**NEW Pathway:**

**“On The Shoulders of Giants”**

“If I have seen further than others, it is because I am standing on the shoulders of giants.” – Isaac Newton

“Euclid alone has looked on Beauty bare.

Let all who prate of Beauty hold their peace…”

-Edna St. Vincent Millay

“I had not imagined that there was anything so delicious in the world.”

–

*Bertrand Russell*“Nothing hinders us from doing justice to the originality of ancient science by allowing ourselves to be guided only by these phenomena to which the Greek texts themselves point and which we are able to exhibit directly. , our different orientation notwithstanding, Greek scientific arithmetic and logistic are founded on a “natural” attitude to everything countable as we meet it in daily life, This closeness to its “natural” basis is never betrayed in ancient science.”

– Jacob Klein,

– Jacob Klein,

*Greek Mathematical Thought and the Origin of Algebra***Pathway Overview:**A slow reading pathway in the primary sources of mathematics and natural science, ancient and modern. The key study is the transformation of ancient science and math into the modern. What is the difference between the ancient and modern approaches? Participants formulate their own opinions based on direct experience with the primary sources – including practicing the skills of argument and demonstration. Meetings two days per week. Open-enrollment: recordings archive will be made available, and participants may join at any point along the course of this program.

**Syllabus for**

**2025**

I.

**Winter/Spring Session**Euclid,

*Elements*II.

**Summer Session**Apollonius,

*Conic Sections*III.

**Fall Session**Plato’s

*Timaeus*Ptolemy,

*Almagest***Syllabus for**

**2026**

I.

**Winter/Spring Session**Copernicus,

*Revolution of the Spheres*Kepler,

*Astronomia Nova, Epitome of Copernican Astronomy, Harmonies of the World*II.

**Summer Session**Kepler

*(continued)*III.

**Fall Session**Galileo,

*Two New Sciences*________________

**Start date:**January 2025

**Day/Time:**

Mondays and Thursdays, 12 pm EST/11 am Central, 9am Pacific –

*starting*.__January 2025__(Current/Ongoing Euclid Group): Thursday evenings, 9pm EST/8 pm Central/6 pm Pacific.

**Meeting Frequency:**twice weekly

**Session length:**1.25 hours

**Instructors:**David Saussy

**Pathway Duration:**Open, Quarterly

**Cost:**$250 subscription per quarter (3 months of weekly sessions), with all-access pass to any other slow reading courses.

__The Design of the Pathway, “On the____Shoulders____of Giants”__Isaac Newton once said, “If I have seen further than others, it is because I am standing on the shoulders of giants.” This image – and the idea – goes back at least to the 12th century. John of Salisbury wrote: “Bernard of Chartres used to compare us to

*dwarfs perched on the shoulders of giants*. He pointed out that we see more and farther than our predecessors, not because we have keener vision or greater height, but because we are lifted up and borne aloft on their gigantic stature.” (*Metalogicon*, 1159)Our world has been shaped in a decisive way by mathematics and natural science. To understand the world we live in, one must at least come to a genuine understanding of the math and science that helped to shape it. But math and science have become so specialized and arcane – and education derivative – that the liberal learner interested in the fundamental questions, finds it difficult in the highest degree to come to a comprehensive understanding.

To make matters more difficult, most of us have been introduced to mathematics and natural science – as well as many other subjects besides – through textbooks.

Now a textbook is a derivative rather than an original source of ideas and concepts. To the extent textbooks have constitute the core of our basic education, our basic science and math education is or has been derivative in character.

The point is to save time, to offer a short cut. But this means that the crooked byways, cul-de-sacs and the meandering pathways of discovery characteristic of real human thinking, thinking at its best, must be hidden from us. The approach to difficult subject matter, that is, to the actual problems and their ground, is made somewhat more palatable for beginners in a textbook.

Consequently, the story we are given by textbooks – and our education, let alone our popular notions – tends not to be the

*real*story of human thought in mathematics and natural science – but a digested and even misleading representation of the original. What it leaves out may be crucial to developing our*own*understanding.So what is the real story? What has been left out?

The following 14 great thinkers in mathematics and speculation about nature – and their original voices – have been left out.

Euclid, Elements

Apollonius Conic

Plato Timaeus

Aristotle,

*Physics*Ptolemy,

*Almagest*Copernicus,

*Revolution of the Spheres*Kepler,

*Astronomia Nova, Epitome of Copernican Astronomy, Harmonies of the World*Galileo,

*Two New Sciences*Descartes, Geometry

Newton, Principia

Faraday

Maxwell, A Treatise on Electricity and Magnetism

This list is by no means exhaustive, but it is an essential list.

Our intention is not to learn about the ‘history of ideas’, as though it is detached subject matter, but to undergo this very history itself by working through mathematical proofs and arguments.

That is, we want to follow these great thinkers’ footsteps and try to see the world as they saw it, to understand the problems as they understand them; we intend to face the challenges that arise in contact with subsequent thinkers, and to see first hand how the radical transformation in the manner of concept formation from the ancients to the moderns takes place.

For example, Ptolemy was not engaged in physical speculation – like Kepler and those following him – but endeavored to “save the appearances” (Greek) using mathematics. What is the difference between ‘saving the appearances’ and physical theory? By working through the texts of both, and others, you will have first hand – non-derivative or original experience with what this actually means. The result of such an immersive learning experience is not only enlightening, and gives you a fresh perspective, but it places your thinking on firmer, more solid foundations.

**On our starting point, Euclid**

As we know, Euclid’s Elements is something like a compendium of geometric and arithmetic propositions and proofs from other mathematicians of his day – and his work is indebted to Plato and Aristotle.

Euclid’s great Thirteen Books of the Elements has suffered a peculiar fate in the modern world. On the one hand, the text forms in a certain way one indisputable foundation of modern thought (especially his Fifth Book, which presents what later came to be known as a “theory of ratios”). Not only is the formalized language of modern mathematics rooted in a certain interpretation of Euclid, but Euclid is even is used as a foil, over against modern ideas are presented. Sooner or later, the attentive modern reader will come across critics of a so-called “Euclidean space”, as if it were perfectly evident what that is.

On the other hand, we don’t know actually know Euclid’s thinking. Even though it is relatively easy to get a hold of a copy of Euclid’s Elements, his work is foreign to us. We only see Euclid through the lens of familiar conceptions that do little justice to the original. The most common geometry lessons in high school present Euclid in an algebraic manner, which is to say not on Euclid’s own terms. What we call “Euclidean space” is in fact ‘algebraized’ space, space interpreted through the presuppositions entailed in Algebraic analysis.

Yet what are Euclid’s own terms? Something astonishingly approachable. Those of us who come to Euclid, as readers of the original in translation, often have the experience of delight mingled with a touch of bitter disappointment. It is not unusual to feel robbed of something that could have been our

*own*long before. We are now accustomed learning a ‘mathematical language’ in school, but Euclid’s proofs require not a special separate language to learn, but rather our own natural intuitions of space, form and relationships. We have everything we need already to come to the study of the forms and ratios in Euclid. No wonder the Greek word for learning in fact is the root of our word “mathematics”, as if to say, what is to be learned in Euclid is most approachable and learnable matter of all.It is well known that Abraham Lincoln spent much time with Euclid in his younger years, and we might even discern something of the rigor of Euclid in Lincoln’s thought, for example in his Cooper Union address, or the debates with Steven Douglass (known as the Lincoln-Douglass Debates). There is a reason why Lincoln might have spent so much time with Euclid. Not only is Euclid beautiful, but from Euclid we can learn, in a way like no other, what makes a good argument (or a bad one).

If you are wondering about the intellectual climate of our time – namely what is wrong with it – a fruitful place to spend time (that doesn’t involve hand-wringing but pleasure and satisfaction of sinking your teeth into something substantial) is Euclid’s Elements. A study of Euclid could be the business not of a few specialists, but of all educated people everywhere, or all people seeking an education – which as we know, in truth, is a task for a lifetime.

Our work is to learn how to talk and think about what we see unfolding in Euclid’s Elements. We will start slowly with Book I, the definitions and common notions and postulates. We will move as slowly – or as quickly – as our group is capable of doing. No prior experience in Euclid or the history of mathematics is required. Everyone possesses what is needed already.

**Who is it for?**

- All those who wish to get acquainted with a forgotten source of western thought, as we encounter it through reflections on Euclid and our own experience.
- Those who “hated” math or think they are “not mathematical” in school.
- Anyone educated in modern mathematics, who would like to investigate the history of mathematics, through direct contact with the primary sources (rather than through the secondary or intermediary sources).
- Someone looking to fill the blanks missing in their own education, which have likely overlooked classical sources.
- Someone who has spent time with Euclid, but not enough time – would like to revisit, and take their time unconstrained by academic calendars.

**What is the aim of a conversation**

**(what**

**learning experience should I expect)?**

Our approach to Euclid will be conversational rather than lecture. Serious conversation allows discussants to meet the Euclidean text on the basis of discussants’ own questions – thus promoting a more active and intimate relationship to the material, not to mention a better understanding of it.

Conversation aims for

**reflective understanding of Euclid**based upon a sustained work with the text. There are no tests in this learn pathway. Some supporting essays may be offered to help develop an understanding of the difference between ancient and modern mathematics and its relationship to human thought in general. The primary focus of this pathway is the building of reflective understanding based on direct engagement with the text, rather than amassing facts and historical data outside our reflections.Euclid’s work elegantly builds a succession of theorems on the basis of primary definitions. To each one of the theorems is offered a “proof”, which demonstrates the truth of the theorem on the basis of the foundational definitions, common notions and postulates, as well as preceding theorems. Most of us encountered the “Pythagorean theorem” in school in an algebraic formula, but this algebraized interpretation strips away the originality and fullness of Euclid’s actual thinking. As we read Book I, you will get to see that theorem in its authentic form – the 47th proposition, built upon preceding insights and axioms.

Not all the proofs we encounter are entirely convincing, at least not at first; and this is a question which will engage us in a lively way. What is a proof? What would we find convincing in a proof? How does demonstration work exactly? What does it mean to make an argument? What is the mode of being of these mathematical objects – points, lines, circles, triangles etc. ?

As a matter of reading most actively, and getting the most of our engagement, we take turns demonstrating theorems. This means that you try to follow in the footsteps of Euclid’s proofs, by showing all the steps he takes setting out the figure and unpacking the relationships, leading the final conclusion. If you try to do this without the aid of notes or the book, you develop a skill that you can’t get in any other way. You’ll find your mind sharpened, and your ability to think and talk though complex arguments improved.

__Books and Resources__Euclid Elements, Green Lion Press; First Edition (January 1, 2002)

**Interested to see what a seminar on Euclid is like? Sign up below and j**

**oin us for a free introductory seminar on Euclid.**

**When: Date and Time,**TBD

**Where:**Zoom

**Discussion Focus:**Book 1: Definitions, Common Notions and Postulates, Proposition 1 and 2. Bring your questions!

*(*

*Free PDF of reading will be sent to those who sign up.)*