Category: Bulletin

Bulletin

Euclid’s Elements

Instead of our usual excerpt, we have thought it appropriate to give you a taste of Euclid’s Elements. Using the interactive online version created by David Joyce of Clark University, we invite you to sample the 15th proposition of Book I, in which Euclid proves that, when two lines intersect, the angles across from one another are equal. You can see the proposition unfold through this video created by Sandy Bultena. If you want a little more, the next proposition’s proof (text here, video here) that the exterior angle of a triangle is greater than either interior angle is neat (and sets the stage for his proof that the interior angles of a triangle equal two right angle).

Bulletin

Euclid and The Mathematician’s Process

Last year, Dr. Seeley of the Boethius Institute connected me with Rosary College, a start-up junior college looking for an instructor to teach Euclidian geometry. Unsure how to teach Euclid’s Elements in a way that was authentic to both the classical liberal arts curriculum and my experience as a mathematician, I looked through the library in the math department at Colorado State University, my alma mater. To my surprise, I found a text titled Geometry: Euclid and Beyond authored by Robin Hartshorne. This text provided me with both insight into how fellow mathematicians think about Euclid in the modern day, and sympathy for how intimidating Euclid’s Elements might be to students reading them for the first time. 

Robin Hartshorne is a professor emeritus at the University of California, Berkley, and a highly influential mathematician. His 1977 text Algebraic Geometry was one of the first sources that made algebraic geometry – a highly technical and evolving field in modern mathematics born in the 20th century out of the works of Diophantine – “accessible” to graduate students. Yet it was still notoriously difficult, and especially intimidating to me as a first year graduate student. At the time, it seemed just a series of complicated nonsense: an intricate definition, a proposition, another definition, several more definitions, and a long list of proofs I could not follow until I really understood the definitions. Finally, in my third year, I took an algebraic geometry course taught by my advisor. His carefully selected exercises allowed me and my classmates to interact with the content in a meaningful way. Most importantly, these examples were fun to work on together. In a playful spirit, we made progress reading and writing arguments.

That same year, I started to make progress in my research for the first time. Research in mathematics – that is, proving novel theorems – requires a deep understanding of the definitions at hand. Most of the time, building this foundation is a tedious rhythm of reading and taking notes. However, when I went back to basics and played with lower-level examples similar to those my advisor selected for his course, I finally learned the definitions that were previously too complicated to understand.

Such play is not only an essential aspect of modern mathematics, but also of Euclid’s development of the Elements. Starting with five fundamental axioms, Euclid would have played with points, lines, and plane figures until he recognized certain provable truths. Then, using only the axioms and previously proven results, he could construct arguments demonstrating such truth beyond refute. Through the act of reading these arguments, a modern audience might gain an appreciation for Euclid’s use of logic and the axiomatic method but miss the playful discovery of such truths. Euclid’s long list of definitions and propositions may even seem like a series of complicated nonsense like Algebraic Geometry was to me as a graduate student. As I built my syllabus for MATH 101: Euclidean Geometry, I wanted to make the material come to life just as my advisor had done for me.

In the late 1980s, Hartshorne found himself in a similar position – teaching classical geometries as a modern mathematician. In Geometry: Euclid and Beyond, Hartshorne “takes Euclid’s Elements as the starting point for a study of geometry from a modern mathematical perspective,” by walking the reader through Euclid’s most significant propositions and connecting them to newer problems (Hartshorne, 2000, p.1). Although this text ultimately covers material suited for a senior level mathematics major, the first chapter, meant to be read concurrently with the Elements, is filled with accessible discovery-based exercises designed to put any reader in Euclid’s shoes.

Early in the course, I had my students write conjectures based on their explorations in Geogebra, an online calculator that allows for compass and ruler constructions. For example, on a discussion board problem in the first few weeks, I had students answer true and false questions about polygons circumscribed by circles, and write their own conjectures. They had not yet read Books III and IV of Elements which deal with these relationships, but through collaboration and play, they found their own reasoning for why all triangles can be circumscribed by a circle, and used results from Books I and II to justify their reasoning. They also correctly conjectured relationships between angles in a quadrilateral circumscribed by a circle (namely that opposite angles are supplementary). These results were significantly more meaningful when we later discussed their proofs because my students knew them first hand. 

Along these lines, one of Hartshorne’s classical geometry students argued that in order to have a truly authentic experience, it would be better to have no accompanying textbook at all, not even the Elements (Hartshorne, 2000 p.13). Students would live the axiomatic method as Euclid did and discover geometry from scratch. Of course, one pitfall of this idea is that a semester is only 15 weeks long. Exploration takes time, but constructing an argument after the play is done takes even more time. Under an exploration-only model, it would be impossible to cover all of the material in the Elements. Additionally, the act of reading written arguments is an important part of the learning process. How could one write a proof having never read one?

With this in mind, I had my students write their own proofs to problems not found in Euclid’s Elements after reading several similar arguments. For example, the third week, I assigned the following homework problem. 

As to not spoil the problem solving process for the interested reader, I refrain from including a solution here. However, using known results, and other strategies, my students constructed excellent arguments. 

It is worth mentioning that play is sometimes messy. My students did not always find clear-cut solutions like they did for the above exercise. For example, halfway through the semester, I assigned the exercise below.  

While students creatively worked toward a proof, the process for this problem was significantly more open-ended and involved; I could not expect routine answers. For certain problems, all of my students observed and communicated the same ideas. Other times, I received as many different answers as I had students because each individual chose to explore a different angle. In class, having open-ended conversations sometimes meant that I stumbled through the material too. Yes, this process was sometimes frustrating, but it was also exactly the point, and my students rose to the challenge. Because this course included rote activities directly from Euclid’s Elements, structured written homework assignments, and a playful activities adapted from both Geometry: Euclid and Beyond and my own undergraduate geometry curriculum (written by Nathaniel Miller1), my students got to live the entirety of the mathematician’s process. From reading the definitions and proofs to playing with lower-level examples, they persevered through the messiness. Guided by the structure of Euclid’s Elements, my students ultimately communicated their arguments and found success writing proofs. Teaching this course reminded me that people are inherently good at recognizing truth when given the opportunity to reckon with it.

References

Hartshorne, R. (2000). Geometry: Euclid and Beyond. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-22676-7

  1. Nathaniel Miller is a professor at the University of Northern Colorado and has several papers about teaching proof-writing to pre-service secondary teachers. In his dissertation, A DIAGRAMMATIC FORMAL SYSTEM FOR EUCLIDEAN GEOMETRY, he formalizes Euclid’s diagram proofs using computer programming.  ↩︎
Bulletin

Is Algebra a Language?

Note: What follows is an abridged version of a lecture delivered at a conference of the Boethius Institute convened at the Augustine Institute Graduate School of Theology on 6-7 August 2025. 

When it comes to algebra, the conventional approach to teaching it often leaves a lot to be desired. That forces us to ask: how should algebra be taught? Or should it be taught at all? Underneath those questions, there lies what is really the more fundamental question, namely, what is algebra? Is it mathematics, or just a tool that mathematicians use? If the latter, do we really need it, or does it turn out to be more of an impediment to education in mathematics?

…

In order to discuss these questions, and especially the fundamental one concerning what algebra or mathematical symbolism really is, it is perhaps best to first observe the historical process by which mathematics gradually adopted the symbolic form it now has. You all know, I’m sure, that mathematics was for a long time not nearly as reliant on symbolic notation as it is now. Euclid’s Elements, for example, is a compilation of hundreds of propositions, every one of which is stated in words, and then proven in words. That presentation makes it quite clear that the purpose of that treatise and others like it was to say things that are taken to be true and demonstrable in the same manner that other things spoken in words are true. 

…

But many centuries later, once symbolic notation is introduced and becomes customary, striking changes begin to occur. Both enunciations and proofs continue to exist – or at least those names still exist for what mathematicians do – but at times it becomes less clear whether it is still really a matter of saying anything, or equally significantly, of saying something that can be called true without qualification. …

The books of the early analytic school of philosophy bear the distinct mark of having arisen out of the customary habits of symbolic mathematics, but they also show distinct signs of being concerned with more than just a mathematical technique. … A new vision of human reason, if not even a new vision of reality itself is being developed. This vision is associated with a kind of skepticism. It is this skepticism that prompted C. S. Lewis to write his book The Abolition of Man.

 …

A notable example is Bertrand Russell’s famous Principia Mathematica, which undertakes the serious task of reinterpreting the foundations of mathematics. From there the analytic philosophers go on to try to reinterpret the entire world of philosophical pursuit. … Two premises combine to make a single conclusion. The first, that symbolic representation perfects mathematical and logical reasoning, making it much more efficient and efficacious. This was what Descartes had already shown centuries before. But now the second, newer premise is that this perfection derives from what is taken to be not just metaphorically, but really, a language. Putting these two premises together, it appears plausible that what we are now looking at is not just a language, but some sort of refinement or perfection of language itself.

…

And so it is time now to settle this question about what the so-called symbolic language really is. Is it really language in a proper sense, or is it something else? I will start with some fairly humble and concrete observations.

The first observation is that while the words we speak are about, or said of the things we talk about, the symbols of algebra do something different: they do not speak about, but rather represent, that is stand in place of, their objects. Thus there is in both cases a relation between sign and signified, but the relation is quite different in each case.

A good example to illustrate this is the difference between a dollar bill and the word “dollar.” If someone says to you, “I need a dollar,” and you respond “dollar,” that person is going to find you unhelpful and not even very funny. But if you hand him a dollar, he’ll be more content. So what’s the difference? To “represent” means, literally, to “re – present,” or in other words to stand in place of something. The dollar bill has actual value because it is accepted as a substitute for the thing it signifies. But a word isn’t a substitute for the thing. It is, indeed, a representation – but what it represents is not the thing it signifies, but rather a thought about that thing. Immediately we can infer from this that algebra derives its remarkable power from the fact that it does something that words do not do. In fact it is more like the figure in a Euclidean demonstration than like the words we use to describe the figure. It is also similar to the stones that a shepherd might use to count sheep. It’s a lot easier to carry a bag of stones around than a flock of sheep.

…

One further observation will help us confirm the difference between mathematical symbols and words. In the old tradition of logic that existed before symbolic logic came into being, it was understood that words could be universal. A single word could thus signify what was common to many things: cat could refer to any number of cats since every cat is a cat, and likewise with dogs, stars, and so forth. The traditional way of describing this was to say that cat signifies what all cats have in common, which is to be a cat. But in the early analytic school, it was presumed that this universality of words was the same as the universality of symbols. It was either not noticed, or swept under the rug that we have here two different kinds of universality. For the universality of a symbol is a universality of representation, whereas the universality of word is a universality of predication, or in other words a universality in being able to be said of something. A universality of representation is in symbolic representation what we refer to as being a variable. The universal of the old logic doesn’t refer to a variable, because words are not variable substitutes for things, but rather said of them because of what they have in common. When we say Socrates is a man, we are not putting the word “man” in place of Socrates, but rather saying what Socrates is.

So far so good. Now we come to what may prove to be a somewhat steeper climb. To make the climb more gradual, I am going to avail myself of a remarkable moment in the early theorizing of the analytic school of philosophy.

There is a  passage in the writing of that most outspoken of analytic philosophers, Bertrand Russell, in which he castigates traditional philosophers for what he thinks is their primitive understanding of language, logic, and finally even of metaphysics. The foundation of his complaint is that he thinks the traditional idea of a nature or essence is nonsense. Up until around Russell’s time, the ideas of essence, nature, or essential attributes were taken seriously. But now, in the midst of the new development of the methods of analytic philosophy, all of which rely very heavily on the use of symbolic representation, objections and criticisms to the traditional ideas become much more focused. … Russell’s claim is, in effect, that there are no essential natures or species or definable realities that come from the world as it is; what there are is classes that we invent as we please, and we use symbols as marks of these classes that we invent. The symbol thus serves as a kind of seal or mark placed upon a completely arbitrary act of reason not discovering, but constructing. Russell is proposing that this example typifies the whole world of thinkable realities. The alternate view, which holds thinkable reality to be something that derives from the world outside of ourselves, and in fact ultimately from an intelligence that is not our own, is in Russell’s view just old nonsense.

The critical thing to note here, however, is that Russell just assumes gratuitously that there is no difference between symbols and words. And since there is no difference between symbols and words, it follows that what symbols represent is exactly the same as what words signify. If Russell had paused to wonder if symbols and words are really distinguishable, this might have become a teachable moment; an occasion to understand better not only what symbols really are, but also what words are in contradistinction from symbols. The fact that symbols signify something that we make with our minds could, in that case, have been recognized as a distinguishing mark of symbols from words.

… 

To re-summarize, I will say this in another way. Human reason draws all of its insights from the material world. We rise from the observation of material things to more and more abstract and immaterial kinds of knowing. But in doing so we continue to rely on material supports to some extent, and one way we do so is by using sensible sounds, words, to express even our more abstract thoughts. But symbolic construction goes exactly the opposite way: here we can begin from abstract ideas, but we embody them in symbolic constructions. And we do so with the latitude of the whole potential world of artificial construction, as opposed to the world of apprehending. I can say, just because I choose to, “Let x stand for a shoe, a star, or a triangle.” Then just by pure fiat, I have made a class of objects, which exists no otherwise than by an act of my will, ratified and sealed, so to speak, by the symbol x.

… 

If symbolic construction is what I have said it is, namely the expression of arbitrary rational construction, then how in the world can it be relevant to liberal education? Isn’t liberal education supposed to be not about arbitrary construction, but about higher things which we contemplate and do not make? Won’t trying to insert symbolic construction here just destroy the whole endeavor? There are some who have tried to answer that question with a very emphatic “yes!” Yet doing that flies in the face of a great deal of mounting evidence, for example in modern mathematics as well as physics, where what look like dazzling discoveries would have been impossible without the aid of symbolic representation. So how is that possible? 

… 

Symbolic representations do something like what figures do, but often they can do it better. Both exteriorize and complete a constructive act of the mind, so that we can subsequently bring our minds to bear on it if we wish. In making a symbolic representation, if we choose to make it after the image of something which we naturally apprehend, then the symbolic representation can often retain the character of the natural object as we conceive it better than a picture would, partly by dispensing with what is accidental. We don’t have to worry about making our lines good and straight, our angles accurate, and so forth. And so in that case the symbolic representation facilitates contemplative thought, and it can often do so better than a picture would.

One day years ago I walked into the office of a good friend of mine, who happens to be a first class mathematician. Somewhat to my surprise, I saw all of the five regular solids of Euclidean geometry sitting on his bookshelf. He had made them out of paper, evidently with great care and precision. The thought crossed my mind that I was seeing at least part of the reason why he was such a good mathematician. He took the trouble to replicate, artificially, the things he was trying to learn about. Mathematical symbols do the same thing, and they can do it often with the advantage of being immediately the offspring of our own minds without the intermediation of paper, scissors, glue, or anything like that. The mistake, if there is one, is not to use such an excellent aid to thinking; it is rather to think that such aids are themselves the ultimate object of reason itself, or that they are the proper and immediate expression of reason itself apprehending an essential, natural order.

Conclusion

In conclusion, we can now see that there is a balance to strike in how we teach algebra. That we must start with things imaginable and accessible to the young is clearly right. It is also true, though, that we are capable of higher degrees of abstracting through symbolic representation, providing that we see it as what it is, namely a representation of some kind of intelligible reality. I do not mean to suggest that the representation must always be quasi-allegorical, any more than it must be such in literature. Still, it must in some way point our minds to the real.

To this, then, we could add two further caveats when it comes to actual teaching. The first is that although, or perhaps I should say because, symbols represent real things, we must not confuse the symbols themselves with the reality, speaking absolutely. To illustrate this with a symbol outside of mathematics, we should say that the Statue of Liberty represents liberty, and we might even say that it embodies it after a fashion. But if anyone were to suggest that liberty is nothing but that statue, we should have to disagree.

I will put the second caveat in the form of a parable, something that really happened when I was a child. It was the birthday of two of my younger siblings, who were perhaps about 6 or 7 years old. My sweet, dutiful mother had gone to the store to get them a present. She came back with a “fishing kit,” consisting of two small, fake fishing poles, and some plastic fish that one was supposed to put into a bucket of water, so that the children could fetch them out with the fake fishing poles. Upon seeing that gift, my father said rather vehemently to my poor mother, “Take those poles back right now and get them real fishing poles!”

That event has epitomized for me, as both a teacher and learner, what has become ever more apparent over the passing years. There is one thing, and one thing alone, that inspires and really teaches the young or the old. That one thing is reality. Children, or any of us, may enjoy imitations insofar as those imitations are vehicles towards seeing what is real. But no child or student naturally enjoys or is moved by human artifice purely and absolutely for its own sake. No representation, no procedure, and no art is ultimately able to inspire wonder and to become an object of genuine learning except insofar as it is real, or at least refers to the real. And understanding this is a great key to successful teaching. Too often, educators kill the wondering souls of their students, by inadvertently shutting out the real desire of the soul, which is reality itself, replacing it with procedures, rules, or other forms of arbitrary artifice, work, or caprice. 

As for the art of algebra, it certainly has a beauty and a power that can inspire. But to see that, we must make sure that we teach it as something which represents and refers back to what is real. And in order to do that, we should always start, as far as possible, with what is really real.

Bulletin

From the President (September 2025)

Ecco chi crescerà li nostri amori!
Paradiso, Canto V.105

“Behold one who will increase our love,” sang the blessed souls as they lovingly received Dante and Beatrice in the sphere of Mercury. I often shared that feeling as a summer I expected to be restful turned out to be quite the opposite, filled with travel, presentations, preparations, much toil and stressful anxiety. It was very challenging and exhausting and yet throughout the time, sometimes as guest, sometimes as host, I experienced my love increased by many meetings with wonderful people devoted to passing on our beautiful liberal arts tradition. 

Before launching into my travelogue, I want to let you know that this edition of our Arts of Liberty Bulletin corresponds to the launching of our Quadrivium Retrieval Project. The Learnables Conference drew over 50 lovers of mathematics from around the hemisphere to get to know one another and begin to see the challenges and needs facing a liberal arts renewal of mathematics education. Our two feature articles approach one major question facing us from two different perspectives. In an article drawn from his conference lecture Is Algebra a Language?, Dr. Sean Collins discussed whether algebra really belongs in a liberal arts curriculum. Amie Bray describes her experience as a modern mathematician learning to teach Euclid. Since most of us know modern, algebraic approaches better than traditional ones, we offer you a look at the text of a proposition from Euclid’s text along with a video presentation showing something of the format of formal demonstration.

My summer began in Sao Paulo, Brazil, in early May, where I addressed the first national conference of Catholic classical educators. The Brazilians impressed me by their joy, their friendliness, their love of parties and of Instagram, but above all by their hunger for liberal learning and teaching. At the end of the last day of the conference, the organizer was going to take the speakers to dinner, but 40 or 50 people wanted to join us. So we went to what amounted to an outdoor street fair where we had excellent Brazilian grilled meats and tasted their national drink, caipirinha. When the large screen monitors blaring Lady Gaga’s Rio extravaganza got to be too much for us, we quietly slipped off to a cigar bar. What did they want to talk about at midnight in that lovely bar? Aristotle’s Categories and the definition of man! This is just one instance of how strongly they feel that their teaching flows out from their love with learning.

I spoke about the love of teaching at the Adeodatus conference in North Carolina; I experienced it in a particular way at the Institute for Catholic Liberal Education’s National Conference in Lincoln, Nebraska. Will Clerx, a sixth grade teacher from St Jerome Academy in Maryland, lead us through the steps he takes to introduce the Iliad to his students. He modeled love of his subject, love of the liberal arts, and love of his students as he led us to learn by heart the opening lines of Alexander Pope’s poetic translation. Later, a Dominican sister from Australia exclaimed, “I had no idea there could be so much joy in teaching! I wish all my colleagues could have witnessed it.”

My summer activities culminated in hosting three different Boethius groups at the beautiful Augustine Institute campus in St Louis. Each one was an experience of the joy of intellectual fellowship among devoted educators. Most touching for me was hearing from our newly certified fellows how much their experience in our formation program, particularly the quadrivium studies, had changed them. They were excited by the mathematical presentations during our Learnables conference. They were able to encounter Plato’s Republic, the text for our Fellows symposium, with a depth of insight they had not known before, and enter fully into conversation with our senior and associate fellows. We enjoyed rich conversations inside and outside of our discussion room, often late into the evening. 

I’m looking forward to more joy as a new semester begins. I’ll be teaching a history of education course for the Augustine Institute, leading a new cohort of junior fellows into the Trivium and Quadrivium as roads to wisdom, and continuing conversations with our certified fellows in Don Quixote and Aristotle’s teaching On the Soul. So much to learn, so much to teach, so much to love!

Bulletin

The Confessions (An Excerpt)

As the day now approached on which she was to depart this life (which day Thou knew, we did not), it fell out — Thou, as I believe, by Your secret ways arranging it — that she and I stood alone, leaning in a certain window, from which the garden of the house we occupied at Ostia could be seen; at which place, removed from the crowd, we were resting ourselves for the voyage, after the fatigues of a long journey. We then were conversing alone very pleasantly; and, forgetting those things which are behind, and reaching forth unto those things which are before, (Philippians 3:13) we were seeking between ourselves in the presence of the Truth, which You are, of what nature the eternal life of the saints would be, which eye has not seen, nor ear heard, neither has entered into the heart of man. But yet we opened wide the mouth of our heart, after those supernal streams of Your fountain, the fountain of life, which is with You; that being sprinkled with it according to our capacity, we might in some measure weigh so high a mystery.

And when our conversation had arrived at that point, that the very highest pleasure of the carnal senses, and that in the very brightest material light, seemed by reason of the sweetness of that life not only not worthy of comparison, but not even of mention, we, lifting ourselves with a more ardent affection towards the Selfsame, did gradually pass through all corporeal things, and even the heaven itself, whence sun, and moon, and stars shine upon the earth; yea, we soared higher yet by inward musing, and discoursing, and admiring Your works; and we came to our own minds, and went beyond them, that we might advance as high as that region of unfailing plenty, where You feed Israel for ever with the food of truth, and where life is that Wisdom by whom all these things are made, both which have been, and which are to come; and she is not made, but is as she has been, and so shall ever be; yea, rather, to have been, and to be hereafter, are not in her, but only to be, seeing she is eternal, for to have been and to be hereafter are not eternal. And while we were thus speaking, and straining after her, we slightly touched her with the whole effort of our heart; and we sighed, and there left bound the first-fruits of the Spirit; (Romans 8:23) and returned to the noise of our own mouth, where the word uttered has both beginning and end. And what is like Your Word, our Lord, who remains in Himself without becoming old, and makes all things new? (Wisdom 7:27)

We were saying, then, If to any man the tumult of the flesh were silenced — silenced the phantasies of earth, waters, and air — silenced, too, the poles; yea, the very soul be silenced to herself, and go beyond herself by not thinking of herself — silenced fancies and imaginary revelations, every tongue, and every sign, and whatsoever exists by passing away, since, if any could hearken, all these say, We created not ourselves, but were created by Him who abides for ever: If, having uttered this, they now should be silenced, having only quickened our ears to Him who created them, and He alone speak not by them, but by Himself, that we may hear His word, not by fleshly tongue, nor angelic voice, nor sound of thunder, nor the obscurity of a similitude, but might hear Him — Him whom in these we love— without these, like as we two now strained ourselves, and with rapid thought touched on that Eternal Wisdom which remains over all. If this could be sustained, and other visions of a far different kind be withdrawn, and this one ravish, and absorb, and envelope its beholder amid these inward joys, so that his life might be eternally like that one moment of knowledge which we now sighed after, were not this Enter into the joy of Your Lord? (Matthew 25:21) And when shall that be? When we shall all rise again; but all shall not be changed.

Such things was I saying; and if not after this manner, and in these words, yet, Lord, You know, that in that day when we were talking thus, this world with all its delights grew contemptible to us, even while we spoke. Then said my mother, Son, for myself, I have no longer any pleasure in anything in this life. What I want here further, and why I am here, I know not, now that my hopes in this world are satisfied. There was indeed one thing for which I wished to tarry a little in this life, and that was that I might see you a Catholic Christian before I died. My God has exceeded this abundantly, so that I see you despising all earthly felicity, made His servant — what do I here?”

    Bulletin

    The History of Astronomy: A Project Update

    For the past two years, I’ve been quietly working on a project aimed at making science—particularly astronomy—more accessible and teachable, especially for educators who may not have a formal background in the subject. As someone who doesn’t have a science background, I’ve served as the project’s first test subject: Could I learn the material well enough to teach it? Could I find a way to convey these ideas in a way that felt intuitive, compelling, and grounded in evidence?

    The answer turned out to be yes—but only after a great deal of trial and error. And now, I’m excited to share the next stage of the project: a simple, organized set of resources aimed at helping teachers learn the material themselves and feel ready to pass it on to their students. The goal is to make the story of the solar system approachable and teachable, even for those without a science background.

    From Learner to Teacher

    I came into this project without a science background. My goal was simple: could I learn the material and then teach it to someone else? That process helped me relate to what a new teacher might also struggle with. What was challenging to explain? What kind of details did I need in order to be able to pass the knowledge on to the students I was working with?

    One thing became clear early on: having good demonstration tools would be essential. At first, I cobbled together resources I found online, but I eventually realized I would need to build them myself. There were a lot of tools that just didn’t exist, and I didn’t want teachers to have to scavenge for these resources on their own. I wanted everything they needed all in one place.

    The Tools

    The tools I’ve built are meant to make it easy for a teacher to demonstrate key concepts clearly. For example, take retrograde motion. I’ve integrated an open-source tool called Stellarium into the platform. This tool shows the night sky and comes preloaded with a date and time that demonstrates retrograde motion in action. I’ve added the ability to draw directly on the sky, so teachers and students can track the movement of celestial bodies over time. In the image below, one object is marked in red and another in yellow. Students can see how the yellow body moves with consistent spacing between markers, while the red body’s spacing varies over time—a visual cue that helps highlight the difference between stars and planets and their motion over time.

    This is an interactive tool teachers can use with their students to guide them through the kinds of observations they need to make in order to identify two key ideas: first, that planets move differently than stars; and second, how to recognize the phenomenon we now call retrograde motion.

    There are many more tools like this, each tailored to support specific learning objectives. They’re designed to help students move from direct observation to theoretical models by making abstract ideas visible and interactive.

    The result is a platform that brings together these tools along with lesson plans—everything a teacher would need to teach the material, even without prior experience in astronomy.

    What’s Next: Equipping the Teacher

    The next stage of the project is perhaps the most important: equipping teachers themselves with the knowledge and confidence to teach the material well. 

    Starting next month, I’ll be releasing a series called Tracing the Sky. These short, 10–15 minute videos will come out multiple times per week and guide learners through the course content. They’re designed for anyone curious about how we came to understand the solar system—whether you’re learning for your own interest or because you want to teach it.

    I see these videos as the prerequisite step for teachers. Once they’ve followed along and built a basic understanding, they can move on to using the platform and lesson plans I’ve developed to teach the material to their students. These videos aren’t meant to be played in the classroom—instead, they’re designed for the teacher. My hope is that any educator, regardless of their prior familiarity with astronomy, could watch these videos and gain a solid understanding of both the historical development of celestial models and the practical tools available to teach them.

    Each video is designed to be the teacher’s starting point—a way to learn the material themselves, step by step. Once they feel comfortable, they can take the next step and explore the platform and lesson plans I’ve created. These plans tie directly into the demonstrations and provide guidance on how each lesson fits into the broader narrative, what questions to ask, and which observations to emphasize.

    Closing Thoughts

    I’ve come to believe that science education doesn’t need to be intimidating or overly technical. At its best, it should be a process of wonder, reasoning, and discovery—and that’s exactly what the history of astronomy offers. It’s a story of humans looking up at the sky, puzzling over what they saw, and gradually learning to make sense of it all.

    If I, as a non-scientist, can learn to tell that story, then I’m confident that others can too. And with the right tools and support, I believe we can help students not only understand the solar system—but appreciate the incredible intellectual journey it took to discover it.

    If you’d like to keep up to date with the project and follow along with the videos as I release them, you can sign up for my newsletter here: https://mailchi.mp/7be3350a4b33/tracing-the-sky-newsletter

    Bulletin

    Fruits of the Fellows Formation Program

    When I applied to the Boethius Institute Fellows Formation program, I knew it would be a challenging but fruitful opportunity. I had been involved in the classical education movement for over twenty years but was aware of many gaps and deficiencies in my own education, so I was excited to deepen my understanding of not only what is meant by “a liberal arts education” in its historic sense, but also to become better educated in the very arts themselves. There seem to be many opportunities for educators to sharpen their teaching skills, but I wanted to work toward acquiring for myself that education that the classical education movement seeks to impart to students. The Boethius Institute program was offering the opportunity to move toward becoming that which I want my students to become.
    As part of the Fellows Formation program, we each proposed a project to work on throughout the year, and I rapidly realized what mine should be. As a former homeschooling mom, I have long wanted to see a classical learning center in my community – a place where the liberal arts are explored, imaginations are cultivated, and the greatest books, thoughts, and ideas of the past are preserved and passed on to the next generation. My city has a thriving homeschool community, with over two thousand families and several once-a-week classical communities, and I wanted to found a center where those families can turn for guidance, resources, and classes. I also wanted to begin working with several new classical schools in my area, offering workshops and support as they grow. I had already been teaching a series of classes, but I hoped to expand the classes I offer and begin to seek out like-minded teachers whose greatest love is learning to join in my endeavor. And so Legenda Classical Resources began to take shape.


    As I have worked this year to learn about and understand more deeply the liberal arts of grammar, logic, rhetoric, and geometry, I have been changed as a teacher while my vision for a classical education center has become more focused. Delving into Latin and Greek grammar, as well as exploring and contemplating what language is and how it communicates thought, worked with our study of the rhetoric of famous speeches to give me a deeper love and reverence for the beauty of language itself. Logic and geometry, on the other hand, have challenged me with their precision, and this spills over into my teaching; I find my thought becoming more orderly, my questions to students more precise. It has also inspired me to eventually work with students through some of the speeches we discussed. This year I developed my curriculum for a Rise of the Modern World course, and I have incorporated several of Lincoln’s speeches which we analyzed together this year.


    As part of the Fellows program, we met periodically in small groups with one of the Senior Fellows to discuss and receive feedback on our projects. A large part of my work this year consisted in building a website as a springboard to present my learning center to the community, and my small group, under the leadership of Dr. Matthew Walz, was helpful in guiding my choices as it was under construction. Dr. Walz encouraged me to craft a concise mission statement that would convey exactly what I hope to accomplish in my community: “Legenda Classical Resources supports and promotes classical education by nurturing and instructing students, parents, and teachers, encouraging them to pursue the good, true, and beautiful, and joining them in a lifetime love of learning.” I shared with him that there seem to be many in my community who are either unsure or mistaken in their perceptions of what is meant by classical education, and he gave me an idea to address this: to replace my description of classical education with an FAQ page, addressing many misconceptions and concerns sometimes associated with liberal arts education. My answers to several of these were guided and changed as I worked through this year’s fellows training, particularly the question “What are the liberal arts? Why should we study them?” Our working through the first four liberal arts enabled me to hone my idea of what they entail and the effect of studying them on the mind.

    After publishing my website (www.legendaclassical.com) and an accompanying social media page (as this is the main means of communication among local homeschool families), I began developing new classes whose content has been shaped by my year with the Boethius Institute. As we have been reading and discussing Aristotle and Plato, receiving an overview of Latin and Greek, and following Euclid through his methodical propositions, I gained a greater respect for those whose minds, for generations, were formed by the same books and discussions. Three outstanding examples of this education are C.S. Lewis, J.R.R. Tolkien, and G.K. Chesterton (who, although he had less formal education than the other two, still was formed by the greatest literature). I started development, therefore, of a new set of classes, a “Great Authors Series”, in which students at my learning center can spend a year with each of these great minds. Since my desire, as I stated above, is to have a place where imaginations are cultivated, I could think of no better authors to fire the minds and imaginations of students.


    In addition to working with homeschooled students, Legenda Classical Resources also seeks to reach out to and work with classical schools, providing support, guidance, and materials to small, newly-formed schools. This year I laid the groundwork for this by working with one new school and another in the planning stages. For the former, I began coming into the school and offering classes once a week in history, literature, and writing, as well as providing some curricular guidance and materials. I was invited to attend a board meeting of the latter, set to open next year, and I developed a talk discussing “What is classical education?” which I will be giving at their initial meeting introducing the new school to the community. The Fellows program, along with my small group, were also helpful in outlining and developing my talk.


    Additionally, Legenda Classical Resources also seeks to guide and nurture parents as they work to educate their children. To that end, I have begun offering workshops which not only teach students but guide parents as well. As a lifelong lover of Shakespeare, I read with delight Dr. Seeley’s account of a Shakespeare workshop he developed for younger children, and I am currently adapting his ideas and developing a similar workshop for my community this summer in which students and their parents can together discover the joys of his plays. I am currently teaching the second of two spring IEW writing workshops; I have been asked this summer to present an introduction to that writing program for parents.


    As my vision for Legenda Classical Resources includes bringing multiple teachers together so that students can attend classes all meeting in the same location, I also began sharing my vision with like-minded homeschoolers who have a background in and knowledge of classical education. I discussed my plans with parents currently teaching in classical communities meeting once a week, and several expressed an interest in joining me and continuing to teach when their children have graduated. I also plan to launch a classical education reading group for parents and teachers this summer, where we can read, discuss, and grow together as educators and provide support and encouragement.


    And this desire for a community of learning in my city has also been shaped and refined by my year in the Fellows Formation program. As we read through portions of Book I of Aristotle’s Topics, one particular comment he made struck me with special force. Discussing how to delimit the use of a word in any particular case by examining its contrary, Aristotle says that, while pleasure has pain as its contrary when discussing bodily pleasures, “there is no contrary and so no name to the pleasure of beholding that the diagonal of a square is incommensurable with the side.” The pleasure of knowing, of seeing that something is true, of having one’s mind opened – this is the distinctive pleasure we have enjoyed this year. We fellows-in-training have together worked through difficult material; we have learned from our Senior Fellows and from each other; we have grown as learners and teachers. And it is this idea of fellowship that I hope to develop locally – a place where people, children and adults, but all truly students, can come together and celebrate the joy found in exploring the wealth of the past. The groundwork has been laid, and I hope to continue to build Legenda Classical Resources into a home for the pleasure found in learning.

    Bulletin

    From the President (June 2025)

    Last month, our first cohort of Junior Fellows completed the two year sequence of informal courses in the traditional liberal arts that constituted. This issue of our bulletin features the work of those who participated in the fellowship. Michelle Ferguson describes her development of Legenda Classical Resources to foster education renewal in the Quad Cities of eastern Iowa and western Illinois. Lucas Dos Santos writes of his work in Brazil. Joseph Tabenkin announces an exciting series of videos he has developed to help teachers learn and teach the history of astronomy. The conversation between Augustine and his mother, Monica, gives a taste of the conversations we began to have as the fellowship progressed.

    When Jeff and I first conceived the program, I was edified by the response. “We are going to introduce fellows to the seven traditional liberal arts as they are ordered to the life of wisdom.  No degrees, no grades, but we will give you a shiny certificate at the end.”  “Please can I be a part of that?” many responded. I have continued to be edified throughout my work with them. Katie Gillett told me that it has transformed her intellectual life. “I had never studied anything that wasn’t connected to a grade. So liberating!”

    I was especially edified by our final conversation, which focused on the question of whether a life of wisdom was achievable for them or would it always remain an unachievable dream. Michelle raised this question, which was close to her heart; her thirst for learning, for wisdom, has been a source of both great delight and great pain. 

    I would have liked to have offered a comforting white lie. I wish I could have said that wisdom is equally available to all, but I couldn’t. However, I encouraged them by saying that the grounding they had received in the liberal arts had put them in a position to participate more fully in wisdom by imbibing more deeply from what they read and heard. They have also learned how to converse about the best works in an ordered, serious, and fruitful way. They took some encouragement from this, but also expressed how impossible it seemed without continuing the fellowship they had built up over two years of challenging and illuminating conversations. So we began brainstorming about the best ways to keep moving forward. 

    Michelle sent me a lovely email this week, detailing ways in which she benefited from the program. “Finally, and most importantly, I feel that I have been set on a path towards that lofty goal of wisdom – a long-held desire of my heart, but elusive without tools and guidance. I so appreciated your advice at our last meeting to be willing to live with questions and struggles. I think it was Katie who asked me in our final session if I felt ‘wise’ now, and I laughingly told her no – but I DO feel that I am on the path that leads there. That path will, I assume, last for eternity – I will always be learning more about the unity of all things. I have spent much of my life a victim of standard schooling, in which discrete facts and disconnected ideas need to be retained for the test. While my adult years of study and teaching my own children certainly moved away from that model, I have struggled to step back from the details to see the bigger picture – my mental camera is always in close-up and infrequently pulls back for the wider shot. After going through the Fellows Formation program, I am more consistently pulling back to orient all the knowledge we sampled in a wider framework. I look forward to continuing that journey in connection with the other fellows!”

    If you know someone devoted to the classical liberal arts revival who longs to have what they themselves missed, encourage them to consider applying for our next cohort, which begins this August with a symposium on Plato’s Republic at the Augustine Institute in St Louis.

    Bulletin

    A Fellow’s Work in Brazil

    I was gratified and encouraged to receive the email below from Fellow Lucas Fonseca Dos Santos, a resident of Sao Paolo in Brazil. Lucas is one of many Brazilians who are hungry to learn everything they can about liberal education, so that they can revive education in their own country. Lucas’s zeal led him to learn spoken Latin through the prestigious Vivarium program in Rome, to study for the Master’s in Classical Education under Senior Fellow Erik Ellis at the University of Dallas, and to study the Quadrivium with me as part of our Fellows in Formation program. 

    Dear Dr. Seeley,

    I wanted to share with you about a course I taught here in Brazil last week. Since last Thursday, I have been in a city in the interior of São Paulo (a 6-hour drive from the capital), teaching intensive teacher training to a group of teachers I have been working with for a long time.

    I am sending this information, as well as a link to some photos and videos of the course, as a way of thanking you for everything you have done for me and for your generosity in allowing me to participate in the Institute. A large part of the curriculum studied was born from the Fellowship studies. They are enthusiastic about the project. They are also grateful to the Institute, because I try to transmit, even with my limitations, what I learn from you. I want to ensure that the fruits of these studies can spread throughout Brazil.

    In fact, here in Brazil, there is a revival of interest in the liberal arts and classical education, but we have many things related to the Trivium, and almost nothing related to the Quadrivium.

    I have tried, in my training for teachers, to bring the importance of the Quadrivium. Only very advanced teachers participated in this course – some pioneers in Classical Education and Homeschooling in Brazil – and with influential work in the dissemination of these studies.

    I have set the goal that all teachers in this class should master Latin as well as possible, so that in the near future, classes can be taught in Latin – a good portion of the class already has a reasonable command.

    The following were present:

    – Gessica Hellman and Alexei Hellman, together with their 3 children.  They are pioneers in Homeschooling in Brazil, and among the pioneers are those who dedicated themselves to the production of Grammar, Logic and Rhetoric materials.  They are the creators of the website Vias Clássicas and the publishing house Vias Clássicas. In addition to their participation, their son Michael, 14 years old, is a regular student in the adult class, and I follow his studies and writings. He has a personal goal of gradually following the Thomas Aquinas College curriculum.

    – Kemily Rodrigues, a very experienced teacher, who works with Children’s Literature and the importance of reading aloud to children. At the moment, she is delving deeper into John Senior and his vision of the Great Books and Children’s Literature as preparation for the Great Books. In addition, she is competing for a scholarship to take the Latin I course at Accademia Vivarium Novum (online). (https://www.instagram.com/profkemily/)

    Ian Pompeu: he is preparing for a doctorate in Philosophy of Law, and is very dedicated to Aristotle, and is also my Latin student. He has worked as a pedagogical director at a Catholic school in the North of Brazil.

    Felipe Rodrigues: a beginning but talented teacher who has dedicated himself to classical education in the city of Araçatuba, in the interior of São Paulo, and has planned events in this area. He found an investor, who gave us the place where the course was held free of charge.

    A simple place, as you can see from the photos, but it was a place of great hope for the future of liberal Catholic education in Brazil.

    In the curriculum, we had 3 subjects:

    1) Advanced Seminars in Classical Literature

    We had 3 large seminars, each lasting an average of 2h30min-3h, in which we discussed:

    a) Iliad XVIII and Aeneid VIII (study of ekphrasis)

    b) Odyssey XI and Aeneid VI (study of katabasis)

    c) Commented reading of The Lusiads, book I (comparing the similes with Ovid and Virgil).

    2) Introduction to the Quadrivium.

    a) Republic, book VII. We had 3 discussions of almost 2h each on this book, dealing with the subjects of the Quadrivium. Unfortunately, Dr. Lehman’s article was not yet available – it was still pending revision – and I was unable to use it with the students.

    b) Introduction to Arithmetic, book I, chaps. I-VI. We had an introductory discussion on this text, which I had learned about during our meetings, but which was completely unknown to the students – even the most advanced ones. We had a Socratic seminar of almost 3 hours.

    c) Ptolemy, Almagest (excerpts from book I).

    d) Euclid, Elements. I gave an introduction to the elements, and the students had to solve propositions I and II. We had two long classes on the subject.

    e) We would still read Boethius (excerpts from De Musica) and Copernicus, but we didn’t have time for this.

    3) Liberal Arts – Great Works.

    In this course, we dedicated ourselves to the study and meditation on the work of Saint Bonaventure, De Reductione Artium ad Theologiam. We had two Socratic discussions on the work, one at the beginning and one at the end of the course, and it was a memorable and inspiring closing.

    In Christ,

    Lucas

    Bulletin

    Leisure the Basis of Culture

    The concept of intellectual work has a number of historical antecedents, which can serve to clarify it.
    First, it is based on a certain interpretation of the human knowing process.

    What happens when our eye sees a rose? What do we do when that happens? Our mind does something, to be sure, in the mere fact of taking in the object, grasping its color, its shape, and so on. We have to be awake and active. But all the same, it is a ”relaxed” looking, so long as we are merely looking at it and not observing or studying it, counting or measuring its various features. Such observation would not be a ”relaxed” action: it would be what Ernst Jünger termed an ”act of aggression.”1 But simply looking at something, gazing at it, ”taking it in,” is merely to open our eyes to receive the things that present themselves to us, that come to us without any need for ”effort” on our part to ”possess” them.

    There would scarcely be any dispute about this, if we were speaking about an act of sense perception.
    But what about an act of knowing? When a human being considers something imperceptible to the senses, is there then such a thing as mere ”looking”? Or, to use the scholastic technical terminology, is there such a thing as ”intellectual vision”?

    The ancient and medieval philosophers answered, ”Yes.” Modern philosophers have tended to say, ”No.”
    To Kant, for instance, the human act of knowing is exclusively ”discursive,” which means not ”merely looking.” ”The understanding cannot look upon anything.” 2 This doctrine has been characterized, in brief, as ”one of the most momentous dogmatic assumptions of Kantian epistemology.”3 In Kant’s view, then, human knowing consists essentially in the act of investigating, articulating, joining, comparing, distinguishing, abstracting, deducing, proving – all of which are so many types and methods of active mental effort. According to Kant, knowing — (intellectual knowing, that is, by the human being) is activity, and nothing but activity.

    It is no wonder that, starting from this basis, Kant was able to conclude that all knowing, even philosophy itself (since philosophy is at the greatest remove from sense perception), should be understood as a form of work.

    And he said so expressly: in 1796, for example, in an article written to refute the Romantic ”vision” and ”intuitive” philosophy of Jacobi, Schlosser, and Stolberg.4 In philosophy, Kant objects, ”the law of reason is supreme, whereby property is possessed through labor.” And this Romantic philosophy cannot truly be a philosophy because it is not ”work.” This accusation he directs even against Plato, that ”Father of all raving enthusiasm in Philosophy,” while, Kant says with recognition and approval, ”Aristotle’s philosophy is truly work.” From such a perspective, originating from the exaltation of a ”philosophy of work,” the ”recently exalted, privileged tone of Philosophy” is branded as a false philosophy, in which one ”does not work but merely listens with delight to the oracle within oneself, in order to come into complete possession of the whole wisdom promised by philosophy.” And such a ”pseudo–philosophy” thinks itself superior to the strenuous labor of the true philosopher!

    Now, ancient and medieval philosophy had quite the opposite view, without, of course, justifying any charge that philosophy was something ”easy.” Not only the Greeks in general – Aristotle no less than Plato – but the great medieval thinkers as well, all held that there was an element of purely receptive ”looking,” not only in sense perception but also in intellectual knowing or, as Heraclitus said, ”Listening-in to the being of things.”5

    The medievals distinguished between the intellect as ratio and the intellect as intellectus. Ratio is the power of discursive thought, of searching and re-searching, abstracting, refining, and concluding [cf. Latin dis-currere, ”to run to and fro”], whereas intellectus refers to the ability of ”simply looking” (simplex intuitus), to which the truth presents itself as a landscape presents itself to the eye. The spiritual knowing power of the human mind, as the ancients understood it, is really two things in one: ratio and intellectus, all knowing involves both. The path of discursive reasoning is accompanied and penetrated by the intellectus’ untiring vision, which is not active but passive, or better, receptive – a receptively operating power of the intellect.

    And something else must be added: the ancients likewise considered the active efforts of the discursive ratio to be the essentially human element of human knowing; ratio as the decisively human activity was contrasted with the intellectus, which had to do with what surpasses human limits. Of course, this ”super-human” power nevertheless does belong to man, and what is ”essentially human” alone does not exhaust the knowing power of human nature; for it is essential to the human person to reach beyond the province of the human and into the order of angels, the truly intellectual beings.

    ”Although human knowing really takes place in the mode of ratio, nevertheless it is a kind of participation in that simple knowing which takes place in higher natures, and we can thus conclude that human beings possess a power of intellectual vision.” These are the words of Thomas Aquinas, Disputed Questions on Truth.6 This statement means that human knowing is a partaking in the nondiscursive power of vision enjoyed by the angels, to whom it has been granted to ”take in” the immaterial as easily as our eyes take in light or our ears sound. Human knowing has an element of the non-active, purely receptive seeing, which is not there in virtue of our humanity as such, but in virtue of a transcendence over what is human, but which is really the highest fulfillment of what it is to be human, and is thus ”truly human” after all (in the same way, again
    according to Thomas Aquinas, the vita contemplativa as the highest form of human living is not ”properly human, but superhuman”: non proprie humana, sed superhumana).7

    For the ancient and medieval philosophers the ”laboring” nature of the human ratio was likewise a mark of its humanness. The operation of the ratio, its discursive thinking process, really is work, and a difficult activity.
    But the simple act of the intellectus is not work. And whoever thinks, along with the ancients, that human knowing is a mutual interplay of ratio and intellectus; whoever can recognize an element of intellectual vision within discursive reasoning; whoever, finally, can retain in philosophy an element of contemplation of being as a whole such a person will have to grant that a characterization of knowing and philosophy as ”work” is not only not exhaustive, but does not even reach the core of the matter, and that something essential is in fact missing from such a definition. Certainly, knowing in general and philosophical knowing in particular cannot take place without the effort and activity of discursive reasoning, without the ”nuisance of labor” (labor improbus) involved in all ”intellectual work.” Even so, there is something else in it, and something essential to it, that is not work.

    Excerpt from Leisure
    The Basis of Culture
    Josef Pieper
    Introduction by Roger Scruton
    New translation by Gerald Malsbary
    St. Augustine’s Press
    South Bend, Indiana
    1998

    1. Blätter und Steine (Hamburg, 1934), p. 202. ↩︎
    2. I. Kant, Kritik der reinen Vernunft, ed. R. Schmidt (Leipzig, 1944) p. 91. ↩︎
    3. Bernhard Jansen, Die Geschichte der Erkenntislehre in der neueren Philosophie bis Kant (Paderborn,
      1940), p. 235. ↩︎
    4. ”Von einem neuerdings erhobenen vornehmen Ton in der Philosophie,” Akademie-Ausgabe 8, pp. 387-
      406. ↩︎
    5. Diels-Kranz, ed., Die Fragmente der Vorsokratiker, frag. 112. ↩︎
    6. Q.XV,1. ↩︎
    7. Quaestio disputata de virtutibus cardinalibus 1. ↩︎