Euclid's definitions of a point and a line

Euclid's definitions of a point and a line—“A point is that which cannot be divided. A line is a length without breadth.”—are the opening words of a book that Smilla reads aloud to Isaiah and, paradoxically, the means by which the two approach one another. The definitions’ language of indivisibility and boundary reflects the book’s austere authority, while the reading scene quietly dismantles the distance Smilla initially tries to impose. Euclid’s Elements becomes not a lesson Isaiah comments on, but a text they can return to together.

The Book as a Barrier

When Isaiah asks what book Smilla has, she answers that it is Euclid’s Elements and shuts the door. Later that evening, when he appears at her apartment, she takes the same book down from the shelf. She reads it as if to send him away, to make clear that she owns nothing a child would find interesting and that they cannot meet through a book—or in any other way. Her choice gives the definitions an immediate dramatic setting: they are offered not as an invitation, but as a boundary.

Yet the gesture cannot accomplish what Smilla intends. Isaiah enters the apartment and her life, and the text she raises between them becomes part of their shared routine. The tension between the book’s role as a barrier and its eventual role as common ground is central to its significance: the reading begins with resistance, but the boy’s presence makes the words into a form of companionship.

Reading Together

On that first August evening, Smilla reads Euclid aloud while she and Isaiah sit on the sofa. The text moves through definitions and propositions, including the line that describes a semicircle as a figure bounded by a diameter and the circumference intersected by it. Smilla’s response passes from irritation at the impractical situation to veneration for a work she regards as foundational, a boundary at the beginning of mathematical thought. As she reads, the sunset light from South Harbor fills the living room; eventually she is no longer attending to the page, and the two sit quietly together.

The scene makes the book’s abstract precision coexist with an intimacy that needs no explanation. At the end, Smilla stops reading. Neither fills the silence. Isaiah later leaves quietly, but the encounter has established a pattern more durable than the evening itself: he returns, sometimes daily, sometimes after a couple of weeks. The book is not presented as the cause of their closeness. Smilla recognizes that she could have read from a telephone book or from Lewis and Carrisa’s Detection and Classification of Ice; Isaiah would still have come to sit with her. Euclid matters as the recurring form their closeness takes, not as its explanation.

A Shared Text, Not a Lesson

The contrast with Smilla’s attempt to read him a children’s book clarifies what Euclid does—and does not—mean between them. When she borrows Rasmus Klump on the Ice Cap, Isaiah listens to the descriptions of its pictures and then asks what the toy-like bear tastes like. The question refuses the expected innocence of the illustrated story. Euclid, by contrast, is the book he never comments on, and the one the two continue to revisit.

Their arrangement is expressly mutual rather than instructional. Much later, Smilla says they can read about Greenland together in Euclid’s Elements. Isaiah calls it “we” reading aloud, understanding that his presence contributes as much to the shared activity as Smilla’s voice. The choice is striking precisely because the book offers no obvious account of Greenland: its role is not to supply local information but to provide a common ritual. Smilla cannot promise him anything, she says; reading together is what she can offer.

Meaning and Recurrence

The definitions’ spare terms—point, line, length, breadth—give the passage a language of limits, while the scene’s emotional movement crosses the limits Smilla tries to establish. The book’s formal exactness is set against the unspoken, unmeasured relation taking shape on the sofa. Its authority matters to Smilla as the foundation from which later mathematics can be traced backward; for Isaiah, the text’s importance lies less in its propositions than in the fact that Smilla reads it with him.

The recurring book therefore carries two meanings without collapsing into either one. It begins as a deliberate defense against connection and becomes the shared object through which connection is sustained. The passage also resists turning that bond into a simple story of teaching or rescue: Isaiah does not need to comment on Euclid or master its contents for the reading to matter. What the two share is the time and presence of reading aloud, and the book’s repeated return marks that companionship as something they make together.