Gérard Desargues: A Genius Who Revolutionized Geometry

🔎 Episode Overview:

In this episode of Math, Science, History, we dive into the fascinating life and work of Gérard Desargues, the 17th-century mathematician, engineer, and architect whose groundbreaking ideas in projective geometry laid the foundation for modern mathematical thought. Despite his contributions, Desargues’ work was largely ignored during his lifetime, overshadowed by more well-known figures like René Descartes. However, his theorem and insights into perspective have since become essential in fields ranging from architecture to computer graphics. Join us as we explore how Desargues’ ideas helped shape the way we understand space, depth, and geometry today!

🧠 What You’ll Learn:

The Birth of Projective Geometry – How Desargues developed the mathematical principles behind perspective and why his theorem remains crucial in geometry.

His Overlooked Legacy – Why Desargues’ work was ignored for centuries and how it was rediscovered in the 19th century.

Modern Applications – How his ideas are used today in architecture, engineering, computer graphics, and even virtual reality.

🏛 Links & Resources:

Oeuvres de Desargues (Cambridge Library Collection - Mathematics) (French Edition) https://amzn.to/4inV9Ve

History of Projective Geometry – Encyclopedia Britannica.

Marin Mersenne and the Mathematical Circle – Stanford Encyclopedia of Philosophy.

Episode Transcript

(00:00):
(Transcribed by TurboScribe.ai. Go Unlimited to remove this message.) Welcome to Math Science History.
Today, I'm going to be talking about a
mathematician from the 16th century whose math was
so advanced that he has a theory that
is now applied to today's photography.
Hi, I'm Gabrielle Burtek.
I have a background in math, science, and
journalism, and by the time you are done
listening to today's podcast, you're going to know

(00:21):
so much more about Gérard Desogues, a French
mathematician from the 16th century.
In the year 415, the infamous philosopher and
mathematician Hypatia of Alexandria Egypt was savagely murdered
by church monks.

(00:42):
This murder shocked the Roman community and its
government leaders.
Hypatia was known far and wide as a
Hypatia, the sum of her life, is a
book that I wrote that looks not just
at the circumstances surrounding her death, but also

(01:03):
at the sum of her entire life.
I weave in the details of her education,
disciples, Neoplatonic philosophies, female contemporaries, and the many
mathematics that she wrote and taught about.
There is truly more to Hypatia's life than
her death.
Hypatia, the sum of her life, written by
me, Gabrielle Burtek, is now on sale on

(01:26):
Amazon.
Buy your copy today.
Lyon, France, was a bustling city in the
late 16th century.
It was a hub for commerce, banking, and
intellectual discourse, and it was here in 1591
that Gérard Desogues was born into a prominent

(01:48):
family.
If you want to look up the name,
it is spelled Gerard, G-E-R-A
-R-D, Desogues, D-E-S-A-R
-G-U-E-S.
He was born into a prominent family, and
his father, Otion Desogues, was a magistrate and
a city official, which meant that young Gérard
grew up in an environment surrounded by influential

(02:10):
thinkers and decision makers.
His mother, whose name is not known, at
least as far as I know, was likely
well educated, as was common in elite families
of the time.
Otion Desogues, his father, was a high-ranking
civil servant, and he had connections with scholars,
architects, and engineers, which provided Gérard with a

(02:31):
natural gateway into the intellectual world.
Unlike many famous mathematicians of the period, however,
there was no direct record of him attending
a formal university.
Instead, he was believed to be privately educated,
possibly through homeschooling or private tutors.
This was not uncommon in Renaissance Europe, especially
for children of the elite, who had access

(02:53):
to some of the best scholars in their
homes.
His education would have included math, engineering principles,
and classical studies, preparing him for a life
in architecture and engineering.
Though he never attended university, he gathered and
collaborated with a group of intellectuals associated with
the mathematical circle of Mersenne, that's M-E

(03:15):
-R-S-E-N-N-E.
It was a group of intellectuals and scientists
who gathered around the French monk, Mérène Mersonne.
This very influential group of intellectuals included some
of the greatest minds of all time.
This included Mérène Mersonne, the French monk, who
had the gatherings at his home, Antion Pascal,

(03:36):
who was Blaise Pascal's father, and René Descartes.
Even as the group grew older, they invited
young, brilliant individuals to join them.
Some younger individuals included Gilles D.
Roberval, Pierre de Fermat, and Blaise Pascal, influenced
by his father Antion.
The Mersenne circle was a hub for mathematical

(03:58):
and scientific discussions, where members exchanged ideas on
topics ranging from geometry to physics and number
theory.
Desargue shared his work on projective geometry with
this group, influencing younger mathematicians like Blaise Pascal,
whose famous Pascal's theorem was inspired by Desargue's

(04:18):
studies.
Through this intellectual network, Desargue's mathematical ideas reached
a broader audience, even though his work remained
underappreciated during his lifetime.
By the early 17th century, Desargue had carved
a place for himself as an architect and
engineer.
But what led him down this path?

(04:39):
Well, one of the key aspects of his
work was his ability to think about structures
and space in ways that others had not
fully explored.
He was particularly interested in perspective, the mathematical
principles behind how we see depth and shape.
This wasn't just a curiosity, but a necessary

(04:59):
skill in architecture, where understanding how buildings appeared
from different angles was essential for design and
construction.
Desargue's engineering work was impressive.
He contributed to waterworks, fortifications, and urban planning
projects, which were critical tasks during a time
when France was expanding its infrastructure.

(05:21):
His architectural works followed Renaissance principles, focusing on
order and perspective, which naturally tied into his
mathematical investigations.
Some historical records indicate that he participated in
planning several private and public buildings in Paris
and Lyon.
However, due to his architectural work, which often

(05:43):
focused on intricate details like staircases, it's challenging
to attribute specific surviving structures solely to his
design.
Consequently, no known buildings today can be definitively
credited to Desargue.
But it was Desargue's work in that would
leave the most significant mark in history.

(06:05):
As an architect, he realized that perspective could
be understood through a new branch of mathematics
called projective geometry.
He published a treatise in 1639 titled Rough
Draft on Events of the Encounters of a
Cone with a Plane, in which he explored
how geometric shapes transformed when projected onto different

(06:26):
planes.
In simpler terms, he wanted to understand what
happened to figures when viewed from various angles,
something that was crucial in art, architecture, and
engineering.
This work led to what we now call
Desargue's theorem, a fundamental principle in projective geometry
that states that if two triangles are in

(06:47):
perspective from a point, their corresponding sides meet
at collinear points.
This idea may sound abstract, but it was
groundbreaking at the time because it helped mathematicians
think about geometry in a way that wasn't
tied strictly to Euclidean rules.
His ideas influenced later mathematicians, including Blaise Pascal,

(07:09):
who expanded on his projective geometry concepts.
Desargue authored several other essential works.
In 1636, he wrote Perspective, a treatise on
mathematical principles of perspective used in art and
architecture.
In 1640, he published Example of a Universal
Method Concerning the Practice of the Trait for

(07:30):
Stone Cutting, which detailed geometric techniques for cutting
stone in architecture.
There is also speculation that he published a
second treatise in 1640 on conic sections titled
Lectures of Darkness or Shadows.
If this piece of work existed, it was
immediately lost.
Three years later, in 1643, his student Abraham

(07:53):
Boas published The Universal Method of Mr. Desargue's
Lyonnais, which included Desargue's lost work on sundial
construction called Mnemonics.
His mathematical contributions were profound, although his writing
style made them difficult for his contemporaries to
grasp.
And as a side note, regarding his work

(08:13):
on sundial construction titled Mnemonics, did you know
that the piece that sticks up on the
sundial is called the gnomon?
It's spelled G-N-O-M-O-N.
So when you see a sundial with that
little triangle sticking up, you'll have something fun
to share with your friends and family.
Unfortunately, Desargue's mathematical contributions were not widely

(08:34):
appreciated during his lifetime.
His writing style was dense and his work
was overshadowed by more well-known mathematicians like
René Descartes.
However, centuries later, his contributions would be rediscovered
and recognized as foundational in the development of
modern geometry.
Gérard Desargue's 1639 treatise titled Rough Draft on

(08:57):
the Events of the Encounters of a Cone
with a Plane faced criticism for its complexity
and unconventional style.
Notably, Jean de Bougrand, a contemporary mathematician, challenged
the originality of Desargue's proposition, asserting they were
derived from Apollonius's conics.
This dispute escalated until Bougrand died in 1642.

(09:21):
And though I can't find this information's original
source, I found something even more interesting.
This is good.
It turns out that Bougrand was also a
member of the Maison Circle.
So he was friends with Descartes, Pascal, and
Desargue.
But all of Bougrand's friends began to distance
themselves from him because Bougrand was on a

(09:43):
mission to throw all of his friends under
the bus.
Descartes even referred to Bougrand's work as impertinent,
ridiculous, and detestable.
Love it.
Finally, after Mersonne had sent yet another letter
to Descartes about Bougrand, Descartes replied to Mersonne

(10:03):
telling him to stop writing to him about
Bougrand because he already had enough toilet paper.
I love the French.
I really do.
However, there was a lot of infighting between
the entire Mersonne group because Descartes also critiqued
Desargue's work, stating it was impossible and implausible

(10:25):
to treat conic sections without algebra.
Descartes believed that advancements in geometry could only
be achieved through algebraic methods, reflecting the prevailing
sentiment among mathematicians of that era.
These critiques contributed to the limited immediate impact
of Desargue's work, which remained underappreciated until its

(10:45):
rediscovery in the 19th century.
So to understand Desargue's theorem, we first need
to talk about projective geometry.
Unlike traditional Euclidean geometry, which focuses on properties
like distance and angles, projective geometry describes how
objects appear when projected onto a different surface.

(11:06):
It's the mathematics behind perspective drawing, the same
principles that Renaissance artists like Leonardo da Vinci
used to create depth in his paintings.
Projective geometry considers points at infinity where parallel
lines meet and examines how shapes transform when
viewed from different angles.
This way of thinking was revolutionary because it

(11:28):
broke free from rigid constraints like measurement and
instead focused on relationships between points, lines, and
planes.
Desargue was one of the first to formalize
these ideas mathematically.
His work provided a framework that allowed mathematicians
to study shapes and their transformations in a
way that was independent of specific measurements.

(11:50):
This concept became a foundation for modern computer
graphics, engineering, and physics.
So this brings us to Desargue's theorem, one
of the cornerstones of projective geometry.
What exactly does Desargue's theorem say?
Well it says if two triangles are arranged
in space so that the lines connecting their

(12:12):
corresponding vertices meet at a single point, then
the intersections of their corresponding sides will always
lie on a single straight line.
So that single point is called being in
perspective from a point and that single straight
line is called in perspective from a line.
This concept directly applies to Desargue's theorem which

(12:34):
explains how two triangles in perspective share a
unique geometric relationship.
So imagine yourself on a long straight road
looking ahead at two identical triangular signs, one
closer to you and the other further away.
Even though they are at different distances, their
corresponding points such as their top and bottom
corners appear to align from your viewpoint.

(12:54):
Now if you were to extend lines connecting
their matching corners top to top, bottom to
bottom, left to left, all these lines would
eventually converge at a single point in space.
This is known as your perspective center.
This is a fundamental principle of projective geometry
where distant objects appear smaller but maintain proportional

(13:15):
relationships.
So let's examine what happens when you look
at the ground where two triangular signs seem
to overlap visually.
If you trace the points to where their
edges intersect on the road, those points will
always form a straight line.
This phenomenon known as collinearity is the key
result of Desargue's theorem.

(13:36):
It explains why parallel train tracks appear to
converge in the distance or why perspective drawings
maintain a sense of depth and realism.
These geometric principles are fundamental in photography, architecture,
and even 3D rendering, ensuring that our perception
of depth and proportion remains consistent in real

(13:58):
and simulated environments.
This could be used in interactive museum exhibits,
escape rooms, or even a fun science demonstration
showing how projective geometry can manipulate perception in
unexpected ways.
This principle is instrumental in architecture, engineering, and
even computer vision.
However, when Desargue introduced this idea in the

(14:19):
1600s, it was so different from traditional Euclidean
geometry that many mathematicians simply ignored it.
He was ahead of his time.
Despite its elegance and power, Desargue's work was
overlooked during his lifetime.
His book, Bouillon Project, published in 1639, was
challenging to understand because it used unfamiliar notation

(14:43):
and was written in a highly technical style.
Mathematicians were focused on algebra and calculus at
the time, and Desargue's idea about projective geometry
seemed abstract and impractical.
However, as history has shown, groundbreaking ideas sometimes
take time to gain recognition.
In the 19th century, mathematicians like Jean-Victor

(15:06):
Poncelet, P-O-N-C-E-L-E
-T, rediscovered Desargue's work.
He built upon it, leading to the formal
development of projective geometry as a significant branch
of mathematics.
Today, his theorem is a fundamental principle in
geometry used in everything from 3D modeling to
perspective drawing in art.

(15:26):
Desargue's theorem might seem like an abstract mathematical
concept.
Still, it plays a significant applications is in
architecture and engineering, where projective geometry helps design
structures with accurate perspective and symmetry.
Architects rely on these principles to ensure that

(15:48):
buildings appear proportional from different viewpoints, while engineers
use them to create stable bridges and mechanical
structures.
Computer-aided design, CAD, software, widely used in
construction and manufacturing, also depends on projective transformations
to generate 3D models, making it possible to

(16:09):
visualize and adjust complex designs before they are
built.
Another field deeply influenced by Desargue's theorem is
photography, which, like computer graphics, relies on projective
geometry to accurately represent three-dimensional scenes on
a two-dimensional surface.
The principles of perspective correction in camera lenses

(16:31):
stem from projective transformations, ensuring that architectural photography
and panoramic shots maintain proper proportions.
In computer graphics and 3D rendering, video games
and CGI in movies use these mathematical ideas
to create realistic depth and spatial effects.
Ray tracing, a technique that simulates light reflections

(16:54):
and shadows in computer generated imagery, is also
rooted in these geometric principles.
Additionally, technologies like augmented reality and virtual reality
utilize projective transformations to create immersive digital environments,
ensuring that objects appear correctly positioned from different

(17:15):
viewing angles.
Without Desargue's theorem, these advancements in visual technology
would not be as precise or realistic as
they are today.
Gérard Desargue may not be a household name,
but his contributions to mathematics continue to shape
the world around us.
His theorem laid the groundwork for projective geometry,

(17:37):
influencing fields as diverse as architecture, engineering, computer
graphics, and even space exploration.
His story is a reminder that sometimes genuinely
revolutionary ideas take time to be appreciated.
Even if the world isn't ready for a
new way of thinking, the impact of a
great idea can last not just for centuries,

(18:00):
but for millennia.
So the next time you see a 3D
movie, admire a beautifully designed building, or play
a video game with stunning graphics, remember the
name Gérard Desargue, the mathematician who saw the
world from a different perspective.
Thank you for tuning in to Math Science
History, and until next time, carpe diem.

(18:21):
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