical study of perfect numbers. those
numbers that equal the sum of their proper divisors,
like 6, where 1 plus 2 plus 3 equals 6. This
was a fascination since Euclid. It was, and Euler
solved a long -standing mystery by proving the
one -to -one correspondence between even perfect
numbers and Mersenne primes. This result is known
as the Euclid -Euler theorem, a perfect example
of his ability to synthesize ancient geometry
with modern number theory. Okay, moving on to
section five. If Euler's work in analysis proved
his theoretical mastery, his contributions to
geometry, physics, and engineering really demonstrate
his role as the ultimate applied genius. He wasn't
just working with abstractions, he was shaping
the physical world. And the best way to introduce
this shift of perspective is through the story
that literally founded an entirely new branch
of mathematics. Yeah. The Seven Bridges of Konigsberg
Problem. From 1735. Retell that problem for us
because it sounds like a simple riddle, but it
completely changed math forever. The city of
Königsberg, which is now Kaliningrad, was built
on the Pregel River, and it contained two large
islands connected to each other and the mainland
by seven bridges. And the citizens wondered if
it was possible to take a walk and cross each
of the seven bridges exactly once. Euler proved
it was impossible, but the true genius wasn't
the no answer. It was the methodology he used
to reach it. Exactly. Euler realized that the
solution had nothing to do with the physical
distances or the size of the islands or the length
of the bridges. All that mattered was the connectivity.
He replaced the land masses with points vertices
and the bridges with lines or edges. Right. And
he determined that for a path to exist that crosses
every bridge exactly once, a Eulerian path, the
graph could have at most two vertices with an
odd number of edges connected to them. And since
Konigsberg have four landmasses with an odd number
of bridges connecting them, such a path was impossible.
This approach, focusing only on structural relationships
and ignoring measurement, is recognized as the
first theorem of graph theory. It's a complete
shift in mathematical perspective, moving math
from continuous measurement to discrete connectivity,
a shift that is crucial for modern computer science
and networking. He made a similar profound contribution
to topology, which is the study of spatial properties
that remain the same even when an object is stretched
or deformed. That was through the Euler characteristic
formula. For any convex polyhedron, a 3D shape
like a cube or a pyramid, the number of vertices,
V minus the number of edges, E, plus the number
of faces, F always equals 2. V, E plus F, L equals
2. It's just beautiful in its simplicity. So
why is that the origin of topology? Because it's
a property inherent to the structure of the shape,
not its size or its curvature. You can distort
a cube into any shape you want. But as long as
you don't break or merge edges, the count VE
plus F will always do too. The study of this
formula and its generalization to other objects
is recognized as the starting point for modern
algebraic topology, a field that seeks structural
invariance in space. And turning to physics and
engineering, Euler didn't just apply Newton's
laws. He refined them to make them more useful
for the coming industrial age. In his two -volume
work, Mechanica, he reformulated classical mechanics,
providing new analytic laws to better describe
the motion of rigid bodies, which are far more
complex than the idealized point masses Newton
often considered. And in structural engineering,
he gave us two formulas that are absolute cornerstones
for anyone building things today. First, the
Euler Bernoulli beam equation. That equation
is foundational for designing any structure that's
subject to bending, whether it's a bridge, a
building frame, or an aircraft wing. It describes
the relationship between the beam's deflection
and the loads applied to it, allowing engineers
to calculate stress with precision. And second,
and perhaps even more famous in engineering,
is Euler's critical load formula. This formula
calculates the exact compressive load at which
a long, slender column will buckle and fail.
Before Euler, structural collapse due to buckling
was often unpredictable. His formula gave engineers
the definitive boundary, the critical load, telling
them precisely when the geometry of a column
will cause catastrophic failure under compression.
It saved countless lives and billions of dollars
in future construction. His work in fluid dynamics
is equally foundational. even if it's often less
visible. In 1757, he formulated the Euler equations
for inviscid flow. Inviscid flow being the motion
of an idealized fluid with no viscosity, no internal
friction. Right. Which, while theoretical, provides
the simplest, most powerful starting point for
understanding how water and air move. These equations
are still the first tools used in computational
fluid dynamics. And here's a fact that just shows
his predictive genius. In 1754, based purely
on mathematical modeling, Euler was the first
person to predict the phenomenon of cavitation.
Cavitation is the formation of vapor bubbles
in a rapidly flowing liquid when the pressure
drops below the fluid's vapor pressure. It's
what causes noise and damage in pumps, propellers,
and giant turbines. And Euler mathematically
predicted this phenomenon, these very tiny bubbles
that challenge giant engineering systems, long
before it was physically observed in the late
19th century. That is true scientific prophecy.
He even weighed in on one of the great historical
scientific debates in optics, disagreeing with
Isaac Newton's prevailing theory that light was
made of particles or corpuscles. Euler was an
advocate for Christian Huygens' wave theory of
light. His papers in the 1740s provided strong
mathematical arguments in favor of light propagating
as a wave. And while science eventually reconciled
these two views with quantum theory, Euler's
work was crucial in pushing the wave theory to
dominance for a long period, ensuring the scientific
debate remained dynamic and productive. And beyond
physics, his influence extended into visualization
and logic. In 1768, he introduced the Euler diagrams
to illustrate syllogistic reasoning. These are
the closed curves we still use today to visually
represent set relationships, whether they intersect
or subsets of each other or are entirely disjoint.
While modern Venn diagrams are a refinement,
Euler diagrams are the visual starting point
for thinking about logical relationships and
set theory. the nascent field of demography,
he essentially launched modern population modeling
150 years too early. He did. In the 1760 paper,
A General Investigation into the Mortality and
Multiplication of the Human Species, he created
a comprehensive mathematical model. He showed
that a population subject to constant rates of
mortality and fertility would eventually grow
geometrically, establishing a fixed age structure.
And that model, the stable population model,
is fundamental to how demographers study population
dynamics today. But it wasn't rediscovered and
widely formalized until Alfred J. Lochte did
so in the 20th century. He was 150 years ahead
of the curve, defining the start of formal demographic
modeling. Finally, we get to one of his most
unusual and personal interests, music theory.
In 1739, he wrote Tentamen Novae Theoriae Musicae,
which was an attempt at a new theory of music,
trying to ground aesthetic concepts entirely
in mathematics. And this work was notoriously
difficult. It was described as too mathematical
for musicians and too musical for mathematicians.
It was a beautiful failure, but a failure of
communication, not insight. Euler was trying
to quantify harmony. He mathematically derived
the gratis suavitatis, or degree of agreeableness,
for musical intervals and chords. He based this
sweetness on the arithmetic of small prime factors
found in musical ratios, specifically three and
five. For instance, a perfect fifth, with a ratio
of three to two, is harmonically simpler than
a diminished fifth. His belief was that aesthetic
pleasure was rooted in the quantitative simplicity
of ratios. He even devised a graph, the speculum
musicum, to classify musical genres based on
their complexity, which has recently drawn renewed
interest in advanced music theory. It just shows
his relentless quest to unify knowledge. If mathematics
governs the stars, mechanics, and light, well,
why not beauty itself? As we synthesize the life
of Lee and Mart Euler, we have to just pause
and consider the sheer magnitude of the mind
that produced all of this. The volume of his
output, especially while blind, suggests an organizational
and computational ability unlike almost anyone
else in history. The famous tales about his memory
are just awe -inspiring. By all accounts, he
possessed a computational memory that was staggering.
Nicholas Fuss, who was one of Euler's disciples,
chronicled these abilities. He had memorized
Virgil's massive epic poem, The Aeneid, in its
entirety. And in his later years, totally blind,
he could reportedly recite the poem and instantly
state the first and last sentence on any given
page of the exact edition he had learned from
decades prior. And his mathematical memory was
equally flawless. He could instantly recall the
first hundred prime numbers and their powers
up to the sixth degree, and he used those facts
as computational scaffolding inside his head.
This ability to run complex equations solely
in his memory is why the loss of his sight barely
slowed him down. And his general knowledge wasn't
just limited to math and the classics. He had
comprehensive knowledge of history, Roman writers,
medicine, botany, chemistry, far more than a
specialist would be expected to hold. This extraordinary
mental catalog allowed him to synthesize fields
and draw connections where others saw only separation.
It's what makes him a truly universal genius.
The scale of his legacy sometimes needs a human
touch, which brings us to a famous, persistent,
but ultimately false anecdote involving the French
philosopher Denis Diderot. Right. This legend
supposedly takes place during Euler's second
tenure in St. Petersburg under Catherine the
Great. Diderot, who was known for his atheistic
leanings, was influencing the Russian court,
and the empress allegedly asked Euler to publicly
silence him. The story goes that Euler approached
Diderot and, with perfect conviction, announced
a mathematical proof of God's existence. He states
the non -sequitur. Sir, a plus b to the n divided
by n equals x, hence God exists, reply. And Diderot,
allegedly knowing nothing of mathematics, was
left speechless, mocked by the court, and soon
asked to leave Russia. It's a wonderfully dramatic,
if perhaps a bit anti -intellectual story. But
as historians confirm, it's a legend popularized
later, and it simply isn't true. Not at all.
Diderot was a capable mathematician himself,
and the incident as described never occurred.
The story is perhaps best appreciated as a cautionary
tale about the intimidation factor of using specialized
jargon rather than a factual account of Euler's
rhetorical style. His real rhetorical power was
in the elegance of his written proofs, not in
courtly verbal trickery. His career came to an
abrupt yet strangely fitting end. On September
18, 1783, Euler was in St. Petersburg discussing
the newly discovered planet Uranus and calculating
its orbit with his student, Anders Johan Luxell.
He suffered a brain hemorrhage and died instantly.
The famous eulogy, written by the French mathematician
and philosopher Marquis de Condorcet, provides
the perfect epitaph for this immense intellectual
engine. He wrote, Il cessa de calculer de vivre.
He ceased to calculate and to live. His legacy
is undisputed. John von Neumann called him the
greatest virtuoso of the period. And Henri Poincaré
went even further, simply calling him the god
of mathematics. And for those who want to grasp
the sheer scope one final time, the publishing
effort to collate his opera omnia, his complete
works, is still ongoing. The current total is
80 volumes, cataloged systematically by the Innistrum
Index from E1 to E866, covering every aspect
of analysis, mechanics, physics, and correspondence.
This body of work is the best school Gauss spoke
of. He appears on the Swiss 10 -franc banknote
and numerous stamps. His influence isn't just
academic, it's embedded in the foundational...
we all use to describe the universe. So what
does this all mean for you? Euler's life shows
the triumph of disciplined intellect, overcoming
political instability and even total physical
blindness through the power of a perfectly structured
mind. But let's connect back to that unusual
passion we discussed earlier, his attempt to
quantify harmony, the grotus suavitatis. He believed
musical sweetness could be derived precisely
from the arithmetic of small prime numbers, the
simplicity of the ratios three and five. It was
a mathematical definition of aesthetic beauty.
So here is the final provocative thought for
you to mull over. If aesthetic concepts as subjective
as musical harmony can be precisely derived from
elegant, underlying quantitative laws, if beauty
has a numerical root, then consider whether all
forms of aesthetic pleasure or natural beauty,
whether it's the structure of a flower, the flow
of a river, or the shape of a seashell, might
also be rooted in similarly elegant, universal
quantitative laws that are just waiting to be
discovered. Euler's work suggests that knowledge
is ultimately not a collection of separate subjects,
but one universe. Thank you for joining us on
this deep dive into Leonhard Euler. We'll see
you next time.