See the Bold Shadow of Urania s Glory, Immortal in his Race, no lesse in story: An Artist without Error, from whose Lyne, Both Earth and Heaven, in sweet Proportions twine: Behold Great Euclid. But Behold Him well, For tis in Him Divinity doth dwell. - George Wharton (1617-1681) Chapter 2) The Euclidean Tradition Fig. 1.1) Euclid {from Thevet??} 1
Introduction: Number, Form, and the Promise of Mixed Mathematics Wharton s ode suggests something of the veneration Euclid enjoyed in the early modern period as well as the richness of the scholarly activity associated with geometry: astronomy, art, geography, astrology, and even theology. Although honored here as a vessel of Divinity, neither Wharton (a well-published astrologer and poet active at the height of the English civil war), nor his contemporaries, nor even we have much solid information about the life of Euclid. His dates of birth and death are unknown. We know, however, that he moved from Athens, where he probably studied at the Academy sometime after Plato s death, to Alexandria in Egypt where he established a school of mathematics and completed his Elements around 300 B.C.E. Despite the near anonymity of its author, the Elements itself became the most widely-known scientific work from antiquity; it has been in continuous publication for more than two millennia and, by one estimate, has undergone more redactions than any other book in the Western tradition, the Bible excepted. The enduring fascination with the Elements surely lies in the intellectual beauty of its axiomatic-deductive structure. Commentators as distant in space and time as Augustine, René Descartes, Thomas Hobbes, and even Albert Einstein have repeatedly emphasized the aesthetic experience of seeing the logical necessity of a given proof. To understand the Euclidean tradition, then, it is important to experience for oneself such an aesthetic moment. Beginning with a handful of self-evident definitions, postulates, and common notions, Euclid gradually builds up a body propositions deduced from these axiomatic statements. His arrangement of propositions gradually takes the student to higher and higher levels of sophistication in a crescendo of interconnected proofs. In the example chosen here, Proposition 47 of Book I (the Pythagorean theorem) calls upon Propositions 46, 14, 4, 41 and Common Notion 2 for its proof. Proposition 46, in turn, requires Proposition 31, 34, 29 and 34. And so on. In 2
all, the proof of the Pythagorean implicates three Definitions, all five Postulate, four Common Notions, and twenty-six Propositions. The other propositions presented here have historical as well as geometrical significance: according to legend the pentagram was the secret insignia of the Pythagorean community; the mean extreme ratio has been widely employed in artistic and architectural composition; and the geometry of the five regular (or Platonic ) solids figure both in Plato s theory of matter and Kepler s theory of the cosmos. But as Wharton s poem implies, the deductive power of technical geometry is only a part of the Euclidean legacy. Well before the composition of the Elements, the certainty and beauty of mathematical demonstration inspired flights of philosophical speculation. That tradition of speculative mathematics has its origin in the life and work of Pythagoras of Samos (fl. 6 th century B.C.E.). We know even less about Pythagoras life than we do about Euclid s, and none of his writings have survived. Much of what we know comes from Iamblichus (c. 270-330), who wrote on the Pythagorean life in the early 4 th century C.E. Iamblichus was a Greek-speaking Syrian trained in the neo-platonic tradition. He hoped to revive Pythagoreanism as a religious movement to counter the growing influence of Christianity (the Roman Emperor Constantine, c. 280-337, had converted to Christianity in 312). Whatever his personal motivations, his work still provides us with the most complete, if idealized, picture we have of Pythagorean thought. One of its central tenants is the notion of the well-order soul as the medium for comprehending a well-ordered universe or cosmos, a term which, along with philosophy, Pythagoras coined. The chief goal of the Pythagorean life is to bring his or her the Pythagorean school as almost unique in admitting women individual soul into harmony or attunement with the world soul. Harmony in the universal soul came about by the imposition of well-order limits and proportions, expressible in terms of number and figure; and harmony in the human soul arose in part from the comprehension of these ordering principles. Music, therefore, was not merely the pleasing sound of voice or instrument. Rather, it became a metaphor for universal 3
order and intelligibility. Hence the discovery of the numerical relations underlying musical harmony carried with it both spiritual as well as scientific meaning. For Pythagoreans, music is geometry for the ear just as geometry is music for the eye. Thus geometrical demonstration, insight into numerical relations, and the close observation of natural phenomena were understood to be essential elements in Pythagorean spiritual life. In the work of Leon Battista Alberti (1404-1472), we find a very different line of influence deriving from the Euclidean tradition. After receiving an excellent humanist education at Padua and Bologna, Alberti distinguished himself as a Latinist at an early age. He moved to Florence while in his twenties and quickly became friends with several of the leading Florentine painters, sculptors, and architects. He also worked as an architect and wrote, in broad imitation of the recently recovered architectural treatise by Vitruvius, his own treatise De re aedificatoria. Although he held an administrative position in the papal court for most of his adult life, his tastes tended much more toward the humanistic and naturalistic sensibilities characteristic of the Italian Renaissance. In contradistinction to the speculative, theocentric, and somewhat austere use Cusa made of geometry, Alberti saw in it the means of advancing artistic expression. His treatise On Painting (1436) contains the earliest published account of vanishing-point perspective, one of the great discoveries at the heart of Renaissance naturalism. The challenge artists of his generation set themselves was to treat the canvas as a window upon which they sought to capture what the eye actually saw. In order to accomplish this aesthetic goal, Alberti argues, the artist must master the geometrical proportions that govern the nature of things seen. Although Nicholas of Cusa (1401-1464), or Cusanus, was an almost exact contemporary of Alberti, he was of a very different mind with regard to the utility of geometry. Nicholas was from north of the Alps from the town of Cusa in {??}, he was less touched by the humanist stirring than was Alberti but of deeper religious conviction. Indeed, he became a cardinal {??}, and his writings place him among the most 4
important theologians, jurists, and philosophers of the fifteenth century. What we find in his writings is evidence of the intimate association in his mind not just of natural philosophy and scripture which we have seen before ut of geometry and theology. If there is a dominant theme in Cusa s writings it is his doctrine of learned ignorance. The human mind, when confronted by the vastness of the world and the infinite power of god, shrinks to near nothingness. Although Cusa himself was an erudite man active who greatly valued learning, his message emphasized the depths of human ignorance when compared to the knowledge possessed by god. Despite his appreciation of the certain and true knowledge that one could extract from geometrical demonstrations, he uses geometry especially the geometry of infinites to illustrate his doctrine of learned ignorance. His discussions of the infinite straight line or the circle or sphere of infinite diameter leave behind the technical challenges of deductive demonstration. Nor does he see geometry as a tool for understanding relations and measures in the physical material world. Rather, he makes geometry into a tool of the imagination and uses the mathematical entities of figure, ratio, and proportion as aids in his contemplation of the incomprehensibility of god. Thus as a theologian, Cusa recommends geometry because it is "good to think with" when contemplating god. In the hundred and fifty years following Cusa's death, from roughly 1475 to 1625, the European tradition in Euclidean geometry experienced an unprecedented period of expansion and inventiveness. While the reasons for this explosion in mathematical activity are complex, two structural changes in the world of learned culture stand out from the rest. First, the invention and spread of the printing press made possible the mass production of books of all sorts, including mathematical works. Second, the rapid increase in the number of colleges and universities in the sixteenth and early seventeenth centuries led to an increase in demand for copies of primers in geometry. The efforts of humanists to recover the literary heritage of the Ancients extended to the mathematical sciences as well as the humanities. The first printed edition of Euclid s Elements appeared in 1482. But since it was based on a defective medieval copy, it 5
was soon replaced by a new Latin translation from the Greek published in 1505, and a Greek edition followed in 1533. With scholarly versions readily available, the stage was set for a series of translations into vernaculars and for the publication of the Elements edited for use in the classroom. Translations into Italian (1543), German (1562), French (1564), English (1570), and Spanish (1576) all went through multiple editions, while numerous editions in Latin made their way into virtually every Protestant and Catholic university in Europe. It was Sir Henry Billingsley, naval captain, merchant adventurer, and future Lord Mayor of London, who published the first English translation of the Elements in 1570. And it was John Dee (1527-1608), court mathematician to Queen Elizabeth, natural philosopher of the occult, and future conjurer of angels, who wrote the Mathematical Preface and annotated the translation. Writing the Preface (ostensibly) for the young gentlemen to whom the book was directed, Dee took the opportunity to philosophize on the nature of mathematical knowledge and flaunt his classical learning. He also provided his readers with the most thorough-going inventory of Renaissance mathematics published in the sixteenth century. While his review included the speculative branches of mathematics, like theosophy the study of the ways the human soul is able to apprehend mathematical forms and thaumaturgy the uses of mathematics in the practice of magic Dee was keen to celebrate the fields of mixed mathematics. Roughly equivalent to the modern notion of applied mathematics, mixed mathematics covered all fields in which the abstract, imaginary entities of geometrical figure and number were used to describe concrete, physical phenomena. The term mixed derived from traditional Aristotelian philosophy, which defined mathematics and physics as two distinct and separate activities since their objects of study were of different ontological status, one ideal the other real. In his breathless tour of mixed mathematics, Dee summarizes every conceivable mathematical field the names of many of which he seems to have invented for the occasion from anthropography to 6
zography. 1 His seemingly curmudgeonly objection to the term geometry is in fact well grounded; Euclid himself never used the term in his work since in his day it pertained to the surveying of land. Alas, Dee s suggested alternative, Megethologia (from the Greek for study of magnitude ), seems not to have caught fire, for his is the only recorded usage of the term in the English language. Dee concluded his Preface with a table or Groundplat to show the hierarchical relationships among his many divisions of mathematics. In the decades following Dee s death, political turmoil had intensified, religious and doctrinal divisions between Protestant and Catholic had deepened, and among European scholars there was a growing sense of despair about the long-held belief in the unity of knowledge. Humanists recovery of works from a number of minor philosophical school from antiquity, like the Epicureans, Stoics and especially the Skeptics, made the task of reconciliation seem impossible. The one fixed star in this turbulent sky was geometry. Solidly founded upon self-evident axioms and guided at each step by Reason, the system of rigorously demonstrated propositions seemed to offer a paradigm for true and certain knowledge. {TBA: Section on Giuseppe Biancani to be inserted here} And thus Biancani Catholic priest, Jesuit theologian, and committed mathematician believed he had convincingly defeated those who would doubt the certainty of geometrical demonstration. Now, if only philosophy could be so securely grounded. This was the dream of René Descartes (1596-1650). In fact, the power of the geometrical way seems to have revealed itself to Descartes in a dream. After completing his education at the Jesuit College of La Flêche near Paris, Descartes took up the life of a gentleman-soldier. He 1 See Chapter 2) The Euclidean Tradition, 2-P5, pp. 000 below for Dee s colorful definitions of mathematical fields. 7
found himself in Bavaria in the autumn of 1619, and on the night of November 10th, 1619 Descartes was just 23 years old he had a series of three dreams. These dreams deeply impressed him. He understood them to be both divinely inspired and decisive for the subsequent course of his intellectual career. In the first dream, Descartes found himself walking the streets of a town, but because of a weakness in his right leg he was unable to walk upright. A strong wind whirled him around several times. He tried to reach the Jesuit church but changed his mind. A strong wind attempted to throw him against the door of the church, but unsuccessfully. Descartes awakened and was deeply fearful of his sins, his spiritual infirmity, and his need for moral justification. In the second dream, Descartes heard what he believed to be a thunderclap. When he awakened, his room was filled with sparks of light. In the third dream he found a book on his table. In it he read, "What road shall I follow in life?" Descartes himself tells us what these dreams meant to him. The first showed him the errors and sinfulness of his past. The thunderclap and sparks of light of the second dream signaled the "descent of the Spirit of Truth." And the third told him that his mission in life was to construct a new philosophy. At the heart of Descartes new philosophy, which he felt he understood intuitively and completely, was what he later called the mathesis universalis, or universal science. He believed this to be a sort of mathematical philosophy capable of uniting all the sciences into a single grand, comprehensive and comprehensible system. Descartes felt he had grasped the proper method of the new science, based on mathematics and with the entire universe as its domain. Descartes unfinished Rules for the Direction of the Mind was his earliest attempt to systematize the program for the reform of knowledge that he laid out more succinctly in his Discourse on Method and Meditations on First Philosophy. 2 On the level of technical mathematics in contradistinction to speculative philosophy inspired by mathematics Descartes s major contribution was his method of 8
analysis into geometry proper, known to use as analytic geometry. Descartes s Géometrie was a remarkably fertile contribution to a century during which mathematics bloomed. Within a generation of his death, Newton and Leibniz had synthesized the calculus, both integral and differential, from the extraordinary products of their contemporaries. In the two hundred years since the appearance of the first printed edition of Euclid s Elements, mathematics had undergone enormous transformation. By the end of the seventeenth century, the spirit of the humanists who had sought wisdom by returning to the sources, had given way to a growing awareness of the conceptual distance that separated Ancients from Moderns. Writing at the very end of the century, William Wotton (16??-17??), a member of the Royal Society of London and an accomplished naturalist, offered a comprehensive assessment of ancient and modern learning. Unsure of his own grasp of recent developments in mathematics, he commissioned his friend, John Craig (16??-17??), to write the section on ancient and modern mathematics. 3 A very different sort of inventory from Dee s, Craig s comparison point up the limitations of ancient geometry and arithmetic while clearly summarizing the chief advances of the sixteenth and seventeenth centuries: the invention of logarithms as an add to calculation, the utility of conic sections in optics and astronomy, the union of algebra and geometry to solve problems involving curves, general methods for finding solutions to equations of the third degree and higher, and the method of indivisibles (part of the calculus). 2 See Chapter 10) Epistemology & Methodology, 10-P5, pp. 0000 below. 3 See Chapter 13) Ancients & Moderns, 13-P4, pp. 000 below. 9