The river of time

Detail from 'Studies of water passing obstacles and falling' (c. 1508-9), one of Leonardo's many studies of the dynamics of water.

Leonardo da Vinci loved water. Eddies, currents, and whorls—he studied them obsessively, sketched them in torrents. As Walter Isaacson discusses in his recent biography, the artist had planned a fifteen-chapter treatise devoted to the subject. Perhaps it is not so surprising, then, that in this aquatic fixation Leonardo found a font of analogies. He likened the branching of rivers to the branching of blood vessels and tree canopies; he intuited parallels between flows of water and flusters of wind; he connected curls cascading down a neck and water plunging down a cataract.

He also sensed an elemental analogy for time itself. “The water you touch in a river is the last of that which has passed, and the first of that which is coming,” he wrote. “Thus it is with time present.” As Leonardo imagines the scene, you are on the bank of the river, dipping a hand into the current. The water that has not passed yet—the water still to come—is upstream and the water that already gone by is downstream. Here, Leonardo alights on an old analogy—the river of time—and, more specifically, what we might call the “on the bank” version of it.

In another version, you are not on the bank but in the water, on a boat, floating into the future. Jose Luis Borges nods to this alternative—let's call it the “in a boat” variant—when he writes that “time is a river that sweeps me along.” And, indeed, countless others—upstream and downstream of Leonardo and Borges—have intuited this same basic analogy, in these same two versions. Marcus Aurelius, in his mediations, written between 161 and 180 AD, opts for the on-the-bank version: “Time is a river, a violent current of events, glimpsed once and already carried past us, and another follows and is gone.” A well-known hymn opts for the in-a-boat version, likening time to “an ever-rolling stream” that “bears all sons away.” Ursula Le Guin famously elaborated this watercraft version in her suggestion that “story is our only boat for sailing on the river of time.”

Such metaphors seem to tug firmly on our intuitions. In a 1949 essay, the philosopher J.C. Smart remarked: “They are not the result of some wild flight of poetic imagination, but are, in some way, natural to us; at first sight, at any rate, it seems difficult to see how we could avoid them.” Nabokov, in Ada, notes this inescapability. His narrator muses that humans are “so keen on liquefying every lap of life, that we end up by being unable to speak of Time without speaking of physical motion.”

But why would such metaphors be “natural to us”? Why the urge to “liquefy ever lap of life”? There are two questions here, one nested inside the other. The more general one, which Nabokov alludes to, is the question of why we like to think of temporal change as motion. The more specific question is why rivers in particular are one of of our preferred images for visualizing time’s metaphorical motion.

Of the two, the more specific one is the more tractable. Part of the answer has to do with the nature of time’s seeming "flow" and the nature of river flow. Francesco Petrarch, literary luminary of the 1300’s, offered a first hint. He observed that: “[O]ne unalterable speed is the course of life. There is no going back or taking pause. We move forward through all tempest and whatever wind. Whether the course be difficult, short or long, through all there is one constant velocity.” Key here are two of Petrarch's phrasings: “move forward” and “constant velocity.” Rivers are not just a general image of motion, they are an image of motion (a) in one direction and (b) at a roughly constant rate.

This is a highly schematized notion of rivers, of course. In reality, they kink, fork, sometimes even reverse course; in some patches they flow briskly, in others they laze imperceptibly. But from any particular view out onto a river—whether from a rocky bank or the prow of a boat—the water seems to move with remarkably constancy, whatever its local pace may be. Other candidate images of motion from the natural world simply do not share these two key properties. A bird’s flight, a squirrel's scamper, a gust of wind, the crash of waves—all these candidates moves in fits and starts and exhibit unpredictable changes of direction.

These considerations help us explain why the river of time endures millennia after its first recorded uses. And these same considerations may explain why some have gotten a little carried away with it. Girolamo Andrea Martignoni, Italian scholar of the 18th century, well-meaningly incorporated river imagery into his monumental chart of the Roman empire—a visualization that was both a map of actual space and a metaphorical map of chronology. The result—according to Stephen Boyd Davis, who discusses the chart in some detail in a highly recommended 2017 article—was "fraught with difficulties."

 Detail from Girolamo Andrea Martignoni's 1718 chronographic map of the history of the Roman empire,  Imago Romani imperii .

Detail from Girolamo Andrea Martignoni's 1718 chronographic map of the history of the Roman empire, Imago Romani imperii.

And yet, rough patches aside, the metaphor rolls on. In some ways it may actually be enjoying new life. Upstream and downstream have recently become favored ways of talking about temporal and causal relationships in fields such as neuroscience and economics. Geneticists talk of cascades of genes and the petroleum industry distinguishes upstream, midstream, and downstream sectors. In all these cases, other drier, options for making such distinctions are readily available. But we seem drawn to water-based terminology. To fully understand why, it may help to go back to Leonardo and to the sources of his own fascination with water—its liveliness and elementality. We don't all express such a fascination as as torrentially as Leonardo did—but, in many of us, at least—it's still there.



1. Isaacson notes that Leonardo, in one of his characteristic manic bursts, made a list of 67 words used to describe water’s flow (see Isaacson, 2017, p. 429). Unfortunately, I have not been able to track down the full list.

2. The on-the-bank and in-a-boat variants of this metaphor correspond to two general models of time that Rafael Núñez and I referred to in our review of spatial metaphors for time as an “external” and “internal” view, respectively. 

3. In addition to Smart's 1949 essay, a good summary discussion of the 'time as a river' metaphor can be found in James Gleick's Time Travel (pp. 125-8).