The Spatial Reformation

Euclid Between Man, Cosmos, and God

Michael J. Sauter

• Publisher: University of Pennsylvania Press
• Year: 2018
• ISBN:
  9780812295559
• Topic: Histoire
• Subtopic: Histoire des sciences

In The Spatial Reformation, Michael J. Sauter offers a sweeping history of the way Europeans conceived of three-dimensional space, including the relationship between Earth and the heavens, between 1350 and 1850. He argues that this "spatial reformation" provoked a reorganization of knowledge in the West that was arguably as important as the religious Reformation. Notably, it had its own sacred text, which proved as central and was as ubiquitously embraced: Euclid's Elements. Aside from the Bible, no other work was so frequently reproduced in the early modern era. According to Sauter, its penetration and suffusion throughout European thought and experience call for a deliberate reconsideration not only of what constitutes the intellectual foundation of the early modern era but also of its temporal range. The Spatial Reformation contends that space is a human construct: that is, it is a concept that arises from the human imagination and gets expressed physically in texts and material objects. Sauter begins his examination by demonstrating how Euclidean geometry, when it was applied fully to the cosmos, estranged God from man, enabling the breakthrough to heliocentrism and, by extension, the discovery of the New World. Subsequent chapters provide detailed analyses of the construction of celestial and terrestrial globes, Albrecht Dürer's engraving Melencolia, the secularization of the natural history of the earth and man, and Hobbes's rejection of Euclid's sense of space and its effect on his political theory. Sauter's exploration culminates in the formation of a new anthropology in the eighteenth century that situated humanity in reference to spaces and places that human eyes had not actually seen. The Spatial Reformation illustrates how these disparate advancements can be viewed as resulting expressly from early modernity's embrace of Euclidean geometry.

Chapter 1

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From Sacred Texts to Secular Space

Arithmetic and geometry are, therefore, the wings of the human mind.

—Philipp Melanchthon, “Praefatio in arithmeticen” (1536), in Corpus Reformatorum Philippi Melanthonis operae quae supersunt omnia, ed. Bretschneider and Bindseil

The Protestant Reformation and the Spatial Reformation were interconnected in a fundamental way: both movements cultivated a sacred text that was assiduously translated into the vernacular and evangelized by true believers. The texts in question were the Bible (naturally) and Euclid’s Elements, which may seem less obviously sacred. Although intellectual historians do not usually juxtapose these works (and not in this way), there are good reasons to do so. First, the Bible and the Elements entered print roughly concurrently, as the former appeared in 1454 and the latter in 1482, with flurries of editions blanketing the continent thereafter.¹ Second, their respective narratives intersect in an overlooked way, as both begin with and build on nothing. The Bible’s first book, Genesis, recounts God’s creation of the cosmos ex nihilo and affixes to that moment a providential history that assures humanity’s destiny in the next world. The Elements, meanwhile, begins with substanceless points and perches on top of them a system of projection that makes idealized space applicable to real space, that is, to the physical realm.²

The express juxtaposition of God with Euclid reveals that beneath early modern thought’s main currents swirl profound changes in space’s intellectual context. Both the Bible and the Elements offer coherent ways to think about humanity’s position not only within the physical cosmos, but also in reference to an ideal realm. Yet their respective spaces were incompatible, since Christianity emphasized hierarchy and discontinuity, whereas Euclidean geometry insisted on uniformity and homogeneity. Before continuing in this vein, however, I summarize Christianity’s reception of hierarchical space. Between roughly 350 and 1350, Christian thinkers applied biblical cosmology to antiquity’s remnants. In doing so, they overlaid onto pagan philosophical constructions of space their own views on the hierarchy that suffused Creation. According to the Bible, God placed the Heavens “above” and fashioned humanity from the elements that rested “below.” The classical heritage, in turn—including especially Neoplatonic and Aristotelian traditions—justified the same physical arrangement, if along more philosophical lines. The ensuing synthesis yielded, in turn, a geocentric cosmos inside which humanity was situated both religiously and philosophically. Any imagined journey “upward”—whether from profane to sacred, or from Earth to Heaven—was defined, thus, by hierarchy and discontinuity.

The reappropriation of Euclid’s Elements after 1350 (and in Nominalism’s wake) upset this exquisitely balanced applecart. First, by moving from nothing to perfect figures that were suspended within idealized space, the Elements suggested that perfection was not necessarily limited to realms above. Second, as human beings applied this idealized space to the real world, including in the production of material culture, it seemed to them that humanity might not have been so utterly inferior to God, after all. The Renaissance Cardinal Nicholas of Cusa, whom I have already discussed, was a student of geometry and arithmetic and, not coincidentally, groped his way to a strikingly positive anthropology. In the mid-fifteenth century, he wrote: “Therefore, as the human world emerges, all things are in fact explained humanly (and with respect to the universe itself, universally). Indeed, all things are folded up humanly in themselves, since the human is [also] God.”³ And this is to say nothing of homogenous space’s implications for medieval cosmology and theology, both of which had previously absorbed fractured space. We can thus begin to understand why the Western triad came under such stress, when European spatial sense shifted toward geometric homogeneity.

In spite of Euclidean space’s prominence within early modern thought, there is currently no broadly conceived analysis of its effects on European intellectual history. Classic intellectual-historical works on the late Medieval, Renaissance, and Reformation eras do not concentrate on Euclid’s Elements and its attending spatial sense.⁴ The situation is similar for work on the seventeenth and eighteenth centuries, which emphasize early modern thought as a radicalizing prequel to the French Revolution.⁵ As for the nineteenth century, homogeneous space has hardly been an issue. University-level surveys, for example, still serve traditional historical-philosophical cocktails of Hegelianism and Marxism, without covering geometry’s influence on Hegel himself.⁶ Concomitantly, classic works on Hegel and Hegelianism foreground neither Euclid nor ancient geometry.⁷ Yet, as I noted in the introduction, Hegel studied geometry as a youth and, later, wrote the minor work Geometrical Studies, in which he analyzed Euclid’s arguments.⁸ Even in the simplest biographical terms, it is clear that Hegel’s mature thought arose within the context of a billowing sense of homogenous space. The scholarship has, however, largely overlooked this point.⁹ In addition, those intellectual-historical works that do confront space in the nineteenth century concentrate on the fin de siècle, by which point the Spatial Reformation was dissolving.¹⁰ Thus, on the whole, it seems safe to say that the contemporary scholarship has elided Euclidean space from intellectual history.

With space’s absence in mind, I turn to the quote from Philipp Melanchthon in this chapter’s epigraph. Coming from one of the Protestant Reformation’s driving forces, the idea that arithmetic and geometry form the wings of the human mind casts light on how the Elements came to serve as not only an ersatz sacred text but also a corrosive philosophical agent. In natural philosophical matters Melanchthon was a thoroughgoing Aristotelian.¹¹ Yet, in spite of the spatial commitments that went with that, the praeceptor Germaniae also dabbled in geometry in a way that undermined Christian Aristotelianism’s hierarchical space. A particularly good example comes from a preface that Melanchthon wrote for a 1536 Latin edition of the Elements: “No one without some knowledge sees enough of this art, which is life demonstrated. No one without it will be a maker of method. . . . There is here great praise of geometry, which did not cling to inadequate and inferior [human] constructions, but flew into Heaven and transported human minds, which were stuck in the mud, back up to the heavenly throne.”¹² If we interweave this quote with the epigraph above, we catch another glimpse of geometry’s implications for European thought: any journey “upward” strained the theological and philosophical bonds that had long tethered God’s children to His cosmos’s center—and Euclid’s Elements gave European thought the means to cut them.

Homogeneity’s Sacred Vessel

In the wake of medieval Nominalism’s spread to the Continent, European thinkers became increasingly interested in incorporating mathematics into their knowledge of both the cosmos and God. By the end of the fourteenth century, the most important center of mathematical study was the Papal curia in Rome, which was slowly confronting the need to recalculate the calendar.¹³ Rome was also becoming Europe’s chief repository of ancient mathematical texts and, not coincidentally, was attracting the continent’s best mathematicians, as in the example of Johannes Regiomontanus, who was recruited from the University of Vienna. The continual enlistment of foreigners produced a powerful mixture of intellectual currents, as the northern universities from which these people hailed often cultivated the Nominalism that had spread from Oxford to Paris. Of equal significance, however, was the continual arrival in Italy of Greek manuscripts from the city of Constantinople, which was then being threatened by Ottoman forces. Among the most important among the returning works were Greek-language copies of Euclid’s Elements. None of these manuscripts had ever been translated out of the original tongue (medieval Latin versions of the Elements usually came from Arabic translations), and this made it seem that “uncorrupted” copies of the original had finally arrived. Of course, more than a millennium of copying had corrupted them, too. This aspect of Euclid’s second “return,” however, is another story.

After coming under intense study in the fifteenth-century Papal court scene, the recently returned Elements took the first step toward cultural ubiquity, when a Latin translation entered into print in 1482 in Venice.¹⁴ Although medieval thinkers had enjoyed access to Latin versions of Euclid’s work since the twelfth century, on the whole the medieval encounter with geometry did not compare to the early modern experience in terms of either its breadth or its intensity. From the late fifteenth century until the early nineteenth century, waves of editions appeared in multiple countries and in multiple languages, producing a web that not only reached across Europe but also extended to its colonial settlements. By 1650, at the very latest, there was no way that an educated European could have been ignorant of the Elements’ lessons, any more than he or she could have been ignorant of the Bible. Indeed, in the history of the book, the Elements may be second only to the Bible in the number of editions, with some scholars estimating the total to be over a thousand.¹⁵

The Elements’ print diffusion was the precondition for a development that I will call spatial secularization. Spatial secularization constituted the continual distancing of the divine from both humanity and the cosmos, with homogeneous space filling the resulting gap. My approach breaks sharply with regnant views. Whereas I emphasize humanity’s imagined relationship to the divine, other scholars generally concentrate on the retreat of religious institutions from society. Along these lines, traditional approaches chronicle how the church receded from public life, only to be replaced by more secular institutions—whether economic, political, or social.¹⁶ I read secularization, however, with respect to changes in the divine that emerged from space’s advance.¹⁷ In this respect, spatial secularization is a successor to the fourteenth-century Nominalist reappraisal of God’s will, which separated God from the cosmos in a way that allowed humanity to apply mathematics to the whole. Not coincidentally, the projection of a regularized space onto the cosmos became the dominant mode of knowing things.

To contextualize secularization’s history via homogeneous space is important precisely because we moderns stand beyond weighty changes in spatial thought. We have a peculiar perspective on space, because we understand Euclid’s geometry, even as we have learned to see it in post-Euclidean terms. This situation has profound implications for the history of secularization. First, the Elements remains present in contemporary education, in a way that the Bible simply no longer is—which means that we see no inherent conflict between space and God, although there long was one. Second, we do not take the Elements seriously as an historical force, precisely because its geometry and, above all, its sense of space have become so innocuous. As Dostoevsky noted, there are other geometries out there, whose implications are more radical and profound than anything that one finds in Euclid; and as Dostoevsky showed us, in this context Euclidean thought can become a refuge for the religious mind.

I illuminate further the differences between my view of secularization and contemporary approaches by looking at our own experience with geometry. The mathematical sequence in secondary education expressly includes a year’s study of Euclidean geometry. Here, I call attention to the year 2000 edition of the high school textbook Geometry by Ray C. Jurgensen. The text begins by emphasizing geometry’s utility and notes that we can use the discipline to find the distance between things on earth, such as a pole, a fountain, and a tree, before also appending an illustration and a proof-like discussion.¹⁸ It then explains how the position of a real thing can be reduced to an imagined point: “Each dot on a television screen suggests the simplest figure studied in geometry—a point. Although a point doesn’t have any size, it is often represented by a dot that does have some size.”¹⁹ Now, compare this rendition to the Elements’ opening line: “A point is that which has no part.”²⁰ Although Jurgensen included contemporary referents in his explication, the sense of the original remains, given that the modern version begins with nothing and heads toward something. Consistent with what I said about the Bible’s space, it is important to underline that Jurgensen’s attitudes toward space are predicated on God’s absence. Thus, where the sixteenth-century pedagogue Melanchthon saw in geometry a potential path to God, his twentieth-century counterpart, Jurgensen, associated Euclid with his own television’s pixels.

The Elements’ diffusion in schools makes it a unique intellectual historical phenomenon. If we take the initial 1482 print edition as a starting point, this text has been in print (and in use) for over five hundred years. The same can hardly be said of any other Western work. Historians of mathematics may object that ancient discussions of conic sections are still present in mathematical education, which is true. Nevertheless, three things mitigate this objection’s force. First, today’s students arrive at conic sections only after having studied Euclidean geometry’s basics. Second, in the course of the early modern period, ancient writings on conic sections were never printed as often as the Elements. Euclid is, in this respect, the intellectual tie that binds, with his greatest work serving as the alpha and the omega of the passage from early modernity to modernity.

Euclid’s contemporary ubiquity, however, has contorted...

Table of contents

• Cover
• Half title
• Title
• Copyright
• Dedication
• Contents
• Preface
• Introduction. The Spatial Reformation
• Chapter 1. From Sacred Texts to Secular Space
• Chapter 2. The Renaissance and the Round Ball
• Chapter 3. Divine Melancholy
• Chapter 4. Eden’s End
• Chapter 5. Modest Ravings
• Chapter 6. Strangers to the World
• Conclusion. Prosaic Reflections
• Notes
• Bibliography
• Index
• Acknowledgments