– Jacob Klein, Greek Mathematical Thought and the Origin of Algebra
On the Shoulders of Giants
Syllabus for 2025
- The dramatic shift from “saving the appearances” to physical theorizing about cause
- The evolution of mathematical reasoning from geometric visualization to symbolic abstraction
- The changing relationship between observation, theory, and truth
- The profound philosophical questions that emerge at each stage of scientific development
- In Euclid’s elegant proofs and Ptolemy’s intricate celestial models, we encounter modes of thought that reveal both the limitations and unexamined assumptions of our contemporary scientific worldview.
- Experience the joy of working through mathematical demonstrations together
- Develop the skill of visualizing and articulating complex mathematical relationships
- Discover how the ancients approached problems with tools profoundly different from our own
- Trace the historical transformation of mathematics and science through direct engagement with primary source
**Limited to 15 participants** to ensure quality conversation and collaborative demonstrations
**No prior mathematical expertise required**—just a willingness to engage actively with the texts
Core Texts:
– Euclid’s Elements (Green Lion Press edition)
– Archimedes, Works (Heath translation on Dover)
– Apollonius’ Conic Sections (Green Lion Press edition)
– Plato’s Timaeus
– Ptolemy’s Almagest (Ed. William Donahue, Green Lion Press)
– Additional selections from Kepler, Galileo, and Newton
Euclid’s Optics: The Mathematics of Vision
Summer 2025: Mondays, June 23 – August 11
Discover how the father of geometry first explained how we see the world
Long before modern optics and perspective theory, Euclid created the first mathematical framework for understanding visual perception. This focused 8-week course explores his groundbreaking analysis of how size, distance, and position affect what we see—creating the foundation for everything from Renaissance art to modern computer graphics.
What You’ll Explore:
The Visual Ray Theory (June 23 – July 7) Examine Euclid’s fundamental premise that vision occurs through rays traveling in straight lines from the eye. Follow his systematic development of how these rays interact with objects to create visual perception, establishing the world’s first geometric theory of sight.
Angular Perception & Relative Size (July 14 – July 21) Understand Euclid’s mathematical explanation for why distant objects appear smaller and parallel lines seem to converge—the very principles that would later revolutionize Renaissance painting through mathematical perspective. Discover how he quantified the relationship between distance and apparent size through elegant geometric proofs.
Motion & Positional Perception (July 28 – August 11) Explore Euclid’s sophisticated analysis of how movement affects perception and how objects appear from different viewing positions. These propositions not only explained everyday visual phenomena but laid the groundwork for geometric optics and theories of visual representation that would evolve over the next two millennia.
Pathway Structure:
- Summers, Weekly Monday seminars at 12:00 noon EST
- Each session examines three key propositions in their historical and mathematical context
- No advanced mathematics required—
- Hands on learning experience: Everyone gets a chance to demonstrate proofs of propositions. We work together through each proposition with clarity and insight.
- Work with translation of the original texts
Why Study Euclid’s Optics Now?
As a bridge between Euclid’s Elements and Archimedes’ works in our “Shoulders of Giants” series, the Optics reveals how ancient Greek mathematics extended beyond abstract geometry to explain physical phenomena. This text influenced not only the development of science but also profoundly shaped Western art through its impact on perspective theory.
Whether you’re interested in the history of science, the mathematics of perception, or the foundations of visual representation, Euclid’s Optics offers fascinating insights into how mathematical thinking transformed our understanding of vision itself.
Tuition: Quarterly Slow Reading Subscription. Already a subscriber? You’ll get access to this pathway. New subscribers have open access to this and all other slow reading programs during the subscription period.
Prerequisites: Basic familiarity with Euclidean geometry not required
Registration: Register below!
“The laws of perspective are to painting what the rules of grammar are to writers.” — Leonardo da Vinci
Explore the earliest mathematical theory of vision that laid the foundation for how we understand, represent, and now digitally render the visual world.
Why Read Ptolemy After Copernicus?
A Defense of Ancient Astronomy in the Modern Age
When we invite participants to study Ptolemy’s Almagest in our mathematics and natural science pathway, we sometimes hear a reasonable question: “Why should we invest time in an astronomical system that was ultimately proven wrong?”
This question reflects a fundamental misunderstanding about the nature and value of scientific thought across time. Reading Ptolemy isn’t about retrieving outdated facts—it’s about encountering a fundamentally different approach to understanding the cosmos that continues to illuminate our own.
Beyond “Wrong” and “Right”
Ptolemy was not engaged in the same project as Copernicus, Kepler, or Newton. His goal was not to discover the physical truth of celestial mechanics but to “save the appearances” (sozein ta phainomena)—to create a mathematical model that accounted for observed celestial phenomena with precision and coherence.
In this, he succeeded magnificently. The Ptolemaic system predicted planetary positions with remarkable accuracy for over 1,500 years. Its mathematical elegance and explanatory power represent one of humanity’s greatest intellectual achievements.
What We Gain from Ptolemy
When we work through Ptolemy’s demonstrations together, several transformative insights emerge:
A Different Mode of Scientific Thinking
Ptolemy’s approach reveals that there are multiple legitimate ways to relate mathematics to physical reality. The modern insistence that mathematical models must describe physical causes is not the only valid approach to natural phenomena.
The Foundations of Mathematical Modeling
The techniques Ptolemy developed—using geometric constructions to model complex motion—remain fundamental to scientific practice today. His epicycles and deferents represent early examples of Fourier analysis, where complex motion is broken down into simpler circular components.
The Philosophical Dimension of Astronomy
Ptolemy’s work raises profound questions about the relationship between observation, mathematical representation, and truth. What does it mean for a theory to “save the appearances” versus describing physical reality? These questions remain vital in contemporary philosophy of science.
Historical Perspective on Scientific Revolution
By engaging directly with Ptolemy, we can understand the Copernican revolution not as a simple correction of error, but as a fundamental shift in what questions astronomy was meant to answer. This insight helps us reflect more deeply on current scientific paradigms and their unexamined assumptions.
Learning Without Dogma
Reading Ptolemy as part of our slow reading pathway offers a rare opportunity to encounter scientific thinking on its own terms, free from the distorting lens of modern prejudice. We’re not asked to believe that planets move on epicycles, but to understand why this mathematical construction made profound sense within its intellectual context.
This approach liberates us from what C.S. Lewis called “chronological snobbery”—the assumption that newer ideas are automatically better than older ones. Instead, we discover how different forms of thought can illuminate different aspects of reality, expanding our conception of what science can be.
In a world increasingly dominated by specialization and technological application, this historical and philosophical perspective offers something precious: the ability to step outside our contemporary assumptions and see scientific inquiry afresh.
When we stand on the shoulders of giants like Ptolemy, we don’t just see further—we see differently.
ON THE SHOULDERS OF GIANTS
Archimedes of Syracuse: “On the Sphere and Cylinder”, “Floating Bodies”, and “the Sand Reckoner”
Fall Session 2025: September 4 – December 18
Journey into the mind of antiquity’s greatest mathematical genius
Immerse yourself in Archimedes’ three masterworks that revolutionized mathematics, physics, and astronomy. This comprehensive 16-week course guides you proposition by proposition through the foundational texts that bridge Euclid’s geometry and the Scientific Revolution.
What You’ll Explore:
On the Sphere and Cylinder (September 4 – October 2) Discover how Archimedes determined the surface area and volume of the sphere through ingenious proofs that prefigured calculus by nearly two millennia. Follow his step-by-step demonstrations that the sphere’s surface equals four times its great circle, and its volume is precisely two-thirds that of its circumscribing cylinder—insights so profound Archimedes requested them carved on his tomb.
On Floating Bodies (October 6 – November 6) Unravel the principles of hydrostatics that emerged from the famous “Eureka!” moment. Trace Archimedes’ systematic development of the law of buoyancy and his sophisticated analysis of floating stability—work that established physics as a mathematical science and continues to govern naval architecture today.
The Sand Reckoner (December 1 – December 18) Explore Archimedes’ breathtaking attempt to count the uncountable—calculating how many grains of sand would fill the entire cosmos. Witness his creation of an exponential notation system capable of expressing numbers far beyond anything previously conceived, while gaining fascinating insights into ancient astronomical models.
Course Structure:
- Twice-weekly seminars: Mondays and Thursdays at 12:00 noon EST
- Each session examines two key propositions.
- Hands on learning experience: participants take turns demonstrating proofs. We work together through each proof with clarity and insight
- Work with the original text in translation rather than text book
Why Study Archimedes Now?
Positioned between Euclid and Apollonius in our “Shoulders of Giants” series, Archimedes represents Greek mathematics at its zenith. His works demonstrate the perfect union of theoretical brilliance and practical application that would later inspire Galileo, Kepler, and Newton.
Whether you’re continuing your journey from Euclid’s Elements or preparing for our upcoming exploration of Apollonius’ Conics, Archimedes offers indispensable insights into how mathematical thinking transforms our understanding of the physical world.
Leave me to my circles!
Legend has it that Archimedes, about to be killed by a Roman soldier, said “Leave me to my circles.” In the words of Valerius Maximus, what Archimedes said was a little different:
“at is, dum animo et oculis in terra defixis formas describit, militi, qui praedandi gratia domum inruperat strictoque super caput gladio quisnam esset interrogabat, propter nimiam cupiditatem inuestigandi quod requirebat nomen suum indicare non potuit, sed protecto manibus puluere ‘noli’ inquit, ‘obsecro, istum disturbare’, ac perinde quasi neglegens imperii uictoris obtruncatus sanguine suo artis suae lineamenta confudit.”
“…while he was drawing figures with his mind and eyes fixed on the ground, he could not tell his name to a soldier who had broken into his house for the sake of plunder and who, with a sword drawn over his head, was asking who he was, because of his excessive desire to investigate what he was working on, but with dust spread out by his hands, he said, “Please, do not disturb this,” and just as if he were neglecting the command of the conqueror, he was killed and mixed the outlines of his work with his own blood.” (Valerius Maximus, Facta et Dicta Memorabilia, Book 8, Ch 7, on Archimedes)
There are several lessons we might draw out from this story:
Heroic dedication to the passion for truth transcends self-preservation: Archimedes was so deeply immersed in his mathematical work that even with death imminent, his primary concern was protecting his calculations. His words “Please, do not disturb this” reveal how completely his life’s purpose had become intertwined with his intellectual pursuits.
The enduring value of knowledge over temporal power: The Roman soldier represented military might and conquest, while Archimedes embodied intellectual discovery. Though the soldier could—and did—end Archimedes’ life, he couldn’t diminish the lasting impact of Archimedes’ contributions to mathematics and science, which continue to influence our world millennia later.
Present awareness versus future impact: What might seem like “mere drawings in the dust” to the soldier represented timeless discoveries to humanity. Archimedes’ concern was not for his immediate safety but for the preservation of knowledge that would outlive him.
Finding meaning that transcends mortality: Archimedes’ unwillingness to abandon his work even in his final moments speaks to finding such profound purpose in one’s calling that it becomes more valuable than life itself.
The tragic collision between brute force and intellectual creation: The image of Archimedes’ blood mixing with his mathematical drawings symbolizes how violence can temporarily interrupt the work of the mind, but cannot ultimately conquer it .
Prerequisites to join this pathway: No prerequisites but a willingness to learn.
Registration: Register below!
“Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not.” — G.H. Hardy
Join us in experiencing firsthand why Archimedes stands as the crown of Greek mathematics and one of the greatest minds in human history
Learning Experience:
Participants will take turns demonstrating theorems using collaborative online tools, working through each step of the author’s reasoning together. This active approach develops unique skills of visualization and argumentation while providing direct insight into ancient and modern mathematical thinking.
No prior mathematical expertise required—just a willingness to engage actively with the texts
Open enrollment throughout the pathway: Unlike many sequential courses, our approach allows participants to join at any point in the journey. Each session stands as a meaningful engagement with mathematical reasoning, while comprehensive recordings and careful documentation of previous demonstrations ensure new members can quickly orient themselves within the broader exploration. The collaborative nature of our sessions means experienced participants naturally support newcomers, creating a welcoming environment regardless of when you begin.
Cost:
Subscribe to the Symposium Slow Reading Program
$250 Quarterly/$750 Annual
Your single subscription to the program provides access to this pathway and all of our carefully curated reading pathways spanning Western civilization’s most influential works. Join as many as you like or are able to. This comprehensive program is designed for those seeking sustained intellectual engagement with humanity’s enduring ideas.
In today’s fragmented world of constant notifications and fleeting attention, Symposium creates a sanctuary where people come together to explore humanity’s deepest questions. What might otherwise be a solitary encounter with great texts becomes instead a collective journey of discovery, where understanding deepens through conversation and shared reflection.
Meetings and breaks
We will be meeting twice per week, for an hour and a quarter, instead of once a week like the regular slow reading pathways. But we plan to take time off (approx. 2 weeks) between sessions (see Syllabus above). All meetings will be recorded and archived for use by members only. If you miss a session, you can gain access to these recordings. Some members may only want to join for one day per week. Twice per week meetings allows us to go at a slower pace while moving through the material in a timely fashion.
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Break between sessions: Between Winter/Spring, Summer and Fall sessions, the group will take a 2/3 week hiatus.
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