Optics and Catoptrics
The science of optics (or perspectiva, as it was known during the Middle Ages) had its origin in Greek antiquity, in the mathematical analyses of *Euclid and *Ptolemy, the physical theories of Aristotle and the Atomists, and the anatomical and physiological endeavors of *Galen. All three traditions were transmitted eastward into central Asia and subsequently translated into Arabic in the eighth and ninth centuries. In the twelfth and thirteenth centuries, these same materials, now accompanied by a large collection of original writings by Islamic scholars, were again translated, this time from both Arabic (mainly) and Greek (to a minor degree) into Latin.
Vision
The central problem of ancient and medieval optics was to understand how we see. Everyone who approached the question of vision agreed that some form of contact must be established between the eye and the visible object. On the extramission theory, visual power emanates from the seeing eye to the seen object; on the intromission theory, something travels from the visible object to the eye. The first objective was to determine which of these two theories is correct, and to identify the nature of the entity that passes from observer to object or from object to observer. The ancient atomists were perhaps the earliest to address this question, arguing that convoys of atoms representing the visible object in shape and color flowed from object to observer, thereby accounting for the facts of visual perception. Aristotle shared the intromissionism of the atomists but refused to accept the notion that anything like the atomists’ convoy of atoms will do. In short, he denied the possibility of a flow of material from object to observer. Rather, he maintained, a stationary medium undergoes an instantaneous qualitative transformation, provoked by light or color and received instantaneously by the eye. Thus light (such as that of the Sun as it peeks over the horizon) transforms a transparent medium, such as air or water or glass, from potential transparency to actual transparency. Once the medium is actually transparent, colored objects situated within it or on its periphery produce further qualitative change in the medium—a red object colors the medium (in some undefined sense) red, right up to and including the interior humors of the eye. This is sufficient to cause perception of the object as red.
The extramission theory of vision was defended by Euclid, who revealed little or no interest in the physical nature of the messenger and dealt almost exclusively with the geometry of sight. A cone of rays, he argued, emanates from the observer’s eye, and anything in the visual field encountered by one or more of these rays is seen. The advantages of this theory were perspectival: the size of an object was judged by the angle between visual rays terminating on its extreme points. Move the object closer, and it appears larger because the rays now encountering its extremes are separated by a larger angle. If an object appears in the upper left of the observer’s visual field, that is because it intercepted a ray in the upper left of the visual cone. And if an object falls between adjacent rays, it is not seen at all.
The astronomer Ptolemy defended Euclid’s extramission theory in his Optica, while arguing that the rays were physical things—visual flux—rather than purely mathematical entities. He insisted that the visual cone is a continuous body of visual flux, rather than radiation proceeding along discrete geometrical lines, separated by spaces. Colored objects, he argued, produce a modification in the visual flux, which is read by the eye and brain as color. Ptolemy also developed a geometrical analysis of binocular vision.
Finally, Galen described the gross anatomy of the eye in terms closely resembling modern knowledge. As for the act of vision, he argued that visual power flowing down the optic nerves and out of the eyes transforms the surrounding transparent medium (the air), endowing it with the power of sight and rendering it capable of perceiving whatever it touches.
The first Islamic author to take up visual theory was the philosopher *al-Kindi (d. c. 870), whose goal was to extend and defend Euclid’s extramission theory, including his cone of visual rays. But al-Kindi insisted, with Ptolemy, that the cone must be a continuous body rather than a collection of discrete rays. He also proposed, in what would turn out to be a revolutionary suggestion, that radiation departing from the eye does not go forth as a single visual cone, but rather as an infinity of discrete cones emanating in all directions from each point on the surface of the eye, thereby differentially “illuminating” the medium. Al-Kindi’s purpose in this argument was to explain why objects located near the central axis of the eye are seen with greater clarity. Some one hundred fifty years later, *Ibn al-Haytham would put al-Kindi’s argument to use for a quite different purpose.
Al-Kindi lived and worked in Baghdad, the home of the ‘Abbasid court, which boasted a rich intellectual life. Ibn al-Haytham (d. c. 1039) was born in Basra, on the Persian Gulf, but apparently spent most of his productive scholarly life in Cairo. A towering mathematician, Ibn al-Haytham produced a synthesis of ancient theories, which managed to draw the useful elements from each of the ancient rivals. Although firmly committed to the intromission theory, Ibn al-Haytham sought to adopt the visual cone of the extramissionists for the sake of its mathematical contributions. He did this by arguing that although the “forms” of light or color issue in all directions from every point of a visible object (here borrowing al-Kindi’s revolutionary idea and putting it to use for a different purpose), only one ray from each point falls perpendicularly on both the cornea and front surface of the crystalline lens (concentric surfaces) and enters the sensitive organ of the eye (the crystalline lens) without refraction. All other rays are weakened by refraction sufficiently to be ignored. Those unrefracted rays form a visual cone and endow the intromission theory (until then looked on purely as a non-mathematical theory about the physical nature of the vision-causing entity) with all of the mathematical capabilities of the extramission theories of Euclid and Ptolemy. As for that other major problem—identifying the agents of vision that emanate from the visible object to the eye—Ibn al-Haytham identified them as the “forms” (a very Aristotelian conception) of light and color. Finally, Ibn al-Haytham introduced Galen’s anatomical discoveries into his theory, tracing radiation as it passes through the eyes and optic nerves to their junction, where forms passing through the two eyes are united and judged by the “final sentient power.”
Ibn al-Haytham’s great Optica was translated into Latin near the beginning of the thirteenth century, thus joining previously translated texts by Aristotle, Ptolemy, Euclid, al-Kindi, and *Hunayn ibn Ishaq’s summary of Galen. The first scholar in Christendom to master the whole of this Greek and Islamic legacy appears to have been Franciscan friar *Roger Bacon (d. c. 1292). With Ibn al-Haytham (known to Bacon as Alhacen) as his primary guide, but also taking account of all the other ancient and medieval authorities on the subject, Bacon produced a synthesis of his own, developing at great length his theory of the “multiplication of species” (multi-plicatio specierum, better translated as the “propagation of images”) to explain the agency that bore visual information from the observed object to the observer.
The other problem that Bacon needed to address was that of the direction of radiation, on which his authorities were evenly divided. Bacon perceived that Alhacen had not disproved the existence of extramitted visual rays, but only that if they existed they were incapable of accounting for visual perception. Bacon therefore invented a new function for them, namely, to prepare the medium for receiving the vision-producing intromitted rays. Thus everybody had a piece of the truth. Vision was caused by intromitted rays, but extramitted rays existed and performed an auxiliary function.
Bacon’s ideas were spread by his own publications, but also by a younger Franciscan brother, the future Archbishop of Canterbury *John Pecham, who presented an abbreviated version of the Baconian synthesis in his Perspectiva communis, which became the standard text on the subject in the medieval universities as late as the end of the sixteenth century. Through their influence, and that of an enormous tome by the Silesian *Witelo (also influenced by Bacon), the optical theories of Alhacen and Bacon came to dominate philosophical thought on problems of sensation and cognition to the end of the Middle Ages.
Reflection and Refraction
Vision was the central problem of medieval perspectiva, but it was not the only one. In Catoptrica, Euclid dealt with the phenomena of reflection and refraction and formulated laws of reflection, which we still accept. The two most basic principles were: (1) That angles of incidence and reflection at a plane or curved reflecting surface are equal; and (2) That the image of an object seen by reflection is located where the rectilinear extension of the ray issuing from the eye (entering the eye for an intromissionist) intersects the cathetus (the perpendicular running from the observer’s eye to the reflecting surface). Building on this foundation, Euclid proceeded to prove a collection of relatively sophisticated propositions.
Ptolemy rearticulated much that he found in Euclid’s Catoptrica, but added an empirical approach, experimentally confirming the law of equal angles. Ptolemy proceeded to a geometrical analysis of reflection in concave spherical and convex spherical mirrors, especially sophisticated in the former case, involving claims about image shape and clarity—again, offering experimental confirmation of his results. The latter part of Ptolemy’s Optica is lost, but in the existing portion of his fifth book he undertook a brief analysis of refraction, going well beyond anything achieved by his predecessors. He revealed a full non-numerical understanding of the qualitative geometry of refraction, which he applied to concave and convex spherical surfaces as well as to plane surfaces. And he made a famous effort to discover a quantitative law of refraction by experimental means, arriving at an interesting geometrical series but falling short of discovering the modern law.
When Ibn al-Haytham wrote his great Optica in the eleventh century, he had at his disposal all the principal Greek works on optical subjects. Drawing inspiration from Ptolemy’s Optica, he prepared an exhaustive geometrical (but non-numerical) analysis of image formation owing to reflection and refraction in reflecting or refracting surfaces of plane, spherically concave or convex, and cylindrically concave or convex form. His analysis included attention to the number, location, size, and shape of the images. The approach was empirical—he replicated Ptolemy’s experimental apparatus for demonstrating the rules of refraction but omitted Ptolemy’s numerical results. His approach was also causal—a search for the physical behavior that would explain the phenomena of reflection and refraction, by the use of mechanical analogies. Ibn al-Haytham’s Optica was not to be equaled for mathematical sophistication until the seventeenth century.
Ibn al-Haytham’s Optica (translated into Latin as Alhacen’s De aspectibus) powerfully shaped the optical writings of Bacon, Pecham, and Witelo—the former two of whom offered abbreviated versions of Ibn al-Haytham’s analysis, while the verbose Witelo managed to expand on Ibn al-Haytham (and all the other sources at his disposal). Borrowing from a short treatise by Ibn al-Haytham that circulated in Latin translation, entitled De speculis comburentibus (On Burning-Mirrors), Witelo also dealt with the focusing properties of paraboloidal mirrors.
Late Medieval Developments
The colors of the rainbow were among the most striking optical phenomena and, inevitably, led to causal speculation. Aristotle had attributed the rainbow to the reflection of sunlight in a cloud. That theory was dismissed by *Robert Grosseteste (c. 1169–1253), who doubted that reflection in a cloud could explain the rainbow’s shape. His alternative was to attribute the rainbow to multiple refractions in a cloud, but he provided no detail on how this might come about. Bacon, well-informed on Grosseteste’s theory, was not persuaded. Committed to an empirical approach to the question, he examined rainbows observable in artificial sprays or those accompanying a waterfall. In the case of rainbows in the sky, he noticed that the rainbow moved as the observer moved and must therefore be produced in a different set of droplets for every different position of the observer. It followed, he argued, that the rainbow was the product of reflections not from the cloud as a whole, but from its individual droplets of moisture. One more Baconian discovery was measurement of the maximum elevation of the rainbow as forty-two degrees (a correct value proposed by no earlier author, doubtless obtained by use of an astrolabe).
Finally, early in the fourteenth century, the Dominican Theodoric of Freiberg (d. c. 1310) undertook a remarkable experimental investigation, projecting rays of light through water-filled glass globes, which were meant to simulate the individual raindrop in a cloud. These experiments taught him that the primary rainbow was the result of two refractions (as light entered and departed from the individual raindrops), along with a total internal reflection at the rear surface of the drop; the secondary rainbow from the same two refractions combined with two internal reflections—essentially the modern theory of the rainbow.
A final phenomenon is worthy of mention because of the interest it attracted during the Middle Ages. Ever since Aristotle, close observers had noticed that the light issuing from a spherical object, such as the Sun, passing through a non-circular aperture, projected an image that imitated the shape of the Sun rather than of the aperture—a problem made interesting because it seemed to hint that light was not being rectilinearly propagated. Ibn al-Haytham took up this puzzle and solved it satisfactorily, but in a treatise that was not available to Latin-speaking scholars. Roger Bacon tackled the problem without help from earlier sources, struggling intelligently through three different versions, managing in the end to retain rectilinear propagation, but paying the price elsewhere in his argument. A successful solution in the West, which recognized the problem as one of scale, would have to wait for Kepler in the seventeenth century.
A substantial tradition of commentary on optical topics continued through the later Middle Ages. Much of this was motivated by debates over sensation and cognition, to which both the Aristotelian and Baconian traditions had made contributions. Debates over the number of colors in the rainbow provided additional motivation. The geometrical side of perspectiva also attracted continuing interest, as perspectiva came to serve, in many universities, as a stand-in for geometry in the arts curriculum. Kepler’s establishment of optics on a modern foundation early in the seventeenth century, including his successful theory of the retinal image and a convincing analysis of the problem of radiation through small apertures, represented correction and fulfillment, rather than repudiation of the medieval syntheses of Ibn al-Haytham and Roger Bacon.
See also Aristotelianism
Bibliography
Crombie, Alistair C. Robert Grosseteste and the Origins of Experimental Science 1100–1700. Oxford: Clarendon Press, 1953.
Eastwood, Bruce S. Astronomy and Optics from Pliny to Descartes. London: Variorum, 1989.
Grant, Edward and John E. Murdoch, eds. Mathematics and its Applications to Science and Natural Philosophy in the Middle Ages: Essays in Honor of Marshall Clagett. New York: Cambridge University Press, 1987.
Lindberg, David C., ed. & trans. John Pecham and the Science of Optics: Perspectiva communis. Madison: University of Wisconsin Press, 1970.
_______. Roger Bacon and the Origin of Perspectiva in the Middle Ages: A Critical Edition and English Translation of Bacon’s Perspectiva, with Introduction and Notes. Oxford: Clarendon Press, 1996.
_______. “Roger Bacon on Light, Vision, and the Universal Emanation of Force.” In Roger Bacon and the Sciences: Commemorative Essays. Edited by Jeremiah Hackett. Leiden: E.J. Brill, 1997, pp. 243–275.
_______. Roger Bacon’s Philosophy of Nature: A Critical Edition, with English Translation, Introduction, and Notes, of De multiplicatione and De speculis comburentibus. Oxford: Clarendon Press, 1983.
_______. Studies in the History of Medieval Optics. London: Variorum, 1983.
_______. Theories of Vision from al-Kindi to Kepler. Chicago: University of Chicago Press, 1976.
Rashed, Roshdi, ed and tr. Oeuvres philosophiques et scientifiques d’Al-Kindi, vol. 1: L’optique et la catoptrique. Leiden: E.J. Brill, 1997.
_______. Optique et mathématiques. London: Variorum, 1992.
Sabra, A. I. Optics, Astronomy and Logic. London: Variorum, 1994.
_______, ed and tr. The Optics of Ibn al-Haytham: Books I-III On Direct Vision. 2 vols. London: The Warburg Institute, 1989.
_______. “Sensation and Inference in Alhazen’s Theory of Visual Perception.” In Studies in Perception: Interrelations in the History of Philosophy and Science. Edited by Peter K. Machamer and Robert G. Turnbull. Columbus: Ohio State University Press, 1978, pp. 160–185.
Smith, A. Mark, ed and tr. Alhacen’s Theory of Visual Perception: A Critical Edition, with English Translation and Commentary, of the First Three Books of Alhacen’s De aspectibus. 2 vols. Philadelphia: American Philosophical Society, 2001.
_______. Getting the Big Picture in Perspectivist Optics. Isis (1981) 72: 568–589.
_______, tr. Ptolemy’s Theory of Visual Perception: An English Translation of the Optics, with Introduction and Commentary. Philadelphia: American Philosophical Society, 1996.
Tachau, Katherine H. Vision and Certitude in the Age of Ockham: Optics, Epistemology and the Foundations of Semantics, 1250-1345. Leiden: E.J. Brill, 1988.
DAVID C. LINDBERG
Oresme, Nicole
Nicole Oresme (c. 1320–1382) was born in Normandy, near Caen. Virtually nothing is known of his early life. He studied at the University of Paris where he was a master of arts by the academic year 1341–1342. He studied theology at Navarre, where in 1356 he became a doctor of theology and grand master of the college. After holding a number of church offices, he was made bishop of Lisieux in 1377, partly no doubt because he had the powerful backing of King Charles V, who ruled France from 1364 to 1380. He remained bishop until his death in 1382.
Oresme’s relations with Charles V and the royal court were significant. He first came into contact with Charles’s father, King John II, who used Oresme’s knowledge of finance to aid the state. Using this experience, Oresme wrote a Treatise on Money in both Latin and French versions. Overall, his treatise had a significant influence. At the royal court Oresme came to know the heir apparent, and when Charles V succeeded to the throne he asked Oresme to translate from Latin into French four of *Aristotle’s treatises, three of which were directly relevant to government, namely the Nicomachean Ethics, Politics, and Economics. The fourth treatise, On the Heavens, may have been included because it was Aristotle’s great work on cosmology. Perhaps Charles hoped that it would give his courtiers a cosmic background for their lessons in statecraft. Oresme accompanied his translation of On the Heavens with a detailed French commentary on the text. It was his last known work, completed in 1377, and one of his most important.
Oresme was perhaps the most original-minded, innovative natural philosopher and mathematician of the Middle Ages. The range of his written works is truly impressive, embracing politics, monetary policy, economics, theology, magic, and treatises on the dangers of *astrology. His primary interests, however, were natural philosophy and the application of mathematics to natural philosophy. Using the scholastic format of quaestiones, Oresme left commentaries on all of Aristotle’s books on natural philosophy, and also wrote treatises on various major themes in natural philosophy, often emphasizing the mathematical aspects of the subjects he considered. Although Oresme had great respect for Aristotle, he frequently disagreed with him.
Oresme’s Scientific Ideas and Contributions
Oresme’s approach to nature provides an important insight into his ideas about the physical world. He was convinced that nature operated in a regular manner and that a given natural cause would invariably produce its natural effect. He was strongly opposed to those who invoked magical and supernatural explanations for phenomena that he regarded as wholly natural. But Oresme was equally convinced that human ability to acquire certain knowledge of nature’s workings was limited, and often uncertain, which led him to proclaim that many propositions about the natural world required as much of an act of faith to accept their truth, as did the truths of revelation. This attitude probably explains why Oresme, more than occasionally, presented equally plausible alternatives for some important problems in natural philosophy.
One such problem concerned the status of the Earth at the geometric center of the world: was it stationary at the center of the world while the celestial bodies made a daily rotation around it, as Aristotle and his followers believed; or did it rotate once daily on its axis, while the heavens remained stationary? Oresme presents a series of impressive arguments in favor of the Earth’s axial rotation, most of which were repeated almost two centuries later by Nicholas Copernicus in his momentous treatise On the Revolution of the Heavenly Orbs (1543). Oresme concluded that physical and astronomical phenomena could be equally well explained in either of the two rival hypotheses. No evidence could decide the issue. In light of this stalemate, Oresme opted, on biblical grounds, for the traditional Aristotelian opinion that the Earth was stationary at the center of the world. A second significant issue involved Aristotle’s claim that the existence of more than one world is impossible. Oresme argued, as did many in the Middle Ages, that God could create other worlds if He wished; hence, it was certainly possible that other worlds could exist. Moreover, if God did create other worlds, they would coexist with ours. Each world would have its own center and circumference, and would be a self-contained entity independently of the other worlds. Content to demonstrate the possibility of other worlds and convinced that no scientific arguments could decide the issue, Oresme again chose to adopt the universally held traditional view that there is only one world.
Perhaps more than anyone else, Oresme applied mathematics to a variety of problems in natural philosophy. He gave a mathematical foundation to the ideas first proposed by *Thomas Bradwardine in his Treatise on Ratios (1328). In a treatise entitled De proportionibus proportionum (On Ratios of Ratios), Oresme first presented certain elements of the mathematical theory of proportionality, relying on the fifth and tenth books of Euclid’s Elements. A “ratio of ratios” is actually an exponent that relates two ratios. For example, in the proportional relationship A/B = (C/D)p/q, the ratio p/q is a “ratio of ratios” because it relates the two ratios A/B and C/D. Oresme gives examples in which p/q is either rational, forming a “rational ratio of ratios,” or irrational, forming an “irrational ratio of ratios.” In effect, Oresme had arrived at the concept of an irrational exponent, perhaps for the first time. He also introduced probability considerations into his treatises. Taking one hundred rational ratios from 2/1 to 101/1, he demonstrates that any two proposed unknown ratios are probably incommensurable, as, for example, 3/1 and 2/1, where 3/1 does not equal (2/1)p/q (that is, p/q is irrational). On this assumption, he showed that the odds were 197 to 1 that any “ratio of ratios” chosen at random from this group of one hundred rational ratios would be related by an irrational ratio, that is, by an irrational exponent.
Oresme applied his mathematical ideas on “ratios of ratios” to the motion of bodies, rejecting, as had Bradwardine earlier, Aristotle’s mathematical representation of motions involving motive forces and mobile bodies. In his Treatise on the Commensurability or Incommensurability of the Celestial Motions, Oresme extended his ideas, arguing for the probability that any two celestial motions are probably incommensurable. From this conclusion, Oresme inferred that precise astrological prediction would be impossible. Throughout most of his career, Oresme was a strong opponent of astrology and wrote treatises against it. One of his desired objectives was to weaken Charles V’s belief in astrological prognostication, an objective that was not realized.
Medieval scholars were interested in what they called “the intension and remission of qualities,” a study of the way qualities varied in intensity or in which they lost intensity (remission). At Merton College, Oxford, scholars used a numerical approach to measure the variations, but Oresme used geometrical figures, as we discover in his lengthy treatise titled On the Configurations of Qualities and Motions. Oresme’s significant move was to assume that qualitative intensities were like continuous magnitudes and therefore representable by lines and surfaces, that is, by geometric figures. Qualities were imagined to increase or decrease their intensities uniformly or non-uniformly. All sorts of qualitative intensities were measured: pains, joys, music, colors, and many more. In a two-dimensional figure, the base, or horizontal line, represented the
extension of a quality in a subject; the perpendiculars erected on that base represented the intensity of the quality at that point. A right triangle, for example, could represent any quality that was assumed to increase uniformly from zero degree to any maximum intensity.
A prince mints coins. Illumination from a fifteenth-century manuscript of Nicole Oresme’s Treatise on Money. (AKG Images)
Not only were variations in qualities represented in the intension and remission of forms, but uniform and nonuniform velocities were similarly treated. Within this context, scholars at Merton enunciated the mean speed theorem, but it was Oresme who first proved it geometrically. The mean speed theorem is usually represented as s = 1/2at2, where s is the distance traversed, a represents uniform acceleration, and t is the time of acceleration. Oresme thus anticipated Galileo’s proof of the same theorem in The Two New Sciences (1638). Oresme also anticipated a significant corollary that Galileo drew from the mean speed theorem.
Oresme, it is important to recognize, was really mathematizing fictional qualities and their imaginary intensities. He was not applying mathematics to the real world, but was rather engaged in an intellectual exercise that, in medieval terms, was “according to the imagination” (secundum imaginationem). It remained for Galileo to apply these important theorems and definitions about motion to the motion of real bodies and thereby establish a new science of mechanics. This should not, however, diminish Oresme’s great mathematical accomplishments and his profound use of the imagination. Arriving at some of the basic concepts of the new mechanics—even if applied only to imaginary conditions—is a considerable contribution and achievement.
In On Seeing the Stars, Oresme rightly rejected the traditional idea that the refraction of light can only occur at a single interface between two media of differing densities. In place of this interpretation, which had been held by Ptolemy, Alhacen (*Ibn al-Haytham, the great Islamic author on geometrical optics), and many others, Oresme insisted that light is refracted along a curved path when it is traveling through a single medium of uniformly varying density. To arrive at the curved path, Oresme used his knowledge of convergent infinite series. That is, the successive refractions produced successive line segments, which formed a curved line as they increased to infinity. This important interpretation of the refraction of light was always thought to have originated with Robert Hooke and Isaac Newton. In fact, it was Oresme who, without proof, first proclaimed the concept, while Hooke and Newton were apparently the first to demonstrate its truth.
Although Oresme did not furnish experimental evidence for some of his most important scientific ideas, he deserves high praise for having enunciated the ideas described here long before they were independently rediscovered in the seventeenth century by such eminent scientists as Galileo and Newton.
See also Astrology; Cosmology; Latitude of forms; Optics and catoptrics
Bibliography
Primary Sources
Oresme, Nicole. Nicole Oresme: Le Livre du ciel et du monde. Edited by Albert D. Menut and Alexander J. Denomy. Translated with an Introduction by Albert D. Menut. Madison: University of Wisconsin Press, 1968.
———. De proportionibus proportionum and Ad pauca respicientes. Edited by Edward Grant. Madison: University of Wisconsin Press, 1966.
———. Maistre Nicole Oresme: La Livre de Yconomique d’Aristote. Critical Edition of the French Text from the Avranches Manuscript with the original Latin version. Introduction and English translation by A. D. Menut. Transactions of the American Philosophical Society, New Series 47, pt. 5. Philadelphia: American Philosophical Society, 1957.
———. Nicole Oresme and the Kinematics of Circular Motion: “Tractatus de commensurabilitate vel incommensurabilitate motuum celi.” Edited with an Introduction, English Translation, and Commentary by Edward Grant. Madison: University of Wisconsin Press, 1971.
———. Quaestiones super De generatione et corruptione. Edited by Stefano Caroti. Munich, Germany: Verlag der Bayerischen Akademie der Wissenschaften, 1996.
———. The De moneta of Nicholas Oresme and English Mint Documents. Edited by Charles Johnson. New York: Nelson, 1956.
Secondary Sources
Caroti, Stefano. Quaestio contra divinatores horoscopios. Archives d’histoire doctrinale et littéraire du moyen âge (1976) 43: 201–310.
Courtenay, William J. The Early Career of Nicole Oresme. Isis (2000) 91: 542–548.
Grant, Edward. Jean Buridan and Nicole Oresme on Natural Knowledge. Vivarium (1993) 31: 84–105.
———. “Nicole Oresme, Aristotle’s On the Heavens, and the Court of Charles V.” In Texts and Contexts in Ancient and Medieval Science: Studies on the Occasion of John E. Murdoch’s Seventieth Birthday. Edited by Edith Sylla and Michael McVaugh. Leiden: E.J. Brill, 1997, pp. 187–207.
Kaye, Joel. Economy and Nature in the Fourteenth Century. Money, Market Exchange, and the Emergence of Scientific Thought. New York: Cambridge University Press, 1998. Chapter 7.
Kirschner, Stefan, Oresme on Intension and Remission of Qualities in His Commentary on Aristotle’s Physics. Vivarium (2000) 38: 255–274.
EDWARD GRANT