In the Preface to Being and Time, Martin Heidegger summarizes the aim of the investigation upon which he and the reader are about to embark – an investigation Heidegger himself never completed – as “the Interpretation of time as the possible horizon for any understanding whatsoever of Being”.[2] His first English translators, John Macquarrie and Edward Robinson, immediately warn the English-speaking reader in a note that for Heidegger the word “horizon” has meanings they will find unfamiliar. “We tend to think”, they write, “of a horizon as something which we may widen or extend or go beyond; Heidegger, however, seems to think of it as something which we can neither widen nor go beyond, but which provides the limits for certain intellectual activities performed ‘within’ it”.[3] Time, we might rephrase Heidegger to say, is the one condition – the horizon – beyond which we cannot go, under which the question of Being must inevitably and always be asked.

Phenomenology has been out of style for some time, but the phenomenological metaphor of the “horizon” has persisted, perhaps entering present critical and hermeneutic discussion most directly through the writing of Heidegger’s student, Hans-Georg Gadamer, who in his Truth and Method connects this metaphor not so much with explicit ontology as with the temporality, and more specifically, with the cultural historicity, of the individual subject. Following Gadamer, it became common to speak of activities of interpretation, from encountering people unlike ourselves, or other people taken altogether, to the reading of literature or the viewing of art, as a “fusion of horizons”.[4] That is, to return to the explanation of Macquarrie and Robinson, the limits of the intellectual activities, the “prejudices” in the neutral sense in which Gadamer meant us to use that otherwise charged term, must reach an accommodation, or fusion, with other such limits.

Whether we think of horizons as limits of thought, as culturally specific assumptions or “prejudices” about the “world” in which we happen to live – and of course “world” in this metaphorical sense is closely related to “horizon” – or as limits of experience in some way or another to be extended or exceeded, the horizon metaphor entails limits and boundaries. What are these limits and boundaries? Probably the most common meaning of “horizon” is the apparent meeting of earth and sky, but when we speak, for example, of “opening up new horizons” we mean more than that. We mean entering a new sphere of experience, and even in this case the horizon so to speak exists around us, or between us and the juncture of earth and sky. It is also important that; although there may be “clouds on the horizon”, the “horizon” metaphor, either as the limit of the most authentic thinking, or as a limit constantly to the extended, is positive, and the limit it implies is a definition relative to which these fundamental human activities may be carried out.

However we might spin out these examples, “horizon” remains close to its etymological roots.  The word descends more or less directly from the Greek. The verb horizein  means to mark out by boundaries, lay down, limit or define. It may also mean to mark out for oneself, to set up or dedicate, to determine, appoint of settle. The noun is horos, which may refer to mountains or hills, but also to boundaries, limits, frontiers and borders, especially those defined by a landmark. Horoi were also inscribed stones set up to mark mortgaged property. More metaphorically, horoi might be any rule, standard, limit or measure, or finally, the definition of a word.

“Horizon” is thus an excellent example of what I call a “real spatial metaphor”[5], a metaphor that takes its value from our embodied experience of the world in which we find ourselves, and at its etymological base it belongs with such other metaphors as “precinct” (from praecingere, to belt or girdle), templum, temenos, and a great many words in a great many languages of which I am unaware that refer to limits and boundaries, words, that is, that refer to qualitative differences between inside and outside.

The limits and boundaries implied by horizon and related terms are essentially arbitrary, and so in need of some additional warrant or sanction. If horos may refer to a hill or mountain, that hill or mountain has entirely different meanings if it marks the boundary of a sacred precinct, the border of a nation, or the extent of property. To put this in other terms, the specific forms in which limits and boundaries are encountered, respected, and negotiated always represent cultural choices. It is important to distinguish limits and boundaries as conditions of human social life from those results of cultural choices, the specific limits and boundaries, inclusions and exclusions, understood by members of specific human groups. But if the etymology of horizon leads us to Greek social spaces and their markers, how did “horizon” come to refer to what we call “horizons”? If a horizon is a limit, and might in principle be the walls of a city or the precinct of a temple, why do we reserve the horizon metaphor for the limits of vision? Why is the horizon metaphor so closely associated with perspective, or better, with the modern perspective metaphor? And how did both of these words become so closely associated with the “subject”, and at the same time with “world” and “culture”?

In order to begin to address these questions, and to bring the horizon metaphor into clearer focus, we must inquire after a further essential element. We do not think of horizons as real limits but rather as literally pro-visional limits established in relation to a center, a consciousness, ourselves, for example, for whom the limit is a limit of vision. In itself, this limit may shift, and indeed is constantly shifting; but the center relative to which the limit both shifts and remains the same persists, at least for a time, until that center is no more. The horizon is thus defined as and by the ongoing and cumulative experience of a subject. Where did this conception of center and limit come from?

Once again, it came from the Greeks. Aristotle was probably not the first to observe, as he does in his Meteorology, that “throughout the habitable world the horizon constantly shifts, which indicates that we live on the convex surface of sphere”.[6] This statement occurs in the midst of arguments brushing aside the belief of Anaxagoras that the earth is a disk, which Aristotle thought was preposterous. The earth, he argued instead, is a sphere, and he does not mean that all horizons “constantly shift” because there are mountains here and plains or oceans there, but rather that, wherever the center of ourselves may go, the world is always dropping out of sight at the edges. If the earth were a perfect sphere, and air perfectly clear, then the horizon, the limit of the field of  vision, would be the same in any direction we turned, and, because of the curvature of the earth, it would drop out of sight in all directions at the same distance, thus forming a circle around the center, the seeing subject. Not incidentally, sight might thus measure sections of the earth’s convex surface and its volume.

Toward the end of the Classical tradition, late in the fourth century C. E., Macrobius wrote that wherever you stand, you will seem to see an end to the heavens, and the ancients – perhaps Aristotle by this time – called this the “horizon”.[7] Macrobius expanded these remarks in his Commentary on the Dream of Scipio, writing that there are ten circles in the cosmos after the Milky Way. These include the zodiac and the ecliptic, plus five circles on the earth’s surface, the polar circles, the tropics and the equator. The final two circles are the meridian and the horizon (Stahl, p. 152), which he says, are not inscribed on the sphere, because they have no fixed place.  Because the earth is round, the sun is directly overhead, or it is noon, only in one place at one time. “Similarly”, Macrobius continues, “the circumspection [in the literal sense of looking all around in a circle] of individuals (singulorum) makes a horizon for [each of] them. The horizon is a circular boundary that marks the apparent junction of the sky and the earth, and since our eyes cannot see to the ends of the earth, as much as each one beholds by looking about him in all directions is for him his individual boundary of that portion of the sky above the earth. This horizon, which each one’s vision circumscribes for himself, cannot extend beyond three hundred and sixty stades in diameter [about 35 miles], for vision does not exceed 180 stades in any one direction.  When it reaches this point [vision] fails since what is beyond is concealed from us by the roundness of the earth”. These limits continue as the observer moves, and are once again to be realized only under conditions of near-perfect sphericity, “ on a perfectly flat plain, or at sea in a moment of tranquility”.[8]

Macrobius did not distinguish, as astronomical writers did, between this “sensible” horizon and the “rational” horizon, the second a plane through the center of the earth, perpendicular to the position of the viewer on the earth’s surface, relative to which the varying visibility of the fixed stars could be plotted. Such abbreviations notwithstanding, however, the examples of Aristotle and Macrobius are sufficient to explain why in ancient writers the word horizon often appears with the word kyklos, circle, and, since a circle may be completed as a sphere, why horizon also appears with aer. Macrobius seems to be imagining that each of us has a kind of hemispheric bubble of atmosphere, the circumference of the base of which, and the vertical radius of which, is defined by the limits of sight.

A certain number of tentative conclusions may be offered. First, what must have been the older real spatial definition of horizon as a boundary was expanded to entirely new levels of scale and abstraction as part of Greek natural philosophy, and, more specifically, of cosmography and cosmology. In this expansion, the term retained its precinct-like connotations of qualitative inside and outside, that which is inside being the province of individual experience as delimited by sight. In addition, the horizon circle located human awareness within the embrace of the great circles of the cosmos itself. It also vividly characterized the uniqueness of the physical conditions of human consciousness, giving equally unique dimension to the perennial classical theme of human uprightness.[9] (Animals have no horizons because they are always looking down.) In this scheme, sight itself is subject to geometric description and under ideal conditions is a precise measuring device.

At the same time that it had a more or less stable geometry, reflecting a higher, absolutely stable geometry, the horizon circle also had a radically temporal center, the experiencing (if interchangeable) individual, upon whom it was dependent in the sense that, as the individual moved, the circle also moved, even if the circle always remained the same.

The horizon circle located human experience in the harmony of the circles of the world at large, but it also absolutely associated the microcosm of human awareness, not just with vision, but with optics, that is, with the – once again culturally specific – geometric description and explanation of vision. Although it is a related geometry, optics raises new issues.  When we look at the horizon – when we watch the sun set, for example – we say that the earth at our feet “rises” to the horizon, to the meeting of earth and sky. When we say that, we mean that the earth’s measurable surface on which we stand is the base of a very acute angle, and that along this base measures are seen under smaller and smaller visual angles, rising to the height of our point of view, and of our line of sight, as it meets the horizon. In principle, if the world were a perfect plane, this diminution might go on indefinitely, but according to the argument I am following, it must stop at a certain point because, even though the world is a very large sphere, and looks flat from wherever we might be, it eventually curves away out of sight. In effect, the line of sight becomes tangent to the earth’s sphere as the plane of the earth and the parallel line of sight come together at the horizon.

I will return to the question of the connection of the horizon circle and optics, but first I want briefly to outline the history of the horizon circle in Europe after antiquity. Macrobius was a major source for medieval encyclopedists, but it was only in the 13th century, well into the project of the reclamation of classical science, that the ancient terms and definitions began to reappear. Astronomy itself is key. John of Sacrobosco’s De sphaera mundi, which would remain a basic university textbook through the Renaissance, was circulating by around 1225.[10] In a French introduction to astronomy written around 1250 we read that the horizon is “a circular line where the earth seems to rejoin the sky”.[11] By the time Dante wrote the Divine Comedy, the cosmic geometry of revived astronomy provided a firm armature for his imagination. Here is the beginning of Canto II of the Purgatory.“The sun had now reached the horizon whose meridian circle covers Jerusalem with its highest point”. That is, the sun, standing directly over Jerusalem, in the east, appeared to be just rising from where Dante stood. The term is encountered more and more frequently in European writers through the 14th century. “Horizon” could also take more pictorial forms, perhaps paralleling the rise of landscape.  Around 1375 Geoffrey Chaucer (who wrote a treatise on the astrolabe) described the break of dawn as follows: “And whiten gan the orisonte shene”.[12] In the later 15th century, Lorenzo de’ Medici, converted the definition of “horizon” – "nothing else than the last limit, beyond which human eyes cannot see”[13] – into a meditation on the western horizon of death, about which the sunflower instructs us by turning her last loving gaze to the sun’s disappearance. The center that defines the circle perishes.   When he describes the horizon as “that last place where the setting sun is no longer seen, and, when it rises, the first place the sun appears”, Lorenzo might have been imagining the classical circle and the curvature of the world, or the possibly endlessly planar extent of then-new one-point perspective construction. Leonardo da Vinci certainly meant to refer to perspective construction when he advised the painter not to put “histories” one above the other on the same wall, because their different horizons would look like a bottega di merciaio with all of its little square boxes.[14]

In the 13th century Thomas Aquinas gave a new metaphorical meaning to “horizon” that would be elaborated by Neoplatonic writers like Marsilio Ficino and Giovanni Pico della Mirandola. Aquinas began from the astronomical definition – “horizon is the circle where vision terminates, the lower limit of the hemisphere above and the beginning of the hemisphere below;” and as similar unions of heaven and earth, of soul and body, human beings may properly be called “horizons”. “The intellectual soul [that is, the uniquely human soul, as opposed to vegetable and animal souls] is said to be as if a certain horizon and confinium [a common border] of the corporeal and the incorporeal, insofar as its substance [essential nature] is incorporeal, although it is corporeal in the form of a body.” The virtual meeting of earth and sky is the condition of humanity in statu viae, on the way through the pilgrimage of this life.[15]

If in the very long tradition I have outlined the horizon is a circle, how did it become a straight line, a horizontal line? In ancient writers, sufficient reality seems to have been given to the horizon circle by the confidence that, under ideal circumstances, sight would measure a perfect circle from a single point of view. The concentricity of this ideal circle with the larger order must also have bolstered this confidence. But however we might think about them, the representation of apparent horizons is only possible in virtual space, more particularly, in painting, and the words of Chaucer, Lorenzo de’Medici, and Leonardo da Vinci suggest that painting was an important factor in the transformation from the curved horizon to the straight. To be sure, if we are standing at the center of a circle 35 miles in diameter, the edge of the circle – 17.5 miles away at any point – will appear to be nearly straight. But strictly speaking it is not straight, and in order to understand a crucial – and ongoing – episode in the historical life of the horizon metaphor, we must return to the connection between the horizon circle and optics.

As I have mentioned, classical optics was based on the principle that light travels in straight lines, and that vision could therefore be described geometrically as the relation of straight lines to a point, a “point of view”, as we still say, which might coincide with the point at the center of the horizon circle, connected by the line of sight to a point on the horizon circle. The economy of light or sight with respect to a point provided the basic analytical tool of optics, the visual angle, which, as I have also briefly mentioned, made it possible to explain why equal quantities appear to be unequal at different distances, why, for example, the columns in a colonnade appear to recede: from the same point each column is seen under a smaller and smaller visual angle. And so all kinds of visual appearances might be explained, why, for example, circles appear to be ovals when not seen straight on, and, to return to my earlier discussion, why the ground beneath our feet appears to rise to the horizon.

Although both cosmography and optics were disciplines to which geometric demonstrations were indispensable, optics was by and large not concerned with problems of curvature. The base of the visual angle of optics was a line standing for a quantity, and, if this line was expanded into a second dimension, the shape was a square or circle at the base of a three-dimensional pyramid or cone. I have argued elsewhere that the beginnings of Greek optics coincided closely with the invention of architectural scene-painting for tragic theater by the painter Agatharcus of Samos in Athens in the second half of the 5th century B. C. E.[16] A virtual stage of space was literally fundamental to this fictive architecture, and, because it was lower by construction than the point and line of sight, it appeared to rise with distance. This is an optic plane, and whether or not a whole architectural illusion was constructed, it was possible to make the appearance of a virtual stage of space simply by drawing a horizontal line across the surface, a device that has survived to the present.

When ancient scene-painting and architectural drawing – to both of which Vitruvius gave the same name, skenographia – were re-invented in 15th-century Florence,[17] the optic plane became modular, Leon Battista Alberti’s famous checkerboard. In an Albertian perspective construction the last thing to be drawn is the horizon, which Alberti did not call a horizon. (Although a disk he called a horizon was part of the apparatus he set out in his treatise on sculpture.) Having projected his gridded optic plane, with its recessive diagonals obedient to the centric point and line of sight, Alberti drew a horizontal line through this “centric point”. “When I have carefully done these things, I draw a line across, equidistant from the other lines below [that is, parallel to the transversal lines defining the grid below] , which cuts the two upright sides of the large rectangle [that is, his “window”, or panel] and passes through the centric point. This line is for me a limit or boundary [terminus atque limes], which no quantity exceeds that is not higher than the eye of the spectator. Since it passes through the centric point, this line may be called the centric line”.[18]  Alberti’s centric line is defined in terms like those used for the horizon, but he says that it is a limit in that only things taller than a human being – trees, mountains, buildings – may be placed above it. More generally, however, reference of this limit to the center implies that the view framed by the panel is a segment of the horizon for a viewer. But there are fundamental differences between Alberti and earllier writers.

Alberti’s construction defines only a pencil of space, a tunnel of ratios, so to speak, derived from the modular division of the baseline of the painting. The entire framework of his construction is parallel to that baseline. Since it extends “almost to infinity” [paene usque ad infinitam distantiam][19], the tunnel cannot acknowledge the curvature of the earth, which by implication becomes endlessly planar, perfectly and notionally flat. Nor can the horizon line acknowledge the horizon circle, because it is “almost at an infinite distance” beyond the limits of vision. Even if we think of a much greater horizon circle, Alberti’s centric line, determined by the baseline and its modular division, must be tangent to this greater circle, and so assumes an independence from any possible limit of human vision, thus to constitute an infinity.

This infinity – not only in depth, since the horizon line might also be extended endlessly to either side – comes into view only when the geometry of vision is developed as if it were pure geometry. Then the straight line of the horizon implies the separation of the geometry of space from the geometry of vision, what I have called metaopticality.[20] This separation is entirely consistent with the emergence of an important modern definition of objectivity, achieved by removing the visual angle from the metaoptical matrix, thus to yield the infinite coordinate framework of classical Newtonian physics, which although supplanted, remains the framework of the control and prediction of force basic to modern technological life. Since metaoptical space universalizes the principle of modularity – it is isometric, even if the unit of measure is arbitrary – and static, it is perhaps understandable that perspective, at least from Henri Bergson onward, has been associated with instrumental reason, and opposed to a deeper life principle, thus to become one of the prime villains of contemporary cultural criticism.[21]

Things, however, are not so simple, and perhaps not so bad. When Alberti wrote his treatise on painting, even though he wished to make painting an intellectual and liberal art by associating it with geometry, he reined in his Euclidean impulses a bit when he wrote that his parallel lines recede “almost to infinity”. In general, Alberti wrote of pursuing “ a plumper Minerva”,[22] a plumper wisdom, and he claimed to be writing as a painter. In these terms, a point is not simply notional, or purely geometric, but rather the smallest mark a painter can make, and a contour is the finest line a painter can draw. When he wrote this way Alberti was fully in the tradition of classical and medieval optics.

From its beginnings, optics was regarded as a special, limited case of geometry[23], an example of what in the Middle Ages came to be called a subalternate geometry, or a middle science. Whatever they might be thought to be in themselves, the “lines” of optics were physical; geometry itself was “pure”, general and intellectual in contrast to optics. Simply put, sight is not primarily geometric, it is a physical interaction of light with the organ of vision, an interaction of surfaces that can  be described and explained in geometric terms, leaving aside the questions of just what sort of thing rays of light might be, or just how the eye might work.

If seeing is a physical interaction, it must be thought about in very different terms. The Stoics, for example, compared the visual angle to a blind man feeling his way along a path with a pair of sticks, vividly suggesting that vision is fundamentally related to the sense of touch, even to feeling. Leonardo da Vinci performed a simple demonstration: if vision is adequately described geometrically, then if I put my finger in front of my eye, the apex of the visual angle should be cut off and I should be able to see nothing. Since I do see something, light, the eye and their interaction must also be much more complex. Leonardo developed linear perspective together with what he called the perspective of disappearance, not just diminution in size but loss of distinctness of color and detail, more properly and simply physical limits of vision.

We have seen enough to put together a fairly simple intellectual historical narrative. In Classical science, an idea like “precinct” was expanded in such a way as to make the conscious individual – in principle any individual – the center of a circle nested in the great living circles of the cosmos itself. The horizon circle, as the example of Heidegger at the beginning of the paper shows, retains its classical configuration to a remarkable degree, perhaps much as the ego has remained at the center of Western languages. The concentric cosmic circles validating the original conception have of course dissolved, but the metaphor of the horizon circle persists, together with “perspective”, in a host of metaphors from phenomenology to archaeology to culture to identity.[24] Through all of this, the center, the individual and subject as the locus of distinctively human experience, life and death, has survived, if not without continual challenge and redefinition. The second part of the title of this paper, infinities without end, was meant to indicate the numberless possible human spaces, times, and horizons, and also to indicate the value these absolute particularities should have.