Gerard Desargue: A Genius Who Revolutionized Geometry
Lyon, France, was a bustling city in the late sixteenth century. It was a hub for commerce, banking, and intellectual discourse. And it was here, in 1591, that Gérard Desargues was born into a prominent family. His father, Étienne Desargues, was a magistrate and a city official, which meant that young Gérard grew up in an environment surrounded by influential thinkers and decision-makers. His mother, whose name is less documented, was likely well-educated, as was common in elite families of the time.
But did his family influence his mathematical and engineering inclinations? The answer seems to be yes. Étienne Desargues, as a high-ranking civil servant, had connections with scholars, architects, and engineers, providing Gérard with a natural gateway into the intellectual world. Unlike many famous mathematicians of the period, however, there is 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 to some of the best scholars in their homes. His education would have included mathematics, 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. It was a group of intellectuals and scientists who gathered around the French monk Marin Mersenne. This very influential group of intellectuals included some of the greatest minds of all time. This included Marin Mersenne, the French monk who had the gatherings in his home, Etienne Pascal, who was Blaise Pascal’s father, and Rene Descartes. Even as the group grew older, they invited brilliant young individuals to join them. Some younger members included Gilles de Roberval, Pierre de Fermat, and Blaise Pascal, influenced by his father Etienne.
Mural by Théobald Chartran in the Sorbonne (about 1855, detail)
The Mersenne Circle was a hub for mathematical and scientific discussions, where members exchanged ideas on topics ranging from geometry to physics and number theory. Desargues shared his work on projective geometry with this group, influencing younger mathematicians like Blaise Pascal, whose famous Pascal’s Theorem was inspired by Desargues’ studies.
Through this intellectual network, Desargues’ mathematical ideas reached a broader audience, even though his work remained underappreciated during his lifetime.
By the early seventeenth century, Desargues had carved a place for himself as an architect and engineer. But what led him down this path? 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 skill in architecture, where understanding how buildings appeared from different angles was essential for design and construction.
Desargues’ 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. 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 focused on intricate details like staircases, it is challenging to attribute specific surviving structures solely to his design. Consequently, no known buildings today can be definitively credited to Desargues.
But it was Desargues’ work in geometry that would leave the most significant mark in history. As an architect, he realized that perspective could be understood through a new branch of mathematics: projective geometry. He published a treatise in 1639 titled Rough Draft on the Events of the Encounters of a Cone with a Plane, in which he explored how geometric shapes transformed when projected onto different 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 Desargues’ Theorem, a fundamental principle in projective geometry that states that if two triangles are in 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, who expanded on his projective geometry concepts.
Desargues 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 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.[1]
Three years later, in 1643, his student Abraham Bosse published The Universal Method of Mr. Desargues, Lyonnais, which included Desargues’ lost work on sundial construction, Gnomonics. His mathematical contributions were profound, although his writing style made them difficult for his contemporaries to grasp.
As a side note regarding his work on sundial construction titled Gnomonics, did you know that the piece that sticks up on a sundial is called the gnomon? So, when you see a sundial with that little triangle sticking, you’ll have something fun to share with your friends and family!
But, back to Desargues.
Unfortunately, Desargues’ mathematical contributions were not widely 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 Desargues’ 1639 treatise, Brouillon Project d’une Atteinte aux Événements des Rencontres d’un Cône avec un Plan, faced criticism for its complexity and unconventional style. Notably, Jean de Beaugrand, a contemporary mathematician, challenged the originality of Desargues’ propositions, asserting they were derived from Apollonius’ Conics. This dispute escalated until Beaugrand died in 1642. Though I can’t find this information’s original source, I found something even more interesting. It turns out that Beaugrand was also a member of the Mercenne circle. So, he was friends with Descartes, Pascal, and Desargues. But all of Beaugrand’s friends began to distance themselves from him because Beaugrand was on a mission to throw all his old friends under the bus. Descartes referred to Beaugrand’s work as impertinent, ridiculous, and detestable. Finally, after Mersenne had sent yet another letter to Descartes about Beaugrand, Descartes replied to Mersenne, telling him to stop writing to him about Beaugrand because he already had enough toilet paper.[2] I love the French. I really do!
However, there was a lot of infighting between the entire Mersenne group because Descartes also critiqued Desargues’ work, stating it was impossible and implausible 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.[3]
These critiques contributed to the limited immediate impact of Desargues’ work, which remained underappreciated until its rediscovery in the 19th century.
To understand Desargues’ 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. It’s the mathematics behind perspective drawing, the same principles that Renaissance artists like Leonardo da Vinci used to create depth in their 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 broke free from rigid constraints like measurement and instead focused on relationships between points, lines, and planes.
Desargues 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. This concept became a foundation for modern computer graphics, engineering, and physics.
This brings us to Desargues’ Theorem, one of the cornerstones of projective geometry.
Understanding Desargues’ Theorem
So, what exactly does Desargues’ Theorem say?
In basic terms, if two triangles are arranged in space so that the lines connecting their corresponding vertices meet at a single point (this is called being in perspective from a point), then the intersections of their corresponding sides will always lie on a single straight line (in perspective from a line).
Let’s break this down with an example. Imagine standing on a long, straight road with two identical triangular signs placed at different distances from you. If you draw lines connecting their matching corners, those lines will meet at a single point in space—your perspective center. Now, if you look at where the edges of those two triangles overlap on the road, those intersection points will always form a straight line.
This concept directly applies Desargues’ Theorem, which explains how two triangles in perspective share a unique geometric relationship. 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. Now, if you were to extend lines connecting their matching corners (top to top, bottom to bottom, left to left, etc.), all these lines would eventually converge at a single point in space, known as your perspective center. This is a fundamental principle of projective geometry, where distant objects appear smaller but maintain proportional relationships. It reminds me of when I taught my kids how to look at a person far away, put your fingers on each side of their head, and pinch their head. We always got a kick out of that. Their heads were so small because they were so far away. That is proportion.
Let’s examine what happens when you look at the ground where the two triangular signs seem to “overlap” visually. If you trace the points where their edges intersect on the road, those points will always form a straight line. This phenomenon, known as collinearity, is a key result of Desargues’ Theorem. 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 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 Desargues introduced this idea in the 1600s, it was so different from traditional Euclidean geometry that many mathematicians ignored it.
The Impact and Rediscovery of Desargues’ Work
Despite its elegance and power, Desargues’ work was overlooked during his lifetime. His book, Brouillon Project, published in 1639, was challenging to understand because it used unfamiliar notation and was written in a highly technical style. Mathematicians were focused on algebra and calculus at the time, and Desargues’ ideas 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 Poncelet rediscovered Desargues’ work. They 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.
Desargues’ Theorem might seem like an abstract mathematical concept. Still, it plays a crucial role in many aspects of modern technology. One of the most 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 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 visualize and adjust complex designs before they are built.
Another field deeply influenced by Desargues’ 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 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 and shadows in computer-generated imagery, is also rooted in these geometric principles. Additionally, technologies like augmented reality (AR) and virtual reality (VR) utilize projective transformations to create immersive digital environments, ensuring that objects appear correctly positioned from different viewing angles. Without Desargues’ Theorem, these advancements in visual technology would not be as precise or realistic as they are today.
Gérard Desargues 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, 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 for centuries.
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 Desargues, the mathematician who saw the world from a different perspective.
Thanks for tuning in to Math! Science! History!
FURTHER READING
René Taton, “L’ø Euvre Mathématique de G. Desargues,” Revue Philosophique de La France Et de l’Etranger 148 (1958): 98–99.
[1] Le Goff, Jean-Pierre. “Desargues et la naissance de la géométrie projective. Desargues en son temps.” In From Here to Infinity, 157–206. Springer Nature, 1994.
[2] Lennon, Thomas M. “Beaugrand, Jean de (1595–1640).” In The Cambridge Descartes Lexicon, edited by Lawrence Nolan, 56–57. Cambridge: Cambridge University Press, 2015. https://doi.org/10.1017/CBO9780511894695.025.
[3] René Taton, “L’ø Euvre Mathématique de G. Desargues,” Revue Philosophique de La France Et de l’Etranger 148 (1958): 98–99.