Euclid and the Art of Loving: Reflections on Euclid’s 8th

I was listening to a lecture the other day when I learned a great truth about love. This talk had little to do with ethics, and even less to do with philosophy. It was neither sermonic nor “theological” (in the way that we use the term). Indeed, the word love never made an appearance. But it emerged, albeit veiled; and it did so in a lecture on math.

My professor was explaining Euclid’s definitions when we arrived at the 8th:

“A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.”

In hearing this, it occurred to me that what Euclid observed, and perhaps all mathematicians before and after him, is the pattern of love in nature. I will explain in a moment just how he did so. But it must first be stated how this sort of inquiry, at the intersection of theology and math, is possible (for the premise is strikingly Christian, but dissonant with our mechanistic view of things): that the divine quality of love has a foothold in all the world, and it can be seen in all things that move toward union, including math.

I admit—it seems odd to claim that a geometric shape could partake in the operation of love. But as Christians it is our right, and perhaps our duty, to see the world as such. For if the creator himself is love, shouldn’t we expect to find His hallowed energy—love—blazing throughout the whole of creation? In my own reading of things, a planet, drawn to the sun by gravity, is as plausible a candidate for “being in love” as the young man possessed by a flash of emotions for a young woman. Indeed, if astronomers took the movement of the spheres seriously, I suspect they would find the same desire motivating both the planet and the boy: a desire for union—that curious thing we call love.

The operative action in love, or at least the one I am here describing, is a movement towards union. The great minds of Christendom have always described love in the same or similar terms. When we see two things moving towards greater union with one another, shunning the distance between them, what we are observing is an act of love. This operation of love lies at the heart of being: in the Godhead we have distinct persons, who are drawn together in what St. Gregory of Nazansius called a harmonious “circle of glory.” And this very dynamic trickles down from the Trinity and exists in the whole of creation: it can be found—in far less obvious forms, but found nonetheless—in humans, animals, and even mathematical entities. If we took this metaphysic seriously, I suspect the world would unfurl itself before us. But before we broach the world, let us begin with Euclid’s 8th.

“A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.”

The language may seem verbose but the point is sound. What Euclid is defining is a plane angle, which is formed by two lines, coming from any direction, and meeting at a fixed point. And how does Euclid describe this movement of one line to another? As an inclination, which is the movement of one thing towards a specific end. To incline toward something is to set our eyes upon a thing and then move towards it. Therefore—and this is the essential point—any inclination is an act of love, for it is itself an act of union, and this applies whether we are speaking of the inclination of two lines or of two hearts.

Anyone who suspects this to be mere wordplay should consider where else this word appears in our usage. “I have inclined my heart to perform thy statutes always, even unto the end,” (Ps 119:12). The arena is different but the move is the same: as we incline our hearts toward God, so too does one line incline towards another.

The Prayer Book brings us to this same posture of “inclination.” In the Communion Liturgy we go through the Ten Commandments, saying with one accord, “Lord, have mercy upon us, and incline our hearts to keep this law.” And we say this not once or twice. We repeat it ten times over, asking—begging by the end of it—that God would draw our hearts in obedience in Him.

“A plane angle is the inclination to one another of two lines.”

What I am saying is nothing short of the claim that we can see love in math: that two lines, inclining towards each other, participate in a lesser version of the same rhythm that all lovers know. Immediately it will be objected that lines cannot love. But much of the Christian tradition has ascribed loves and appetites to animate and inanimate things alike. Thus, the Christian is not only the lover, ever inclining his heart towards God, but he is also the one who sees love in all the world. As we are taught to pray in the Book of Common Prayer, “Open, we beseech thee, our eyes to behold thy gracious hand in all thy works; that rejoicing in thy whole creation, we may learn to serve thee with gladness.”

Up to this point, all of my language has suggested that in Euclid’s definition (and in the economy of love) we have one line actively moving and the other passively receiving. But neither Euclidean geometry nor worthwhile relational advice allows such negligence: one party cannot remain stagnant while the other moves with joy. (“A plane angle is the inclination…of two lines.”)

Love is not static; it moves with energy and eros towards its desired end. And this lack of laxity, the seeming inability to pause, can be perceived in all of creation. For it is love which moves all things—including planets, plants, and geometric lines. As St. Augustine said, “My weight is my love, by it I am borne wherever I am borne.” The Bishop of Hippo saw his every decision as an act of love, and we should see creation in the same light: for wherever we have seen movement, we have seen love. That is why hell for Dante is icy and frozen below the flames—still with an eerie pause.

Love works quite the opposite effect; it moves like a dance. (And it is no wonder that dancing has always carried a romantic residue.) In a dance, both parties move toward one another, and both are involved at every step. Sure, one must lead, but whoever suspects that leading asks the other to slavishly follow along has never danced before. A good dance is both formal and free-flowing; it is an ordered movement of multiple parties. Even the Church Fathers, explaining the Triune mystery, could not resist the analogy of a dance. Pericherosis they called it—the image of three persons moving and inclining towards each other, all without obliterating the other.

We have established that love is a dynamic movement towards union. We can see it in God and planets, in dancers and math. Let us continue with the last of these—math, the subject we began with. The study of mathematics began in a very practical arena: architecture. Math was used to build things, but the fact that it involved “math” was accidental. It was instrumental but nevertheless essential: sound math made sound buildings.

Let us consider the inverse. If two lines inclining is an act of love, what of its opposite?— two lines refusing eachother. This may lie at the foundation of hell. I mentioned Dante’s description of hell as frigid atomization. But I wonder if the very architecture of hell is atomized as well—warped and deformed, not merely from Christ’s disruptive entry, but from the total lack of love. I wonder if the tiles refuse to meet; the walls each stand opposed; and the floors reject their ceiling. All refuse to incline as they know nothing of love. I do not know, but perhaps the buildings themselves are in derision.

And while being a hypothetical, this is not a random thought experiment. St. Augustine had a term for this derision in human relations: incurvatus in se—a soul turned inward. (And I do intend for us to see the geometric language in the word incurvatus.) A soul, warped by selfishness, is stuck within himself. He has no no windows to see the world; only mirrors to fancy himself. He does not care to meet another, let alone meet their needs. His vision curves inward. He inclines upon himself. But as Euclid has shown us, to love is to know,

“the inclination…of two lines”

We cited the Psalms and Prayer Book to show our own hearts inclining towards God. But Christianity tells of something far more profound. In the Incarnation, we have the ultimate act of “inclination”: divine condescension. When we could not move He moved for us. He descended when we could not ascend. Our God joined us so that we may join Him. Beneath all this talk of love, Christ is the place—nay, the person—where two inclined lines meet. He is “truly God and truly man,” which is why most icons of Christ depict him in the vibrant garments of red and blue: an undivided dance between the human and divine. It is the greatest act of love, and it is mirrored in a dim, distant, and far less dramatic manner in the inclination of two lines towards each other. By comparing the operations of geometry to the incarnation we do not devalue the latter; rather, we give shape to the former. Truly, our God inclined, and He did so with a gentle ferocity far greater than any shape.

“which meet one another”

Euclid’s mention of this “meeting” is so subtle as to be missed. But the whole definition turns on the meeting actually occurring. Now the lines may meet in any number of ways. They may approach one another in a neat orderly fashion, or they may bend and twist their way forward. One line may be short, the other long. One may be curved, the other straight. The combinations are infinite, but when they arrive, it will always be the same: they will embrace one another at a specific point.

So it is with our conversion—the meeting of man with the “lover of Mankind.” The diversity of testimonies shows the plurality of ways that this can happen, but all testimonies are made the same in the waters of baptism. We may come to the garden like St. Augustine, or the desert like Moses. We may find a church in a crowded city or a cross in an empty town. We may enter the waters as a reluctant convert or as a miserable wretch. But it will always be the same: we come to meet God and He us. And it is a meeting made possible because God has first moved and inclined toward us. As the Prayer book has us repeat, “Turn us, O Lord, to thee, and we shall be turned.”

“which meet one another and do not lie in a straight line

I shall end my speaking of geometric lines with this most striking of lines: that the meeting of two lines does not obliterate either; there is a unity, but it does not destroy personality.

He who looks at the angle that Euclid is describing in this definition would see a union of two lines, but also the shape of each. They are unified but not uniform, similar but not the same. Love always acts in this way. When two lovers marry, a unity is formed—a composite unity—but we can still speak of husband and wife.

Love makes the language of 1st person plural a possibility: I am referring to the bold assertion of “we.” For a great mystery is veiled in that simple word—“we.” How can many “I’s” be spoken of as one “we”? And how can they become one yet remain many? I do not know, but I know it has something to do with love: the force charged with making things whole without killing the parts, bringing persons together without destroying their personhood. God demonstrates this very love within himself. He is Triune; and that remains a mystery—a mystery wrought by His love. This is why the distinct person of Jesus Christ calls himself one with the Father, why a husband is both himself and one flesh with his wife. To be in love is to be both singular and plural: you become one with the object of your love but you are not the object of your love. Like two lines converging, we incline and form a unity far greater than our individual self could ever be.

None of this is meant to imply that love operates according to geometric properties. What I am suggesting is quite the opposite: that geometric properties operate according to love. And we can go yet farther. All of the cosmos operate according to love. All things are moving towards God in accordance with the direction He has given them; it is sin which disrupts the process. We, as mathematicians, theologians, and scientists—indeed, as humans—have the blessing to watch and observe these patterns, scattered throughout nature. If we have eyes to see, as the Prayer book instructs us, we may observe love in things as mundane as the order of math and elusive as the wind. Indeed, if we take Dante seriously, we may say that the love which turns the moon and the stars is the same love which inclines our heart to God; and that the love which moves our hearts, moves one line to meet another.


Caleb Knox

Caleb Knox is studying Political Theory at Patrick Henry College in Purcellville, VA. He attends a Reformed Episcopal Church, the Church of our Savior Oatlands, with his friends and professors. At school, Caleb is captain of the college's Mock Trial team. After graduation he hopes to pass along the Classical Christian inheritance at a secondary school and then pursue studies in Theology and Political Thought.

'Euclid and the Art of Loving: Reflections on Euclid’s 8th' have 3 comments

  1. October 4, 2023 @ 10:02 am Paul Erlandson

    As an ex-Geometry Teacher, this appeals to me very much!

    And here is my favorite quote about Euclid. It is the title of a Petrarchan Sonnet by Edna St. Vincent Millay:

    “Euclid alone has looked on Beauty bare.”


  2. October 4, 2023 @ 11:42 am Philip Enarson

    Insighful, wonderful and powerful. God’s love can be seen in everything, wind, water, sky and even in math.


  3. October 8, 2023 @ 9:51 pm David Danielson II

    Thank you for the thoughtful, well-written essay. I have always loved math, and music and literature too, so it is refreshing to reflect on the presence of beauty and relations in math. (Small editorial note being the spelling of perichoresis.)


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