Could There Be Method Behind Kepler’s Cosmic Music?

Abstract

While Kepler is regarded as a major figure in standard historical accounts of the scientific revolution of early modern Europe, he is typically seen as having one foot in the new scientific culture and one in the old. In some of his work, Kepler appears, along with Galileo, to be on a trajectory towards Newton’s celestial mechanics. In addition to his advocacy of Copernicus’s heliocentrism, he appealed to physical causes in his explanations of the movements of celestial bodies. But other work appears to express a neo-Platonic “metaphysics” or “mysticism”, as most obvious in his embrace of the ancient tradition of the “music of the spheres”. Here I problematize this distinction. The musical features of Kepler’s purported neo-Platonic “metaphysics”, I argue, was also tied to Platonic and neo-Platonic features of the methodology of a tradition of mathematical astronomy that would remain largely untouched by his shift to heliocentrism and that would be essential to his actual scientific practice. Importantly, certain features of the geometric practices he inherited—ones later formalized as “projective geometry”—would also carry those “harmonic” structures expressed in the thesis of the music of the spheres.

1. Introduction

Johannes Kepler (1571–1630) has long occupied an ambiguous position in the history of modern science. He was one of the first to defend Copernicus’s revolutionary heliocentric thesis set out in De revolutionibus orbium coelestium of 1543 (Copernicus 1992).1 Copernicus had challenged the evidence of everyday experience, which had suggested (and surely still does) that the Sun moves daily around a stationary Earth—the conception of the universe inherited from the ancient Greeks. In contrast, the new conception demanded that we picture ourselves as located on a planet that not only rotates daily on its own axis but also, like other planets, circles the Sun. With this “Copernican principle” (Dunér 2023) that questions immediate experience on the basis of a theoretical conception of the conditions under which that experience unfolds, the question of the relation of “empirical evidence” to “theory” would become complex.

Construed in this way, Kepler standardly appears, along with his approximate contemporary Galileo Galilei (1564–1642), as one of the earliest instigators of the “scientific revolution” that would break with scholastic Aristotelianism and its theological interpretation and profoundly reshape Western thought (c.f., Koyré 1973; Kuhn 1995). Kepler’s theoretically guided analysis of years of astronomical data accumulated by the greatest sixteenth-century observational astronomer, Tycho Brahe, would lead to the formation of his three geometrically described laws of planetary motion, laws that Isaac Newton (1642–1726) would subsume within his own universal laws of gravitation.

Such “modernism” in this regard is typically described as on display in Kepler’s Astronomia nova of 1609, where he had introduced a “celestial physics” at the heart of astronomy in the spirit of Newton’s later attempts to unify the explanations of celestial and terrestrial phenomena (Kepler 2015).2 With such a focus, Kepler is understood to have helped redefine astronomy as cosmology qua branch of natural philosophy rather than as simply “mathematics” devoted to “saving the phenomena”, that is, as being construed merely instrumentally as means to predict celestial phenomena such as eclipses of the Sun.

All agree, however, that aspects of Kepler’s astronomical commitments resist being fitted neatly into this progression from Copernicus to Newton. Despite the approach introduced in Astronomia nova, his carefully constructed empirically based laws were linked to what, from a modern perspective, appears as a distinctly unscientific conception of the universe. While Einstein had nominated “Kepler’s marvellous achievement” as “a particularly fine example of the truth that knowledge cannot spring from experience alone but only from the comparison of the inventions of the intellect with observed fact” (Einstein 1954, p. 266), not all of Kepler’s intellectual inventions would be regarded positively. As Gerald Holton noted half a century ago, from our typically modern perspective we see Kepler’s “probing for the firm ground on which our science could later build” as often leading into “regions which we now know to be unsuitable marshland” (Holton 1988, p. 54). The path alluded to had been guided by a “commitment to neo-Platonic metaphysics” that sat beside his otherwise “sound instinct for physics” (p. 55). More recently, David Love asserted that Kepler’s “free-ranging imagination” had “sometimes led him to the truth and sometimes into the realms of fantasy” (Love 2015, p. 57), it being the very “contrast between a deep insight into the nature of reality and a hopelessly wrong mysticism, that makes Kepler such an endearing and fascinating character in the story of the scientific revolution” (Love 2015, p. 12). Typically, as exemplifying his commitment to “neo-Platonic metaphysics”, commentators like Holton and Love have especially in mind Kepler’s embrace of the ancient doctrine of the “music of the spheres”—his concept of the universe as designed by God so as to emit the most beautiful harmonies and melodies possible via the coordinated motions of its parts. Here, however, I want to question this simple and often presupposed partition into the “physical” and the “mystical”.

Kepler was a deeply religious man for whom astronomy was another way of coming to understand the nature of God. It is clear that, in an age of fierce religious discord, he had become gripped by the idea of the “harmony” manifested by the cosmos as expressing the state that God intended for humans living within it (Rothman 2017, Introduction; Methuen 2024). Seemingly, as a student at Tübingen (Methuen 2024, pp. 5–6), he had become influenced by a Christianised version of the theology found in Plato’s dialogue Timaeus, (Plato 1997) in which the harmonically conceived unity of the world-soul was understood as a type of “role-model” for the souls of individuals and for the nature of their lives together (see, for example, Corcilius 2018; Betegh 2020). This idea of an underlying cosmic harmony has been taken by some interpreters as integral to Kepler’s own conception of physics, exemplifying an attitude to the cosmos something like that found in recent physics, which aims at an ultimate unity of the four fundamental physical forces—the so-called “theory of everything” (Osterhage 2020, pp. 100–6).

Here, however, I pursue a different way in which the “musical” elements informing Plato’s cosmology might have played a role in Kepler’s astronomical practice and especially in relation to his methodology. It will be argued that Pythagorean harmonic structures are to be found at the heart of the geometry he employed in the context of his theory of optics, as used within his astronomical practice. In this way, a second path leading from Plato’s harmonics to the sixteenth and seventeenth centuries may have been relevant to Kepler’s practice beyond that of the widely acknowledged path via the neo-Platonist mystico-religious tradition. This second path would lead via the work of the Greek geometer Apollonius of Perga, and it would be expressed in Kepler’s work on optics (Kepler 2000), as influenced by Arab mathematicians of the Middle Ages. This path would also lead, a little after Kepler, to the “projective” geometry of Girard Desargues.

Interpreted in this way, these musico-geometric features of Kepler’s purported “neo-Platonic metaphysics” were, I will suggest, simply features of the ancient geometrically based Greek astronomical models upon which Copernicus, Tycho, and Kepler had all drawn.3 Such models, stretching from that of Eudoxus of Cnidus, a contemporary and colleague of Plato, to Ptolemy of Alexandria five centuries later, had been constructed with the resources of a mathematics very different to those opened up to Europe in the sixteenth century when the non-Greek discipline of algebra was imported from Arabic sources to be seized upon by the likes of Descartes and Newton. Kepler’s adoption of neo-Platonic astronomical models would have two linked phases. In the first, as set out in his book Mysterium cosmographicum published in 1596, he had proposed a model for Copernicus’s heliocentric conception in which the orbits about the Sun of the then known six planets (counting Earth as a planet as per the Copernican hypothesis) correlated with a series of six concentric spheres constructed about, within, and between a homocentrically nested structure of the five regular polyhedra or “Platonic solids”—the cube, tetrahedron, dodecahedron, icosahedron, and octahedron. This structure revealed that, despite the Copernican break with ancient geocentric astronomy, Kepler was, as had been Copernicus himself (Swerdlow and Neugebauer 1984), still reliant upon the ancient approach to astronomy found in the likes of Plato, Eudoxus, Hipparchus, and Ptolemy (Field 2024, p. 65).

Later, Kepler’s Platonic approach would more explicitly embrace the more dramatic musical idea that the crucial parameters structuring the cosmos were derived from the ratios and proportions of musical harmony, as is obvious from the title of his major work of 1619, Harmonice mundi, “The Harmony of the World” (Kepler 1997). This approach had originated in an extension of the mathematics of musical relations to cosmology by Pythagorean natural philosophers, such as Philolaus of Croton and Archytas of Tarentum, and would be most known from Plato’s Timaeus; but, even in Plato’s time, it had been already dismissed by Aristotle as a mere metaphor and untrue (Aristotle 1984, On The Heavens 290b12–14).4

Kepler did not believe that the orbiting planets actually gave off audible sounds, but he nevertheless took the “harmonies” of the planetary movements as much more than a metaphor. The movements of the planets gave expression to the most conceivably beautiful music entertained by the mind of God. Thus, in Harmonice mundi Kepler became involved in contemporary debates over the nature of music theory itself, criticizing the form of music theorized by the ancient Greeks, and the theory based on it, as expressing a poverty of their musical intervals.5 But this did not entirely exhaust the role of musical intervals in a “harmonic” astronomy. The Greek musical intervals were, I will argue, deeply embedded with the most scientific aspects of the practice of Greek astronomy, and they were, as we will see, already implicit in the nested polygon theory at the centre of Mysterium cosmographicum. In fact, while employing harmonic musical intervals in his cosmological model, Plato himself had carefully avoided any reference to the fact of their being musical intervals. Plato’s approach to the “music of the spheres” was, I suggest, not that of the literalist opponent to Aristotle’s “metaphorical” reading: for Plato, the actual Pythagorean musical intervals had a significance well beyond their adequacy for the structure of music itself, being relevant to the very geometry of three-dimensional space.

Thus, this paper is devoted to prying apart Kepler’s actual harmonic astronomy and the overt metaphysico-religious connotations he gave to it in the literalist spirit of the neo-Platonic philosophers, and in this task I will be guided by Stephen Gaukroger’s summary description of science itself as “a complex amalgam of (among other things) theory, engineering, technology, and invention” (Gaukroger 2020, p. 285).6 On such a view, Kepler’s scientific practice can be understood as run through with factors—observational techniques, forms of instrumentation and measurement, and so on—that cannot be easily freed from those Platonic features to which they had been bound in antiquity, features not limited to the explicit conception of the universe as a divine work of art.7 In particular, it will be suggested that such features of Platonic and post-Platonic models were implicit in the geometry that would be inherited, perhaps unconsciously, by Kepler via his appropriation of the methods and instruments of ancient astronomy, methods and instruments fundamental to the astronomical progress made by the ancients and by Kepler himself. This was a conception of geometry different to the classically Euclidean geometry that, with the aid of modern algebra, would be modified, following early leads by Descartes and Fermat, into “analytic” geometry” and that would become the framework for modern Newtonian celestial mechanics.

Kepler seems to have not questioned the fundamentally Euclidean nature of the geometry that he employed (Caspar 1993, p. 273), and the very idea of geometries different to that of Euclid postdates Kepler by centuries. However, as stressed by William Donahue, Kepler seems to have understood Euclid, somewhat unconventionally as a Pythagorean thinker (Donahue 2024, p. 202), and, in line with this, I will argue that the geometry actually embedded in Kepler’s astronomy shows fundamental features of projective geometry, a form of geometry with roots in Pythagorean mathematics, Medieval Arabic astronomy, as well as Renaissance neo-Platonic theories of perspective in painting. Projective geometry would only start to be differentiated from Descartes’s geometry in the decades after Kepler’s death in the work of Gerard Desargues and Blaise Pascal, and it would become eclipsed by the type of analytic geometry to which Descartes’s geometry would give rise and so become largely forgotten for a century and a half. It would, however, re-emerge in the nineteenth century as an alternative to the existing conception of geometry based on Descartes’s extension of the metrical geometry of Euclid’s Elements. As the nature of projective geometry is not widely known, a short historical sketch is warranted before its central features are traced in relation to the Platonic features of Kepler’s astronomy and, from there, to the astronomical tradition upon which he drew.

2. Projective Geometry: A Historical Sketch

The geometry found in Euclid’s Elements is a geometry of distances and angles. What makes one line segment identical to another is that one could imagine both lines on a plane as, say, pieces of wire, such that one could physically move one line next to the other so as to check they were the same length. Similarly with angles, one could superimpose one on the other to see that they were aligned. Until the seventeenth century, there was no way of making explicit the underlying metric being presupposed by such equivalences of length and angle. In 1637, however, Descartes, in La géométrie (Descartes [1637] 1954), would introduce algebra into geometry in such a way as to identify geometric curves with algebraic equations in two variables. This would lead to so-called “analytic geometry”, although many of the characteristic features of this approach were as yet undeveloped in Descartes’s famous text.8 Eventually, however, the x and y rectangular coordinates known from school geometry would be taken as providing a framework within which any two points in a plane stood at some determinate distance from each other such that the lengths of lines could be assigned numerical values within the system of “real numbers”—a number system developed over the sixteenth and seventeenth centuries to include negative numbers, irrational numbers, and so on, all of which had been unknown in the “Golden Age” of Greek geometry. The effect on the emerging sciences of this new analytic geometry together with the related differential and integral calculus of Newton and Leibniz would be massive.

Euclidean geometry, even in its analytic form however, had been found to be not particularly helpful in relation to certain purposes for which geometry might be thought to be useful—for example, that of dealing with spatial relations between two-dimensional perspectival representations of three-dimensional arrays of objects, the problem facing those trying to deal with the issue of perspective that had been introduced into Renaissance painting. Viewed perspectivally, as when projected onto a two-dimensional picture plane, the sides of the tiles of a square-tiled floor appear to shrink when receding into the distance while nevertheless understood as being of the same length, and lines understood to be parallel appear to meet at a “vanishing point” on or just above the horizon, as described by Leon Battista Alberti in 1480.9 What was needed was a type of geometrical optics that went beyond that topic as dealt with by Euclid in his Elements. Kepler would be a transitional figure in introducing this type of alternative approach to geometry that modern theories of perspective would demand. This was “projective geometry”, officially introduced in 1639, albeit relatively fleetingly, by the French mathematician Girard Desargues in his Rough Draft of an Essay on the results of taking plane sections of a cone (in Field and Gray 1987, chap. 6). The idea of shapes resulting from “taking plane sections of a cone” Desargues had taken from the Conics of the Greek geometer of the third century BCE, Apollonius of Perga, a Latin translation of which had appeared in Europe in the sixteenth century. Desargues would, however, approach the space within which geometric figures were conceived in a different spirit to that of analytic geometry. Rather than thinking of space as that within which, say, exist planar geometric figures with their distinct metrical properties, what was now fundamental were the relations existing among different figures. It was in this spirit that Leibniz would later propose the idea of a situational analysis, “analysis situs” (De Risi 2018), in conformity with the relational conception of space, which he would oppose to Newton’s “absolutist” conception (Leibniz and Clark 2000).

While Desargues’s Rough Draft is taken as the first expression of the projective approach to geometry, a few decades before, in 1604, and in relation to the role played by optics within his astronomical work, Kepler had published a work that, in addition to revolutionizing the field of optics, had anticipated some key features of Desargues’s projective geometry. This was his Paralipomena to Witelo whereby The Optical Part of Astronomy is Treated (Kepler 2000), the “Witelo” of the title referring to a thirteenth-century Polish monk who had popularized the optics of the tenth- to eleventh-century Arab mathematician and astronomer, Hasan Ibn al-Haytham, referred to in Europe by the Latinised “Alhazen”, who had influenced approaches to perspective by both Piero della Francesca and Albrecht Durer (Donahue 2024, p. 186).10

Ibn al-Haytham (956–1040) had been a major figure within a branch of Arabic mathematics in which had been developed the geometry of the conic sections—circle, ellipse, parabola, and hyperbola—as earlier theorized by Apollonius of Perga in the third century BCE. But, while Desargues had attempted to capture those “projective” relations explored by Alberti, Durer, and others, within an axiomatizable two-dimensional geometry,11 the Arab mathematicians’ development of geometric science had also been at least partly motivated by the need to understand the use of the astrolabe, an astronomical instrument whose use dated back to Greek astronomers. Thus, in the ninth century, Abmad al-Farghani (c 800–870) had devoted a chapter of a book, The Perfect (al-Farghani 2005), to its geometrical underpinnings, a presentation that has been described as “the first truly geometrical study of geometrical projections” (Rashed 2017, p. 2; c.f., Abgrall 2015, pp. 159–60). The Arabs had apparently conceived of Apollonius’s work Conics as having given rise to a whole new branch of mathematics, “ilm al-tastih”, the “science of projection” (Rashed 2017, p. 2).

Apollonius’s work is at the origin of what I have described above as the second path via which ideas from Plato’s “harmonic cosmology” would be transmitted to the seventeenth century. After Plato’s death, the philosophical approach found in the Timaeus, with its thematizing of Pythagorean harmonic theory, had been taken seriously by followers including Speusippus and Xenocrates, the first and second directors of the Academy (Netz 2022, p. 392). Although this tradition seems to have soon faded, it would be later revived in the first century BCE, from which it would become a feature of late neo-Platonic thought (Netz 2022, pp. 392–96) in which the earlier Pythagorean mathematics would become bound up with religious and mystical doctrines broadly thought of as “neo-Platonism”.

Despite the initial fading of Plato’s late Pythagorean turn, evidence for the persistence of distinctly mathematical interest in attempts to unify the three “musical means”—the geometric, arithmetic, and harmonic means, which had been extended from Pythagorean music theory to cosmology in Plato’s Timaeus—can be found in the century after Plato in the work of two figures important for the development of Greek mathematical astronomy. These were Eratosthenes of Cyrene (c. 276–c. 195 BCE), most famous for his calculation of the diameter of the Earth (Netz 2022, pp. 236–38), and Apollonius of Perga (c. 240 BCE–c. 190 BCE). For his part, Eratosthenes was described by Pappus of Alexandria as having written two books, now lost, in the area of Plato’s mathematics: the Platonikos, an apparently fictional work dealing with the underlying mathematics of Plato’s Timaeus, and a work called On Means, discussing “loci with reference to means”, that some have conjectured to have been geometric attempts to demonstrate systematic relations or a mediated unity holding among the Pythagorean musical means (Heath 1921, vol II, pp. 105–6; Pappus of Alexandria 1986, pp. 598–99).12 In On Conics, Apollonius would similarly construct geometric figures in which the geometric, arithmetic, and harmonic means would be related in distinct ways, as shown in Section 3 below. Apollonius’s work would be taken up enthusiastically by Arab mathematicians from the ninth century onwards.

Al-Farghani, Ibn al-Haytham, and other Arab geometers, as later would Kepler and Desargues, would take Apollonius’s approach to conic sections well beyond that found in the work of Apollonius himself. While Apollonius had conceived the four conics as produced by sectioning a cone at different angles, he still treated them as essentially different figures in the spirit of Euclid’s Elements. The Arabs, and later, Kepler and Desargues, had sought “projective” principles linking these shapes when considered entirely as plane shapes abstracted from their perspectival qualities. All introduced into geometry the idea of a type of movement or transformation between shapes that had earlier, with certain exceptions, been conceived in a static way.13 Importantly, Kepler would appeal to a principle of analogy, linking the various conic shapes in ways that would later be referred to as their “projective equivalence”.14 An ellipse, for example, could easily be thought of as a stretched circle, appealing to the type of intuitions that would be later used in the discipline of topology. Using a type of dynamic vocabulary, Kepler would thus note, “a straight line goes over into a parabola through infinite hyperbolas, and further through infinite ellipses into a circle […] the most obtuse hyperbola is a straight line, and the most acute, a parabola; the most acute ellipse is a parabola, and the most obtuse, a circle” (quoted in Rosenfeld 1988, pp. 134–35). Evaluating Kepler’s contributions to mathematics, Eberhard Knobloch has written that Kepler’s “comments on his classification of conic sections can be read as concepts which later become underlying principles of projective geometry” (Knobloch 2024, p. 316).15

Kepler’s anticipation of aspects of Desargues’s projective geometry would also be noted by Judith Field (Field 1986, pp. 449–53) and H. S. M. Coxeter (Coxeter 1974, p. 3), who have pointed to Kepler’s use of the idea of “points at infinity”. Renaissance theories of perspectival painting had already come up with the conception of infinitely distant “vanishing points” at which perspectivally represented lines that are “objectively” parallel appear to meet, and for Kepler, the existence of points at infinity would be a consequence of stretching an ellipse to an infinite extent. Kepler had determined that an ellipse had two “foci” and if one “stretched” an ellipse far enough, one would be left with a parabola with one focus, the other now residing an infinite distance from it. It was recently claimed (Rashed 2017, pp. 313–21) that, in the work The Knowns, Ibn al-Haytham had also appealed to points at infinity in a projective context. Regardless of the degree to which such ideas had been developed by the Arabs or by Kepler, however, Desargues had integrated the idea of points at infinity more systematically into this new form of geometry than had been previously achieved.

In the early nineteenth century, Desargues’s fundamental ideas would be revived in France by Gaspard Monge and his former students, especially Jean-Victor Poncelet (Gray 2007), and the discipline of projective geometry would blossom during the century, developing its own “analytic” and “synthetic” forms. Monge, a military engineer active during the years of the revolutionary wars, sought techniques analogous to those of Renaissance painters, allowing measurable distances between points on two-dimensional maps to be related to points in the landscapes represented on those maps. The projective features of Desargues’s geometry served such purposes. For this, a central “invariant” of projective geometry would replace the invariance of distance and angle in the metrical geometry of Euclid’s Elements and Descartes’s La géométrie. This had been discovered by Desargues but was seemingly unknown to Kepler and would be revived by the likes of Michel Chasles, Jakob Steiner, and August Möbius in the nineteenth century. The invariant allowing perspectival mapping in space was a double ratio, that is, a ratio of ratios, among distances separating four points on a straight line. This double ratio, later called the “cross-ratio,” was grasped as holding for different projections of such arrays or “ranges” of points, as illustrated in Figure 1.16

Here, two “pencils” of rays, each of four rays, radiating from points P1 and P2, are both sectioned by a line to form on that line a range of four points, A, B, C, and D. Each pencil is also sectioned by a shorter line creating two further ranges, A₁, B₁, C₁, and D₁, and A₂, B₂, C₂, and D₂. Some principle is needed to relate the way the four points are proportioned differently over the three sectioning lines. What Desargues had discovered was that a certain double ratio holding among the points was constant. Specifically, the ratio between the ratios AB and BC and AD and DB (AB:BC::AD:DB) was equal to that between A₁B₁ and B₁C₁ and A₁D₁ and D₁B₁ (A₁B₁:B₁C₁::A₁D₁:D₁B₁) as well as equal to that between A₂B₂ and B₂C₂ and A₂D2 and D₂B₂ (A₂B₂::B₂C₂::A₂D₂:D₂B₂). Furthermore, the same cross-ratio was found to hold between the sines of the angles formed among the four rays of the different pencils radiating from P1 and P2. More generally, this cross-ratio of ranges and pencils would hold for any further line intersecting these pencils of rays, regardless of its orientation, and similarly for any further pencil of rays intersecting lines to produce further ranges.

Regarded in this way, this diagram is meant to be taken as a two-dimensional one—the theorems here hold regardless of any three-dimensional interpretation given to the lines or points involved. Basic projective geometry simply studies various configurations of points and lines on a plane. Nevertheless, we might imagine the diagram as a schematic representation of the viewpoints of, say, two painters, viewed from above, their viewpoints represented by P1 and P2, with the nearer, short sectioning lines representing the picture planes of the canvases on which each is portraying a common scene. Thinking of points A, B, C, and D, as points on objects within a three-dimensional common scene, we can now appreciate how the constancy of the cross-ratio might allow us to think of relations among perspectivally different representations of that scene.

It would take some time for the realization that projective geometry was not simply Euclidean geometry with some added theorems of this kind. For example, it might be thought that such ratios could have determinate values only because such values are able to be assigned to them by the underlying metric made explicit by Descartes’s coordinates. In the second half of the nineteenth century, however, this would be shown to be mistaken, allowing the mathematician Felix Klein to raise the question of “the sense in which it seems psychologically justified to construe projective geometry before metrical geometry and to regard it as the very basis of the latter”.17 Klein was relying on the fact that a special instance of the cross-ratio relation, called the “harmonic cross-ratio”, was able to be constructed within an entirely non-metrical form of geometry. Rather than presupposing Euclidean geometry, the latter, in fact, presupposed it.

In this harmonic cross-ratio, the cross-ratio itself has a value of −1. That is, in absolute terms, a ratio like that of AB:BC in the figure above would be equal to that of AD:DC. However, Carnot had introduced the idea of assigning line segments a value, signified as “+” or “−” on the basis of their relative direction or “orientation”, such that if the segment AB is deemed positive, that of BA would be negative. As the orientation of DC is the reverse of the other three line segments, the overall cross-ratio is given a negative value. Here, the adjective “harmonic” hints at the ultimate Greek “musical” sources of this fundamental invariant of projective geometry. The harmonic cross-ratio was, in fact, a generalization of a numerical Pythagorean structure that neo-Platonists had taken to be the “most beautiful bond” that Plato had regarded as responsible for the coherence of the different parts of the cosmos as a whole, a bond in which the three musical means were unified. We will explore the origins of this in detail in Section 4 below. But, for the moment, we are now in a better position to examine this “harmonic” conception of space and its role in Kepler’s astronomy and the neo-Platonic astronomy upon which Kepler had drawn.

3. Kepler’s Early Polyhedral Model

The regular polyhedra of the astronomical model of Kepler’s Mysterium cosmographium are three-dimensional geometric structures with identical regular polygonal faces that meet each other along edges that themselves converge on points of intersection or “vertices”. Theaetetus, a younger colleague of Plato in his Academy, is the presumed author of Proposition 18, of Euclid’s Elements, Book XIII (Euclid 1956), proving that there are only five such solids, the ones illustrated in Figure 2 below: the tetrahedron, with 4 triangular faces, 6 edges, and 4 vertices; the cube, with 6 square faces, 12 edges, and 8 vertices; the octahedron, with 8 triangular faces, 12 edges, and 6 vertices; the dodecahedron, with 12 pentagonal faces, 30 edges, and 20 vertices; and the icosahedron, with 20 triangular faces, 30 edges, and 12 vertices.

The five regular polyhedra had played a fundamental role in the cosmology of Plato’s dialogue Timaeus but not the same role as that to which they would be assigned by Kepler.18 In Kepler’s model, these five “Platonic solids” are arranged in a nested homocentric configuration19—from the inside out: octahedron, icosahedron, dodecahedron, tetrahedron, and cube—such that, between any two of these shapes, a sphere could be inserted with the vertices of the enclosed polyhedron touching its surface from within and the faces of the surrounding polyhedron touching the sphere as tangents from without. The five solids thus determined the spacing of six spheres with which he correlated the orbits of the six known planets. Kepler gives as a direct quotation the form of words (presumably from God himself, to whom Kepler credited the experience) within which the insight had come to him: “The Earth is a circle which is the measure of all. Construct a dodecahedron round it. The circle surrounding that will be Jupiter. Round Jupiter construct a cube. The circle surrounding that will be Saturn. Now construct an icosahedron inside the Earth. The circle inscribed within that will be Venus. Inside Venus inscribe an octahedron. The circle inscribed within that will be Mercury” (Kepler 1981, p. 69). I suggest that to construct the space of the universe on this model could be considered to already build into it fundamental “harmonic” values, values characteristic of space as projectively conceived.

In relation to Copernicus’s measurements of the relative distances of the planets upon which Kepler relied, this model had, in fact, been surprisingly accurate (Field 1988, pp. 64–69). The most obvious failing of the model would come in 1871, a century and a half after Kepler’s death, when a new planet, Uranus, would be discovered—a mathematical impossibility according to Kepler’s model as the fact that the regular polyhedra were limited to five meant that the number of planets was limited to six. But, within his own lifetime, with the availability of more accurate data, Kepler would see the need to revise the model himself, although its basic structure would remain intact. It is now assumed that the original “fit” between model and observation was simply a happy coincidence and that the discovery of Uranus had put an end to any possible significance for Kepler’s Platonic speculations. I suggest, however, that Kepler’s attachment to the relations among the Platonic solids would remain highly significant for other reasons.

The underlying mathematical principle determining this structure, according to Kepler, was to be found in the relations among the polyhedra themselves, which he classified into three “primary” (cube, tetrahedron, and dodecahedron) and two secondary forms (icosahedron and octahedron) (Kepler 1981, p. 105). Kepler’s reasoning here certainly seems ad hoc. For example, the radii of the orbits of the three planets further from the Sun than the Earth correlate with the primary polyhedra while those of the two closer to the Sun correlate with those of the secondary polyhedra, Kepler declaring that “nothing could be more appropriate than that our Earth, the pinnacle and pattern of the whole universe, and therefore the most important of the moving stars, should by its orbit differentiate between the two classes of polyhedra” (Kepler 1981, p. 105). However, from the point of view of our task of showing how Platonic features of Kepler’s model point to features of its underlying geometry, his starting point of the division of the polyhedra into these two classes is highly relevant.

A fundamental feature of projective geometry is its principle of “duality”—a type of functional equivalence between points and lines, such that for all theorems concerning relations among points, a similar theorem can be found concerning relations among lines.20 Kepler’s partition of the five polyhedra into two groups follows from the role the principle of duality plays among the polyhedra themselves. Minus the tetrahedron, the remaining four polyhedra can be divided into two pairs, such that in each pair one figure is understood as the dual—in the sense of the inverse—of the other. Thus, while the square has 6 faces and 8 vertices, the octahedron has 8 faces and 6 vertices, and while the dodecahedron has 12 faces and 20 vertices, the icosahedron has 20 faces and 12 vertices.

This implies that an octahedron could be constructed inside a cube, such that each of its vertices is located at the middle of each of the six faces of the cube. Similarly, a cube can be constructed inside the octahedron, such that each of its eight vertices is located at the centres of each of the eight faces of the octahedron. An analogous arrangement holds with respect to the dodecahedron and the icosahedron. Each of Kepler’s two secondary solids are thus the “duals” of one or other of the two primary solids. As for the tetrahedron, it can be understood as being dual to itself, or “self-similar”.21

Such duality will prove highly relevant to the principles according to which polyhedra can be homocentrically nested; to see this, we might simplify the idea of nesting them by starting, as had Kepler initially,22 with the nesting of two-dimensional polygons separated by inner and outer circles. Thus, consider, for example, a simple square with its inner and outer circles, as in Figure 3.

Here, a square contains a circle that touches each of its sides and is circumscribed by a second circle that passes through each of its vertices. It is now easy enough to envisage how another regular polygon, concentric with both the square and the two circles, might be constructed such that its sides touch the square’s outer circle as tangents, as does the added octagon, for example, in Figure 4.

Of course, a further circumscribing circle could be now constructed passing through its eight vertices and so on, building up a nested series of polygons from the inside out, and from this simple model we might now try to imagine a three-dimensional analogue by interpreting the square in Figure 2 as a section through a cube and its inner and outer circles as the great circles of inner and circumscribing spheres.

We soon discover, however, that adding this new third dimension has considerably complicated the relations involved. For example, while the octagon in Figure 3 might be considered as a section through the middle of some possible polyhedron, exactly which polyhedron it might be is not clear, as the sides of the octagon can be interpreted in different ways: a particular side might represent an edge joining two faces of the three-dimensional analogue or it may represent a section through a particular face. As Pythagorean mathematicians had learnt in ancient Greece, moving between two- and three-dimensional forms of geometry introduces complex problems.23 In this case, what is needed is some kind of underlying principles accounting for the way two-dimensional faces can be combined into three-dimensional figures. One such principle—simple but with a vast array of applications—had been discovered more than a century after Plato by Apollonius of Perga (c. 240–190 BCE) in the context of his study of conic sections, and this principle would be rediscovered in the nineteenth-century revival of projective geometry in France. It is a basic expression of the duality found in projective geometry, in this context, the inverse relation holding between a point—a “pole”—and a corresponding line—its “polar”. Such an inverse relation between pole and polar exists in relation to a circle (or any conic section, but, for simplicity, let us keep to the circle), as illustrated in Figure 5 below.24

A circle inscribed on a plane separates the area inside the circle from that outside it in a quite peculiar way, such that for any point outside the circle (the pole) there will exist a unique line (the polar) that transects the circle. Similarly, for any point (pole) inside the circle there will coincide a unique line (polar) wholly outside the circle. (Again, for simplicity, we will only consider the situation with a pole P lying outside the circle).

For a pole P, the corresponding polar, the line AB, can be found as illustrated in Figure 5. When two lines are drawn from P so as to tangentially touch the circle at points A and B, the line joining points A and B will be the polar of P with respect to that circle. The point P′, at which the polar intersects the line joining the pole (P) to the centre of the circle (O), is called the inversion of P25 and, it turns out, that a particular relation will hold among distances marked out by the pole and its inversion with respect to the circle, in that the product OP·OP′ = OA². Without much algebra, the Greeks would have expressed this arithmetical relation geometrically—that is, in terms of areas, such that the area of a rectangle with sides OP and OP′ would be said to be equal to the area of a square with a side the length of the radius of the circle.

It is now easy to see how the pole–polar relation might be relevant to Kepler’s nested polyhedra by returning to the simpler example of a square with inner and outer circles. Were we to construct a further square inside the circle contained in the first square and rotated 90° in relation to it, it can be appreciated that the sides of the innermost square are the polars of the poles represented by vertices of the surrounding square, just as, in Figure 6, A is the pole of polar HE, B that of EF, C that of FG, and D that of GH.

Adding a third dimension now eliminates the constraint that the inner and outer polygons be the same (in the figure above, squares). The analogous constraint is now that the inner polyhedron has the same number of vertices as the outer has faces. The dual relation now understood in three dimensions can thus be described as follows: “Take any point within the polyhedron as the centre of a sphere of arbitrary radius. Let the vertices of the polyhedron be poles with respect to this sphere. The dual is defined to be the envelope formed by the planes polar to the vertices. [...] If we restrict ourselves to symmetric polyhedra that have circumscribing spheres, such as the platonic solids, the poles can be taken to be the vertices intercepted by the circumsphere while the polars are tangent planes to this sphere (the inscribed sphere of the dual)” (Kappraff 2001, p. 292).

We will see variants on this figure of Apollonius, relating pole to polar and its inverse, reappear in many guises in relation to the implicit conception of space that had been imported from antiquity into Kepler’s cosmological models, but, significantly, it also has deep direct connections with the harmonic theory that the Pythagoreans and Plato had extended to astronomy, the “harmonic” principle to which Kepler would appeal in his second model in the Harmonice mundi of 1619.

4. Kepler’s Later Harmonic Model

In Mysterium cosmographium, Kepler had flirted with ancient harmonic theory in a somewhat confusing way in relation to trying to justify the partitioning of the zodiac into its traditional twelve constellations or “houses” (Kepler 1981, chap. XII, pp. 131–35). In Harmonice mundi, however, the harmonic model is expanded to effectively become the dominant structure, Book III being devoted to the development of harmonic theory itself, independently of any astronomical application, and Book V, to its application to the orbits of the planets. However, here we must distinguish between the way that Kepler thought of harmony as relevant to astronomy and the way that Plato had envisaged this.

That Kepler had regarded the thesis of the “music of the spheres” as much more than a metaphor is apparent in Book III of Harmonice mundi, which is devoted to a full-blown theory of musical intonation adequate to contemporary European music. Thus, Kepler criticises the limitations of the conception of intonation as theorized by Pythagoreans of Plato’s time and adopts the extension of this earlier approach found later in Ptolemy to be developed in the sixteenth century in a more modern musical context by Gioseffo Zarlino.26

The traditional Pythagorean conception of the musical consonances, Kepler holds, showed the limits of their willingness to trust their senses for discerning consonantal intervals (Kepler 1997, p. 137), as they had restricted concordant intervals to three: the diapason (or octave, as in middle C to the C immediately above it, C4 to C5) and, within the octave, the diatessaron (the fourth, as in C to F), and the diapente (the fifth, as in C to G) (Kepler 1997, pp. 133–34). The Greeks had set great store on the fact that for the tones emitted from a vibrating string, the ratios for dividing the string into these three intervals could be constructed from the tetrad or “tetraktys” of numbers 1, 2, 3, and 4. Thus, the octave coincided with dividing the string in half and, so, with the ratio 2:1, while the fifth coincided with dividing the string in the ratio 3:2 and the fourth, in the ratio 4:3. While the octave, the fourth, and the fifth are still thought of as fundamental concords, for the needs of musical theory, especially one relevant to the complex polyphonic music of the Renaissance, many more consonant intervals needed to be recognized.

The three early Pythagorean intervals had been based on the simple arithmetic of the three ways in which a “mean” could divide two terms: the geometric mean in the case of the octave, the arithmetic mean in the case of the fifth, and the harmonic mean in the case of the fourth. The geometric mean, g_m, of two terms, a and b, understood according to the ratio a:g_m = g_m:b (understood algebraically, g_m = √ab), had been given special status in Greek culture but, with a number system limited to the positive natural numbers with no zero, for the most part, the idea of a square root fell outside of the scope of Pythagorean arithmetic and, so, could not be given a determinate value for non-square numbers such as 2. This meant that the geometric mean, while defining the interval of the octave itself, could not divide the octave, the task of which fell to the arithmetic and harmonic means. Algebraically, the arithmetic mean m_a of terms a and b is ${\displaystyle \frac{a+b}{2}},$ while the harmonic mean had been described as holding between three terms when “that by which the first exceeds the second is the same as that by which the second exceeds the third” (Archytas 2016, p. 247).27 This, in fact, turns out to be the mathematical inverse of the arithmetic mean.28 After Plato, what is known of the books of Eratosthenes (276–194 BCE) on Plato’s mathematics (Heath 1921, vol. II, pp. 105–6) testify to the continuing interest in these three musical means and their mediated unity.

A peculiar arrangement of the three means, called the “musical tetraktys” and comprising the numbers 6, 8, 9, and 12 arranged into a double ratio, 6:9::8:12 (Barbera 1984, p. 200), had been interpreted by a number of important neo-Platonic thinkers, including Nicomachus of Gerasa (Nicomachus of Gerasa 1926, pp. 284–286), as the “best bond” that Plato, in the Timaeus, had described as holding the diverse parts of the cosmos into a whole (Plato 1997, pp. 31b–32a). In particular, Plato had spoken of the need for two mean terms to mediate the extremes of a three-dimensional body while only one mean was required to mediate the relation of two extremes on a plane surface (Plato 1997, pp. 32a-b). In short, the three means of Pythagorean harmonic theory were seen as having properties significant at the level of the application of numbers to the measurement of space itself, quite independently of their specific application to music.29

It can be easily appreciated that a double ratio formed among the numbers 6, 8, 9, and 12 actually instantiates the harmonic cross-ratio of modern projective geometry, as the ratio of 8-6 to 9-8 (2:1) is equivalent to the ratio 12-6 to 12-9 (6:3 = 2:1). That is, the modern harmonic cross-ratio can be thought of as a generalization of the Greeks “musical tetraktys”. Without the resources of a well-developed algebra and an appropriate number system, the Greeks had exploited relations between the ratios of numbers on the one hand and ratios of the continuous magnitudes of geometry on the other, ratios that had been discovered to be incommensurable. The system of means could be employed in ways that paired magnitudes unrepresentable in the Greek number system, “irrational numbers” such as the square root of 2 (√2), with approximations expressed in ratios of natural numbers. In fact, the musical tetraktys can be used as the basis of an algorithm for the calculation of a numerical value for √2 itself.

Crucially, an early instance of a geometric construction of the harmonic cross-ratio would appear within Apollonius’s Conics (Apollonius of Perga 2000). Thus, in Book I, Proposition 34, a construction is described in which a line is drawn from a point P passing through a circle (or an ellipse or a parabola) as its diameter and cutting the circumference at points C and D, as below in Figure 7. If a perpendicular drawn to that line cuts it at point P′ so as to form a harmonic cross-ratio, PD:DC::DP′:P′C, then straight lines drawn from P to points A and B where the perpendicular cuts the circumference will touch the circle as tangents.30 This generalization of the musical tetraktys, it is clear, is closely related to the construction of the pole and its inverse with respect to a circle.

Along with the formulae of the harmonic cross-ratio, these constructions can certainly seem confusing, and a simpler way to think of such a “harmonic division” of a line is to think of there being two ways in which a line segment, DC—here the diameter of a circle—can be divided in a certain ratio. It can be divided internally, as here by P′ (the inverse of pole P) forming the ratio DP′: P′C. But, it can be divided externally in the same ratio, as it is here by the corresponding pole P, to form the ratio DP:PC. In such a case, the four points on the line stand in the harmonic cross-ratio, the fundamental invariant of projective geometry as indicated by Desargues and validated by Felix Klein.

Returning to our polygonal simplifications of Kepler’s nested Platonic solids, were we now to construct diagonals through the corners of the outer square in Figure 8 below such that they transect the circumference of its inner circle (as AC transects it at points M and N) and transect the sides of the inner square (as AC transects HE A′), it can be appreciated how deeply the harmonic cross-ratio is implicit within the relation of the two concentric circles that circumscribe their respective squares31 and, by extension, how deeply it is embedded into the three dimensional equivalents of such two-dimensional structures, the nested dual polyhedra of Kepler’s model.

Kepler was committed to the idea of finding the proportions found in the intonation of his contemporary music in the cosmos, believing it to express those most beautiful proportions entertained in the divine mind and, in 1619, believed he had found them with the ratios of the orbital speed of the planets. Were space itself assumed to be Euclidean, finding such music features would surely be in need of some external explanation. But, what if such harmonic relations were features of the structure of the space itself, as seemingly presupposed by the Greek astronomers in their empirical investigations? Then, such an extraneous explanation might be thought of as unnecessary. Kepler’s commitment to the music of the spheres might then be thought of as a variant of the “transcendental illusion” that Kant had believed responsible for the more usual belief that space was “in itself” Euclidean when, in terms of his transcendental idealism, the form of space was a construction imposed on sensory experience by the mind itself (Kant 1998, Transcendental Aesthetic, sct. I). That is, Kepler’s musical cosmos might be thought of as an illusion created by the projection of features of the explanatory resources employed, here, those of projective geometry, onto the explanandum.

As the inheritor of an established astronomical tradition, Kepler could not help but incorporate various aspects of the “amalgam” of diverse factors constituting it. In the following section, it will be shown how various features of what would later be known as “projective geometry” itself had been a part of that amalgam constituting Greek mathematical astronomy. The space of the observed and measured universe in Greek astronomy would thus be “harmonic”.

5. The Role of Projective Geometrical Relations with the Amalgam of Astronomical Practices from Eudoxus to Kepler

When Plato established his Academy in Athens in 387 BCE, Greek astronomy was already at a relatively advanced stage, but it would benefit by the rapid growth of geometry among Plato’s associates such as Theaetetus (c. 417–369 BCE)) and Eudoxus of Cnidus (c. 390–340 BCE). The Earth, by this time, had come to be conceived as a stationary spherical body, “floating” unsupported at the centre of a closed spherical cosmos,32 and, so, the cosmos was fundamentally structured by the idea of a relation between two homocentric spheres: an outer, celestial sphere, onto the inner surface of which the fixed stars were attached, able to be viewed by observers on the surface of the Earth, and the second, much smaller sphere, located at the centre of the celestial sphere. Such homocentric spheres allow the idea of any point on either sphere to be projected onto an equivalent point on the other.33

The outer sphere was understood to revolve around the Earth daily, based on observations carried out at night of the westward movements of the constellations in their counter-clockwise circling about the pole star. A similar east–west rotation, although a little slower, characterized the Sun, which rose each morning, a little behind the position relative to the stars that it had risen the day before. In relation to the celestial sphere then, the Sun appears to complete an annual circuit against the heavens, tracing a circular path, the “ecliptic”. The ecliptic, however, was tilted in relation to the celestial equator, that is, the axis on which the Sun revolved around the Earth was on a tilt (approximately 23 degrees) to the axis on which the celestial spheres rotated. This coincided with the observation that the Sun did not rise in the east at the same point on the horizon each day but, throughout the year, moved between north and south solstices in a way that coincided with the seasons. From the Babylonians, the Greeks had learnt to partition the stars close to the ecliptic on either side into the twelve constellations of the zodiac, and, for such agricultural economies, knowledge of the location of the sun with respect to the zodiac at any time was crucial for the practical know-how of when to plant and harvest.34 In addition to the movements of the Sun, those of the Moon and the five planets or “wanderers” could be similarly traced against the starry landmarks of the zodiac.

A sophisticated geometric model of the universe capturing these phenomena was put forward by Plato’s colleague, Eudoxus of Cnidus, as described by Aristotle (Aristotle 1984, Metaphysics 1073b18–30; c.f., the summary in Crowe 2001, p. 22). Eudoxus’s complex model involved 27 homocentric spheres,35 and the underlying idea of such homocentric spheres would persist with modifications through Ptolemy ultimately to Kepler’s specific version. After Eudoxus, however, more and more observational evidence would be accumulated, showing its need for further elaboration. Hipparchus of Nicaea (c. 190–120 BCE), the first to apply trigonometry to astronomy, had access to a vast collection of celestial observations from the Babylonians and attempted to deal with the anomalies of the existing approach that these data revealed. For example, the annual orbit of the Sun had now been shown to be irregular,36 and, to explain this, Hipparchus located the Earth excentrically, some distance from the geometric centre of the spherical celestial space around which all else rotated (Crowe 2001, pp. 30–31). On the other hand, irregularities had been discovered in the paths of the Moon and the planets, with planets such as Mars and Venus capable of “retrograde motion”, observed at points of their celestial circling as stopping and reversing direction, before resuming their paths.37 In relation to this, Hipparchus employed the “epicycle-deferent” system (Crowe 2001, pp. 31–37; Timberlake and Wallace 2019, chap. 4.3). In this system, a planet, instead of revolving directly around the Earth, revolved in an “epicycle” around a point that itself revolved on a circular path, the “deferent”, around the Earth. Ptolemy would continue the combination of eccentric and epicycle-deferent systems in a model that would be the accepted until Copernicus.

The measurements taken in mathematical astronomy fundamentally related to changes in the measurement of angles subtended by objects over time in relation to the observer, and calculation of relative distances between planets would rely on the phenomenon of parallax, the difference in the apparent position of an object against a background when viewed from different locations. Thus, an object such as the Moon, when observed at a specific time, would appear in a different place in relation to the stars than it would appear when viewed from some distant point on the Earth at the same time or at the same place at a later time. Here, the fixed stars provided the points of reference, like marks on a ruler, because it would not be until the nineteenth century that there existed instruments powerful enough to reveal parallax among the stars themselves.38 Measuring the angles between celestial objects of course required some sort of instrument, and the predominant instrument upon which all astronomical observation depended was the astrolabe, which, as the Arabs would later show, was constructed on “projective” principles.

In his Almagest, the most influential astronomical work from antiquity, Ptolemy had described a spherically shaped instrument—the spherical astrolabe or armillary sphere (Ptolemy 1998, pp. 217–19). It consisted of a central sphere, representing the Earth, surrounded by a series of metal rings,39 each representing one of the astronomically significant circles inscribed on the celestial vault, such as the celestial equator, the ecliptic, and the paths of the Moon and planets.40 The outer moveable band provided positions of prominent fixed stars situated around the ecliptic, sometimes divided into the houses of the zodiac, while the positions of the Sun, Moon, and planets were represented on further fixed bands that could move independently so as to be aligned positions of the fixed stars. In short, this instrument was essentially a simplified metal instantiation of Eudoxus’s theoretical model of the cosmos with its multiple homocentric spheres from which the bands were, as if, cut out. A more common and practical flat form, usually referred to simply as the astrolabe, had also been described by Ptolemy in his Planisphere or Flattening the Surface of the Sphere (Sidoli and Berggren 2007), a work lost in the Greek but which had been preserved in Arabic translation. In addition to reproducing the modelling functions of the armillary sphere, the astrolabe contained a type of quadrant for taking angular measurements and, suspended vertically, it could be used in order to find the user’s bearings.41 The astrolabe would later be perfected by Arabic scientists and craftsmen and in relation to which they would develop their “science of projection”. From Muslim Spain, it would be reintroduced into the rest of Europe in the twelfth century.

The astrolabe consisted of a series of flat plates that could turn around a central fulcrum, each performing tasks performed by the parts of the armillary sphere. The base plate, the “mother”, sat inside a raised annulus divided into 360 degrees against which angles could be measured, and on the mother was engraved a coordinate grid.42 On top of this, and able to be rotated on the mother, was a mesh-like disc, the “rete” (web), containing a ring representing the ecliptic, as well as various pointers representing the positions of prominent stars. Around these discs, and attached from the back, rotated the “alidade” or sighting arm, something like a doubled long hand of a clock, at the end of which small holes pierced raised processes, allowing visual alignment of the instrument with distant objects as with the quadrant. With this coordination of parts, the astrolabe has been described as combining the functions of “a compass, a clock, a calendar, a measuring tape, a sextant, a planetarium, an astrological ruler” (Aterini 2019, p. 242).

In order for the relative positions of lines or points as represented on a sphere to be now represented on a two-dimensional surface, a form of “projection” had to be used, but this was of a different type to the “linear” projection that would be explored in the Renaissance and that would lead into Desargues’s style of projective geometry. It was rather a “stereographic projection”. While the Greeks used the geometry of stereographic projection (in the Almagest, Ptolemy describes it and attributes it to Hipparchus, while Virtruvius had earlier attributed it to Eudoxus)43, there is no evidence that they had found proofs that would justify this use. That advance would await the Arab mathematicians of the ninth and tenth centuries (Abgrall 2015).

In stereographic projection, rays are taken as emanating from a pole of a sphere, passing through the surface of the sphere at points in such a way as to map those points onto points on a plane, conceived as passing tangentially through the opposite pole, as in Figure 9 below.

Analogous to linear projection, equalities of distance are not preserved in stereographic projection, but certain other relationships between points are preserved, analogous to the preservation of the cross-ratio in linear projection. Most importantly for the operation of the astrolabe is the preservation of circles on the sphere when projected onto the planar surface, as in Figure 9, where the circle C is projected onto the circle C′. If a circle on the sphere, like the meridians, pass through a pole, they are onto straight lines.44

The proof of the invariance of the circle within stereographic projection would be first given by a mathematician working in Bagdad, Abmad al-Farghani, in a book devoted to the constructure of the astrolabe (Al-Farghani 2005; Abgrall 2015, sct. II). Not surprisingly, given the basically conical nature of the projection of circles onto circles, it seems clear that al-Farghani had built upon the approach to conic sections in Apollonius’s Conics. That is, the principles of stereographic projection were effectively grounded in the same geometry as that of the cross-ratio of linear projection.

After the use of the astrolabe had spread into the rest of Europe from Arabic Spain around the twelfth century, a simpler but more robust version suitable for use at sea, the “mariner’s astrolabe”, would be used by Portuguese navigators in the fifteenth century (Mott 2007). By the sixteenth century, different variants of the astrolabe, including cheap paper constructions, were being widely used for surveying and navigation as well as astronomy and astrology. In the last third of that century, another form of non-linear projection would be introduced by Geradus Mercator, a Flemish globe, map, and instrument maker. In 1569, Mercator published the first map using the now familiar “Mercator projection”, which, somewhat analogous to the astronomer’s stereographic projection, projected points on a spherical surface—in this case the terrestrial sphere—onto a flat surface but now conceived as wrapped around the sphere to form a cylinder from which it could be “unfurled”. In the Mercator projection, great circles on the globe were mapped onto straight lines on the plane, in this case, the longitudinal meridians meeting at the poles on the sphere appearing as parallel lines. As with stereographic projection, Mercator projection is “conformal” or angle-preserving, and with a Mercator map, sextant, ruler, and protractor, navigators could plot their journeys as straight lines (“rhumb lines”) on the map and with a mariner’s astrolabe, at any time, locate their own position on the map.45

In the last quarter of the century, Tycho Brahe would be able to take advantage of this great development of instrumentation. Having been given control of an island, Hven, located between Denmark and Sweden, he had there established a castle/laboratory filled with such types of instruments, often designed by himself and capable of measuring angles with unprecedented accuracy (Timberlake and Wallace 2019, p. 154).46 After the death of his patron, the King of Denmark, Tycho would relocate to Prague, where he would soon be joined by Kepler. The patron there of both Tycho and Kepler was the Holy Roman Emperor Rudolf II, being said to have possessed eight astrolabes. Figure 10 below shows an instrument combining an armillary sphere at the top with an astrolabe below, constructed by the Swiss watchmaker, Jost Bürgi, who had worked with both Tycho in Hven and Kepler in Prague. This instrument, currently located in the Nordiska Museet in Stockholm, is described as having been used by Kepler.

In his Optics, Kepler had shown himself to be working within the spirit of a tradition that, starting with the Greeks, had become developed into a systematic science in the Arab world in the centuries of its great scientific development. Even had Kepler been unconscious of the way that the tradition he worked within was shaped by projective geometry, this could still be described as implicit in his practice. Nevertheless, it seems clear from his conception of the geometry underlying his innovations in optics that he would have had the “feel” of such projective features of space. In conclusion, we might turn to differences separating Kepler’s attitude towards Copernicanism to that of Giordano Bruno in order to see their relevance for Kepler’s actual calculations.

6. Conclusions: Two Senses of the “Copernican Principle”

A sign that Kepler was self-consciously relying on those projective features of astronomical space implicit within the ancient tradition of mathematical astronomy might be found in his rejection of the cosmology of another infamous Copernican at the time, Giordano Bruno, as Kepler’s Copernicanism was not that of Bruno’s. From a contemporary point of view, Bruno’s seems distinctively more modern, as he had hypothesized that the fixed stars were suns like our own, with their own exoplanets perhaps supporting life. As Field points out, however, in neither Mysterium cosmographium nor Harmonice mundi did Kepler embrace such Brunoean features of the universe, even though he was certainly aware of Bruno’s claims (Field 1988, p. 17).

While this conservatism has been attributed to religious factors, Field assesses Kepler’s rejection of Bruno’s infinite universe in terms of his adherence to what could be established at the time by empirical methods. “If the Universe is immense or infinite, in the sense that we must believe it to contain objects we cannot observe, such as stars that are sometimes too far away for us to see … then we cannot know how to construct theories to explain what we observe.” “[I]n modern terms” she adds, “the word ‘Universe’ must be taken to mean ‘observable Universe’” (Field 1988, p. 18). In a similar spirit, Christopher Graney positively contrasts Kepler’s heliocentric universe with Bruno’s centreless one: “[I]n the early seventeenth century, science reveals the Copernican universe to consist of exactly that which Kepler describes in the sixteenth chapter of De Stella Nova: a vast shell of huge but dull stars, surrounding a tiny but brilliant sun and its lively planets, and at least one of these planets teems with life; and among that life are these motes of dust called human beings, which beget themselves, clothe themselves, and so forth. A universe of Sun-like stars, on the other hand, is the creation of those who do not do their science carefully enough…. It is not the universe observed by careful astronomers” (Graney 2019, p. 166). If science is an amalgam of various techniques, forms of instrumentation that cannot be entirely separated from the explicit consciously held theories with which science is often identified, the components of that amalgam will clearly be subject to historical change. We now have observational evidence that supports Bruno’s belief that the stars are other suns, but that evidence was not available to Bruno and his contemporaries. While regions of the universe unimaginable to Kepler or Bruno have been opened up by instruments such as the James Webb Telescope, we are still not in a situation entirely different to Kepler and we can conceive of regions of the universe that are still beyond observation—for example, regions from which light has yet to reach the Earth.

Field notes how, in a diagram in Ch. XIV of Mysterium cosmographicum, a narrow space separates the “circle of the Zodiac” from the orbit of Saturn, but this space is described as being “like infinity” (Field 1988, pp. 41–43). That is, for the purpose of calculating the relative distances of the planets from the Sun, the distance of the stars from the Sun can be considered as infinite—an infinity meant in the sense of “immeasurable” or “immense”, and, in comparison to which, the orbits of the planets would be considered “insensibly small” (Field 1988, p. 43). In relation to the original geocentric model, Aristotle had argued that the fixed stars could not be at an infinite distance from the Earth as this would imply that, in their diurnal movement, they covered a similarly infinite distance and did this in a finite time (Field 2024, p. 25). The move to heliocentrism, however, undermined Aristotle’s argument, as the stars no longer move. Might not such an infinite distance of the stars from the Sun and the Earth be understood on the model of those “points at infinity” within projective space that Kepler had come up with in relation to his theory of optics? These are points that Kepler had conjectured to exist from his understanding of the transformational relations among the conic sections, and they were points that Desargues had related to the vanishing points of perspectival representations. Might not the fixed stars provide a similar three-dimensional analogue of such vanishing points?

In projective geometry, points at infinity can be understood in a determinate way because they come to stand in determinate relations to finite points because of the peculiarity of projective geometry’s central “invariant”, the harmonic cross-ratio, which, as we have seen can be conceived in terms of the harmonic properties of a pole and its inverse, with respect to points on a circle, as indicated in Figure 11a.47

The Arab mathematicians, as well as Kepler and Desargues, had thought of diagrams of the sort found in Apollonius’s investigations of conic sections as dynamic rather than static, and this idea of movement is relevant here. If pole P is thought of as moving to the left away from the circle, its inverse, P′, will move to the right towards the circle’s centre. A consequence of the harmonic cross-ratio will be that, as P′ approaches the centre of the circle, P will approach infinity. Thus, when P′ reaches the centre, the tangents radiating from P will be parallel—in projective terms, they will meet at infinity, exemplifying the difference between the Euclidean and projective planes. Nevertheless, the equivalence of the ratios involved in the cross-ratio will still be thought of as holding. Here, CP′: P′D will be equal to −CP:PD.48

Kepler does not seem to have ruled out the possibility that, sometime in the future, observations might provide evidence for Bruno’s hypothesis, but restricting “the universe” to the then observable universe was methodologically crucial. Accepting Kepler’s immeasurable, or “immense”, as equivalent to a projective sense of points at “infinity” allows us to understand how the fixed stars could be used as reference points for the measurement of the distances of the planets via the phenomena of parallax. With the limitations of observational instruments available to Kepler, the stars, showing no parallax among themselves, could be thought of as equidistant from the Sun and at an infinite distance in comparison to those other wandering heavenly bodies. Despite this distance, they could still be regarded as “observable”, albeit only as luminous points. With this application of projective geometry’s “points at infinity”, we might now think of the “Copernican principle” as being employed in different ways by Kepler and Bruno.

Without Kepler’s distinction between the universe and the observable universe, Bruno is presumably to be understood as presupposing the space of the universe as uniformly Euclidean. Thus, applied without any projective sense of space, the Copernican principle would be interpreted by him to give a conception of the difference between how the universe appears from some specific point of view within it (the Earth) and how it really is, or, in a more seventeenth-century idiom, how it appears from God’s “point of view”. And, as no human observer can be understood to be located at such a “view from nowhere”, this interpretation of the Copernican principle could not be a part of empirical astronomy itself. Rather, it could only play the role of a type of superadded “metaphysical” gloss. With Kepler, however, the Copernican principle operating within projective space does not oppose a view of the universe from somewhere within it to that from God’s point of view. Rather, it conceives of space more in terms of the reciprocity of inversely related points of view within it—a reciprocity given expression in the harmonic cross-ratio, the invariant providing the underlying metric of projective space, the space of the observable universe bounded by the sphere of the fixed stars.

Kepler may have strayed into the territory of such non-empirically based metaphysical conceptions with his embrace of the thesis of the music of the spheres. This cosmic “music” had to be perfect because it was the artistry of the divine and, hence, perfect musician. This clearly had a “neo-Platonic” provenance, and the neo-Platonists certainly had also extended Plato’s ideas about space from being astronomical notions to “metaphysical” or “mystical” ones. However, I have suggested another and different way in which the mathematics of “harmonic” musical intervals had been implicit in Plato’s cosmology and central to Kepler’s empirically based astronomy. Ideas about the mathematical unity of the three Pythagorean musical means had been transmitted to modernity by the neo-Platonists of late antiquity in the thesis of the music of the spheres. However, they had also been transmitted along a different path from Eratosthenes and Apollonius in the century after Plato to Kepler via Arab mathematicians and astronomers. Moreover, they had been transmitted not simply in the form of concepts and theories said and written down but in ways of carrying out observations and in constructing instruments built to aid it.49