Euclid
EUCLID (c. 300 BCE) was a Greek mathematician. Plato described mathematics as a discipline that turns one's gaze from the Becoming of the sensible world to the Being of the intelligible. The great value of mathematics is to prepare the mind for the apprehension of pure ideas. After Plato's death, geometry flourished among his students. One of the few details known about Euclid's life is that he studied under Plato's followers. Subsequently he founded the great school of mathematics at Alexandria, Egypt. He wrote on mathematics, optics, and astronomy.
Euclid's Elements is the most influential work in all of mathematics. Though other "Elements" were produced before Euclid, his work organized and completed that of his predecessors, who are now known chiefly by reference. As the letters (Gr., stoikheia; "elements") of the alphabet are to language, so are the Elements to mathematics, wrote the Neoplatonist Proclus in the fifth century CE. The analogy is apt. In thirteen books Euclid goes from the most elementary definitions and assumptions about points, lines, and angles all the way to the geometry of solids, and he includes a theory of the proportions of magnitudes, number theory, and geometric algebra. His procedure epitomized the axiomatic-deductive method and became a paradigm for philosophical and scientific reasoning. The greatest works in the history of astronomy imitate the Elements: Ptolemy's Almagest (c. 150 CE), Copernicus's De revolutionibus (1543), and Newton's Principia (1686). There is no greater example of Euclid's influence in philosophy than Spinoza's Ethics (1675), which scrupulously reproduced Euclid's method of definitions, axioms, and propositions.
The Elements became the elementary introduction to mathematics in Hellenistic civilization. Translated into Arabic in the ninth century and into Latin in the thirteenth, it became the foundation of Islamic, medieval, and Renaissance mathematics. It standardized the body of mathematical knowledge well into the twentieth century. The Elements was not translated into Sanskrit until the 1720s, though there is evidence of some prior knowledge. The Chinese may have known Euclid in the thirteenth century, but it did not affect the development of their mathematics until 1607, when the Jesuit Matteo Ricci produced a highly praised translation of the first six books of the Elements as part of the Jesuit missionary strategy in China. The use of the Elements as the textbook of mathematics over millennia is the source of the often repeated claim that, second only to the Bible, the Elements is the most widely circulated book in human history.
Euclid's religious significance can be seen in two ways. First, Euclid fulfilled the value Plato saw in mathematics. Euclid's masterpiece remains the enduring testament of the human capacity to construct a transparently intelligible system of relations grounded in logic and capable of extension to the physical world, though not derived from it. He demonstrates with lucid brevity how reason can successfully operate with purely intelligible objects such as points, lines, and triangles, and discover new and unforeseen truths with them. Such exercise frees the mind from the appearances of the senses and initiates it into an intellectual realm that Plato referred to as the realm of Being. In Neoplatonism such exercise had a paramount spiritual value. Augustine of Hippo, in his Soliloquies (386), written the year before his baptism, esteemed mathematics as a preparation for the soul's ascent to God. The mind perceives necessary truths first in mathematics and is then prepared to pursue eternal, divine truth. Having tasted the sweetness and splendor of truth in mathematics and the liberal arts, the mind actively seeks the divine. A millennium later, the Christian mystic Nicholas of Cusa wrote in his Of Learned Ignorance (1440) that the most fitting approach to knowledge of divine things is through symbols. Therefore he uses mathematical images because of their "indestructible certitude" (bk. 1, chap. 11).
Second, Euclid's geometry implicitly defined the nature of space for Western civilizations up to the nineteenth century. That "a straight line is drawn between two points," Euclid's first postulate, is also a statement about the space that makes it possible. Conceptions of space have religious repercussions because they involve matters of orientation. Isaac Newton (1642–1727) reified Euclidean space in his physics. He identified absolute space and absolute time, which together constitute the ultimate frame of reference for cosmic phenomena, with God's ubiquity and eternity. Euclid's fifth postulate stipulated the conditions under which straight lines intersect, and, by implication, when they are parallel. To his continuing credit, Euclid presented the conditions as assumptions. For millennia mathematicians tried unsuccessfully to prove them. But, because Euclid's postulates were only assumptions, other conditions were possible. Thus in the nineteenth century Nikolai Lobachevskii, Farkas Bolyai, and G. F. B. Riemann were inspired to develop non-Euclidean geometries. These were crucial to Einstein's theories of special and general relativity (1905, 1913) and, hence, to the present cosmology, wherein a straight line cannot be drawn between two points. The conclusion that space and time are inseparable in the mathematical and physical theories of the nineteenth and twentieth centuries owes its existence to the force of the Euclidean tradition.
Bibliography
The classic English translation of the Elements is Thomas L. Heath's The Thirteen Books of Euclid's Elements, 2d ed. (New York, 1956). It includes an introduction to Euclid's place in the history of mathematics and a thorough commentary on the text. A more recent account of Euclid and his achievement, as well as the history of the Elements, with comprehensive bibliographies, is found in Ivor Bulmer-Thomas's "Euclid" and John Murdoch's "Euclid: Transmission of the Elements," in the Dictionary of Scientific Biography (New York, 1970–1980). A discussion of the historical and philosophical antecedents to Euclid and how his methods incorporate Platonic and Aristotelian developments in the philosophy of mathematics is provided in Edward A. Maziarz and Thomas Greenwood's Greek Mathematical Philosophy (New York, 1968). The importance of mathematics in the education of the philosopher is addressed in Werner Jaeger's Paideia: The Ideals of Greek Culture, 3 vols., translated by Gilbert Highet (Oxford, 1939–1944).
New Sources
Gray, Jeremy. Ideas of Space: Euclidean, Non-Euclidean, and Relativistic. New York, 1989.
Lloyd, G.E.R. The Ambitions of Curiosity: Understanding the World in Ancient Greece and China. New York, 2002.
Mlodinow, Leonard. Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace. New York, 2001.
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