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Science (in the Middle Ages)

SCIENCE (IN THE MIDDLE AGES)

The term "Middle Ages" will be taken, somewhat arbitrarily and for convenience, to extend over the period a.d. 500 to 1500, and to embrace the science of three different, but more or less intimately related, centers of civilization: Byzantine (Eastern Roman) Empire, Islam, and West Europe. China and India, while important, are outside the scope of this article.

Introduction

Greek theoretical science had always lacked popular appeal, for it was the work of a relatively few gifted individuals. To satisfy curiosity about the physical world, a tradition of popular science emerged in the form of handbooks, the purpose of which was to communicate the results of the more technical, theoretical treatises. The Romans enthusiastically adopted the handbook form translating, paraphrasing, plagiarizing, and diluting Greek treatises. As W. H. Stahl puts it, "[A]t Rome there was only one level of scientific knowledgethe handbook level" [Roman Science (Madison 1962) 71].

From this tradition handbooks were produced in the Roman Empire period that were to be enormously influential throughout the Latin Middle Ages (e.g., those of Seneca, Pliny the Elder, Solinus), especially in the period from a.d. 500 to 1150, when they constituted the major source of scientific knowledge. Neoplatonic cosmography and interpretations were transmitted to the Middle Ages by three Latin encyclopedists: Chalcidius (fl. a.d. 4th century), who translated into Latin and commented upon a large portion of Plato's Timaeus; macrobius (fl. a.d. 400), author of the commentary In somnium Scipionis; and Martianus Capella (5th century a.d.), who wrote the famous allegory De nuptiis Mercurii et Philologiae, which fixed the number of liberal arts at seven.

Christian Encyclopedists. Of the previously mentioned encyclopedists none is known with certainty to have been a Christian, but a number of Christian encyclopedists were to follow in their footsteps, of whom boethius (c. 480524), cassiodorus (fl. c. 540), isidore of seville (c. 570636), and the Venerable bede (673735) deserve mention. Of these, Boethius was the most significant since his knowledge of Greek enabled him to comprehend, utilize, and translate Greek treatises. Two of the most comprehensive scientific works that were available prior to the 12th century were provided by Boethius. His Arithmetica, a free translation and slight rearrangement of Nicomachus of Gerasa's (fl. c. a.d. 100) Introductio arithmetica, served as the basic textbook of theoretical arithmetic throughout the Middle Ages; his work De Institutione musica, based on earlier Greek treatises, was fundamental. Boethius is said to have translated Euclid into Latin (no extant translation exists; a pseudo-Boethian geometry, Ars geometriae, is not by him and is illustrative of the decline of mathematical thought) as well as Archimedes and Ptolemy (Cassiodorus, Lib. Var. 1.15), but no translations have been found. Of great importance were his commentaries on and translations of Aristotelian logical treatises, which rank him as the most significant of the Latin authors who shaped the scientific and intellectual tradition of the early Middle Ages.

The other three authors mentioned above utilized science as an aid to Christian life and a better understanding of Scripture. Indeed, Bede's principal scientific treatises, De temporibus and a later expanded version De temporum ratione, were each essentially a computus i.e., a work on calendar reckoning, which explained how to calculate the variable date of Easter and frequently had tables associated with it. Bede used a 19-year cyclethe best availableand provided a lucid and well thought out explanation. He also observed that the tides were constant for a given port ("the establishment of the port")i.e., the time interval between the meridian passage of the moon and subsequent high tide is approximately constant. This time interval, Bede observed, differed for different ports along the coast.

Decline of Science. Bede, however, must be set down as an exception, for in western Europe the period from 500 to c. 1150 was a period of scientific decline. This is borne out, for example, by the level of understanding of Greek mathematics. In his Etymologies, Isidore of Seville devotes to geometry a scant two pages, consisting largely of confused definitions and concepts. Similar confusions are found in Cassiodorus's account of the quadrivium. The meagerness of geometrical knowledge and understanding is illustrated in a treatise written about 1050 by Franco of Liège called De quadratura circuli. Squaring the circle was one of the three classical problems in Greek geometry whose solution had to be achieved by ruler and compass. In this insoluble problem the objective was to equate the area of a quadrilateral figure, usually a square, with that of a circle. Franco "solves" it by employing a rule formulated by Roman surveyors that the area of a circle equals 11 d2 /14, where d is the diameter of the circle. Assuming that d is 14 feet, Franco calculates the area of the circle as 154 square feet and then "squares the circle" by constructing a rectangle whose sides are 14 and 11 feet, respectively. Actually, most of his treatise is devoted to finding a square equal to a rectangle of 154 square feet. His solution is hopeless, since he does not utilize, and probably did not know, the Euclidean proposition for constructing a mean proportional line between two given straight lines (Elements, 6.13). However, Franco was convinced that he had solved the problem.

At best, the Latin encyclopedists preserved some scientific knowledge, usually a pale reflection of what once was. Incompatible concepts were often included without the compiler's awareness. A typical illustration is found in Macrobius's Commentarium in somnium Scipionis. Macrobius accepts a fixed order of the planets and yet speaks of Venus and Mercury as sometimes above and sometimes below the planets, thus destroying the fixed order.

Medicine and materia medica were at a higher level, largely as the result of translations into Latin (5th to 9th centuries) of certain treatises by galen, hippocrates, Dioscorides, and some of the late Greek medical writers. Practical considerationsthe need to cure the sickno doubt lent impetus to this translating activity. The literature available was adequate to permit the development of the Italian medical school of Salerno in the 9th century.

Byzantine Empire

In the Byzantine (Eastern Roman) Empire, where Greek was the common language, commentaries on available Greek scientific classics played a significant role. For example, Eutocius (fl. c. 520) revised and commented upon the text of the first four books of the Conics of Apollonius of Perga (fl. c. 250 b.c.), one of the greatest geometrical works of antiquity. He commented also upon at least three treatises of Archimedes and reawakened an interest in Archimedian manuscripts. But for his efforts, these works might have been lost.

Philoponus. Among Aristotelian commentators, john philoponus, a Christian, and Simplicius, a pagan (fl. early 6th century), were especially significant for later centuries. In his commentary on Aristotle's Physics, Philoponus repudiated Aristotle's law of motion, which is sometimes represented as V S/T F/R, where V is velocity, S distance, T time, F motive force, and R resistance of a medium or object, or both. Aristotle had insisted that local motion required a force capable of moving an object through a medium, and denied the existence of void space. Philoponus insisted that motion in a vacuum was possible and would occur in a certain original time; however, the same motion of that same object in a medium would be of greater duration because of the resistance of the medium. Philoponus held that in a vacuum the ratio of weights is inversely proportional to their times of fall, i.e., W 2/W 1 = T 1/T 2 when S 2 = S 1 and W is the weight of a body. In a medium, additional time must be added, which is taken as proportional to the resistance of the medium. Philoponus also appealed to experience, insisting that two unequal weights dropped from a height would reach the ground in almost equal times. Not only does this refute Aristotle, but it is also incompatible with Philoponus's emended version of that law. It is noteworthy that the dropping of unequal weights to refute Aristotle was done long before by Simon Stevin (15481620) and Galileo galilei (15641642).

In the same Physics commentary, Philoponus repudiates Aristotle's explanation that air is a continually acting motive force that maintains the motion of a projectile no longer in contact with its initial mover (for Aristotle, a motion would automatically cease if contact was broken between motive force and moving body). Rejecting the role of a medium producing motion, Philoponus assumed that an incorporeal motive force is imparted from the projector to the projectile. Thus motion in a vacuum would be possible since the medium is not required as a moving force. This interpretation was widely adopted by Arab and Latin authors. Simplicius, in his commentary on Aristotle's De caelo, which was later translated into Arabic and Latin, cited three different explanations of the acceleration of falling bodies. These were to form part of the medieval discussion of this problem.

In an exclusively Christian framework, commentaries on the account in Genesis of the first six days of creation (In Hexaemeron ) dealt with concepts of matter, astronomy, cosmology, and zoology as they bore upon creation. Drawing upon Greek science, St. basil of Caesarea (329379) produced one of the most influential of these works, which was translated into Latin in the 5th century.

Alchemy and Astrology. The pseudosciences of alchemy and astrology developed markedly in late antiquity as an intermingling of Greek science, mysticism, and superstition. Alchemy was a fusion of the Aristotelian theory of transmutation of elements, practical metallurgy, and allegorical mysticism. But in this vain quest, several chemical discoveries were made, along with a steady development of apparatus and technique. In his Tetrabiblos, ptolemy bequeathed to the Middle Ages its greatest astrological treatise. His contention that the celestial bodies could determine human behavior was opposed by the Church, but the influence of celestial bodies on natural phenomena (e.g., drought, flood, tides) was accepted by many Christians even during the Middle Ages.

Arab Science

A fair portion of the Greek scientific corpus found its way eastward through the intermediary of Syriacspeaking Christians who, beginning in the 4th century, translated Greek logical and scientific treatises especially medical worksfor use in their schools.

The Arabs were beneficiaries of this rich legacy. But for approximately 150 years after the death of Muammad they displayed little direct interest in science. During the 9th century, however, there was unparalleled translating activity, and works from Syriac and Greek were translated into Arabic. By the 10th century the Arab world had almost all the extant Greek scientific corpus for study and assimilation. Although they wrote in the Arabic language, the contributors to Arab science included Muslims, Jews, Christians, and pagans who resided in lands governed by Muslim rulers.

Mathematics. In mathematics, the Arabs made significant contributions. Although algebra, which was wholly rhetorical, was established independently of geometry, geometrical proofs were often added for numerical problems solved algebraically. The geometrical proof was deemed more causal and fundamental. Equations were classified by the number of their terms rather than by powers of the unknown. Equations with only two terms were classified as binomials (e.g., ax 2 = c ); those with three terms were trinomials (e.g., ax 2 + bx = c ). Later, Omar Khayyām (c. 10381124) extended algebraic equations to four terms (tetranomials) by adding the cube of the unknown. In Arabian algebra negative roots were ignored as unintelligible (Bhāskara, a Hindu of the 12th century, first clearly asserted that a positive quantity had both a positive and a negative root; in Europe negative roots were finally acknowledged in the 16th century by Cardano in his Ars magna of 1545).

The Arabs developed the six trigonometric functions in conjunction with their astronomy. Arab geometers followed the best of Greek geometry and dealt with problems such as the trisection of an angle, finding two mean proportionals, and construction of regular polygons. Omar Khayyām solved cubic equations by means of conic sections and in so doing was conscious of his originality.

Astronomy. Astronomy was held in high esteem, since it permitted the Arabs to calculate the time of religious observances. Observatories were built and excellent instruments constructed (al-bĪrŪnĪ, 9731048, mentions a sextant with a radius of 40 cubits) to gather more accurate observations for improving astronomical tables and correcting some of Ptolemy's results. Both al-Battānī (d. 928) and al-Bīrūnī revealed discrepancies in the data concerning the longitude of the solar apogee, which pointed to a special motion of the solar apogee itself. This important discovery of the motion of the line of apsides was unknown to Ptolemy. Arab astronomers also redetermined with greater accuracy the value of precession of the equinoxes, and in lieu of Ptolemy's 1° in 100 years Thabit ibn Qurra (c. 826901) and al-Battānī offered 1° in 66 years, and al-Bīrūnī, 1° in 68 years and 11 months. In the reign of Caliph al-Mamun (813833), a degree of latitude was paced off north and south of a chosen position and averaged out to 56 2/3 Arabian miles. The Arabs also engaged in a controversy that had echoes in the Latin West in the 13th century. Was the function of astronomy to represent real motions and true cosmology, or merely to save the appearances? Could the epicycles and eccentrics used in Ptolemaic astronomy have actual counterparts in the heavens, or were they mere computational devices?

Mechanics. In physics the Arabs were interested in both theory and experiment. Apparently following Philoponus, a number of Arabs challenged Aristotle's account of projectile motion. The concept of an impressed force, called mail (inclinatio or impetus in Latin), was clearly expounded by avicenna (9801037) and abŪ al-barakĀt (d. c. 1164). Both authors accepted a natural inclination (mail tabi'i ) in natural upward motion (for fire and air) and downward motion (for earth and water) and an unnatural inclination (mail qasrī ) in motion away from natural placei.e., in violent motion. Avicenna insisted that the mail imparted to the projectile by the motive force was a permanent entity but destructible by the resistance of the medium through which any body moved. Void space was impossible because all motions were terminated by resistant media. Abū al-Barakāt favored a naturally self-expending non-permanent mail and argued that projectile motion in a void was conceivable because any motion would terminate upon the total disappearance of the impressed force. In the natural downward accelerated motion of heavy bodies, both authors accounted for the acceleration by assuming that the gravity, or heaviness, of the body continually produced additional natural mail, which increased the speed accordingly. For Abū al-Barakāt an additional factor was the continually lessening resistance of violent mail (he assumed its coexistence with natural mail ) to the successively produced natural mail. Scholastic commentators in the Latin West were to consider independently many of the same viewpoints in their discussion of impetus theory.

A number of other views appeared that did violence to the accepted interpretation of Aristotle. Thabit ibn Qurra rejected Aristotle's doctrine of natural place. He enunciated a qualitative theory of gravitation based on the concept that like attracts like. When a piece of earth is displaced from the earth itself, it is attracted back to the earth. Of two separate quantities of earth in void space, the greater quantity would attract the lesser; if they were equal in magnitude, they would meet at the midpoint. Nasr ibn Khosraw (c. 100389) rejected the notion that bodies are absolutely light or heavy and opted for relative weight. There were also atomists who defended their position by theological arguments and some who used arguments more akin to those of Greek atomism.

The Arabs determined specific gravities of liquids and solids and used such data to detect fraud and to distinguish alloyed from unalloyed metals. Al-Khāzīnī (fl. a.d. 1100), in his elaborate Book of the Balance of Wisdom, gives specific gravities for numerous substances (e.g., gold, mercury, and silver). It was recognized also that ice had a larger volume than the same weight of water; its specific gravity was put at 0.965. The "Balance of Wisdom" was a hydrostatic balance utilizing Archimedian hydrostatic principles and the concept of center of gravity.

Optics. In optics the foremost Arabian author was Ibn al-Haitham (c. 9651039), known to the Latins as Alhazen. His treatise on optics, which was translated into Latin in the late 12th or early 13th century, surpassed anything that has survived in Greek science. He demonstrated conclusively that light rays come to the eye from luminous objects (Avicenna also adopted this position) rather than emanate from the eye to the object, as Euclid and Ptolemy had assumed. Among a number of arguments, he observed that when one looks into a mirror reflecting solar light, the dazzling light compels one to close his eyes. This was best explained on the assumption that light comes to the eye. Using the camera obscura, he showed that light and colors are not mixed in air. Al-Haitham manufactured his own plane and parabolic mirrors, discovered spherical aberration, and determined the point of reflection of a concave spherical mirror when the positions of the eye and observer were known. To explain refraction he concluded that light rays travel more slowly in denser media (Newton incorrectly insisted that the velocity of light was greater in a denser medium), and he analyzed both incident and resultant rays into two components, one parallel, the other perpendicular, to the surface separating the two media. The velocity was made dependent on the parallel component, which was retarded in a denser medium.

Alchemy. Alchemy was pursued with vigor. A large collection of treatises written by the Brethren of Purity in the 10th century was ascribed to Jābir ibn Hayyān (fl. c. a.d. 760). These works classified substances as (1) spirits or volatile bodies (e.g., mercury, sulphur, arsenic); (2) metallic bodies, embracing all metals except mercury; (3) bodies, or all substances omitted from the first two categories. The formation of metals resulted from a compounding of pure mercury and sulfur. By altering their proportions and purity one could theoretically change one metal to another and, hopefully, base metals to gold.

Al-Rāzī (Rhazès), c. 825925, a Persian, was a practical chemist who possessed apparatus to carry out the processes of distillation, calcination, solution, evaporation, crystallization, sublimation, filtration, amalgamation, and ceration. The Arabs discovered how to prepare ammoniac, sal ammoniac, mineral acids, and borax. Mention of mineral acids occurs in the Latin West in the 13th century in a collection of alchemical works ascribed to "Geber" (the Latin name for Jābir) and based on Arabic sources.

Medicine. Arab medicine was hampered by prohibitions against human dissection; anatomy had to be learned from books. But the Arabs achieved a high level of excellence in clinical medicine. Al-Rāzī, their greatest clinical physician, was a true Hippocratic who distinguished between measles and smallpox in his Liber de pestiliencia. His Liber continens was a great influence in Europe. Scholastic medicine received its greatest organization from Avicenna's Canon of Medicine, which attempted to coordinate systematically the medical doctrines of Hippocrates and Galen with the biological concepts of Aristotle. The breadth of learning and authoritative tone made it the most widely used medical book in Islam; in Europe it was used until the 17th century. It was in a Commentary on the Anatomy in the Canon of Ibn Sina (i.e., Avicenna) that Ibn al-Nafis (d. 1288) enunciated his theory of the lesser circulation. He rejected Galen's view that blood flowed from the right ventricle to the left through pores in the ventricular septum. He insisted that blood flowed from the right ventricle to the lung via the pulmonary artery and, after mixing with air in the lungs, through the pulmonary artery to the left ventricle. This was an important step leading to the correct theory of the circulation of the blood.

The Arabs introduced many drugs, vegetables, and chemicals into their medicines and dispensed drugs in pharmacies, some of which were associated with excellent hospitals that had special wards for women, for eye diseases, and for fevers.

By the middle of the 12th century Arab science was in decline. In 1258 the Mongol sack of Baghdad and the concomitant destruction of numerous books dealt a great blow to science.

Latin West 1100 to 1500

By the 12th century western Europe was in close contact with the Muslims at a number of points. As Europeans became aware of the superior scientific knowledge possessed by the Arabs, they sought eagerly to make it their own. Translations from Arabic and Greek into Latin during the 12th and 13th centuries produced a true renaissance of Greek science. By the close of the 13th century there were available a significant portion of Greek science (though not as much as the Arabs had acquired in the 9th and 10th centuries) and many treatises by Arab authors. The new science engulfed the old handbook tradition, which, nevertheless, continued. Whereas previously Plato had been more influential than Aristotle, the latter now became the dominant influence in shaping medieval physical, biological, and logical thought.

Great universities began to develop, first at Paris and Bologna, and subsequently at Oxford, Cambridge, and many other places. Science was essentially a university enterprise embodied in special forms of literature. Commentaries on the works of Aristotle and others were commonplace. More important was the questio form in which problems of special interest were thoroughly discussed. Of particular significance were Questiones on individual Aristotelian works. In such treatises it was customary to present at the outset the arguments against one's own position, followed by arguments in support of it. Even doubts about one's own position were raised and then resolved. At the end of each question the contra arguments mentioned at the beginning were disposed of in order of presentation. Emphasis was on logical consistency. Appeals to experience or experiment, though sometimes included, were usually of secondary importance. (see education, scholastic; scholastic method.)

Until the latter part of the 13th century scholastics adhered rather closely to Aristotelian physical and philosophical concepts, convinced that most of his principles and demonstrations were necessarily true. Alarmed at a trend that seemed to restrict God's absolute power to have done things differently, Étienne tempier, Bishop of Paris, condemned (March 7, 1277) 219 articles, many of which were deterministic or claimed to be necessarily true. The effects of this action were to inhibit claims that Aristotle's principles and demonstrations were necessarily trueat best they were to be considered only probableand to encourage the formulation of possible alternative explanations, many of which were in direct conflict with Aristotle. Aristotle's influence was undermined further because some of his arguments were found to be inconsistent and unsatisfying. Finally, where Aristotle had been vague, scholastics interpreted him as they saw fitoften in novel ways. Thus in the 14th century many non-Aristotelian positions became quite respectable, and a lively interplay of scientific ideas developed.

Motion. The Aristotelian law of motion, V F/R, was abandoned by many because its critics noted that by repeatedly doubling R, it could be made greater than F; when R was greater than F, no velocity could result, but this was not what the formula expressed. To obviate this defect, thomas bradwardine proposed a new exponential law of motion that is usually represented as F 2/R 2 = (F 1/R 1) V2/V1. Therefore if F 2/R 2 = 8/1 and F 1/R 1 = 2/1, then the ratio, or exponent, V 2/V 1 = 3/1, and F 2 moves R 2 with a speed three times that with which F 1 moves R 1. This function was widely adopted and designated a ratio of ratios (proportio proportionum ). In his De proportionibus proportionum, nicholas oresme provided an elaborate mathematical foundation for this function, and even considered exponents, or ratios of velocities, that were irrational (the concept of an irrational exponent may have originated with Oresme).

Existence of Void Space. The possible existence of void space and motion in such a space was widely discussed. St. thomas aquinas and many subsequent authors held that if void space existedalmost all authors, including Thomas but excepting nicholas of autrecourt, denied its existencemotion through it would be of finite duration, not instantaneous as Aristotle held. Thomas insisted that void space would be a magnitude; and like any extended magnitude, its parts could be traversed successively so that motion would occur in some definite time interval. Although void space within the Aristotelian cosmos was denied by almost all scholastics, Bradwardine and Oresme asserted that beyond the cosmos there extended an infinite, dimensionless, indivisible void space. The attributes of this space were associated with God, who is infinite, dimensionless, and indivisible. Oresme said that "this space is infinite and indivisible and is the immensity of God and is God himself." The relationship between God's immensity and an infinite void space was a hotly debated problem in the 17th and early 18th centuries, involving Isaac Newton through the famous Clarke-Leibniz correspondence of 171516.

The existence of other worlds beyond our cosmos Aristotle had rejected thiswas made a moot point by article 34 of the condemnations of 1277, which, in effect, asserted that God could create more than one world. Almost all scholastics accepted the uniqueness of our cosmos, but some began to examine the conditions that might obtain if a plurality of worlds did exist. Oresme maintained that the elements of every world would behave just as those of our worldi.e., heavy matter would tend toward the center of its own world and light matter toward the circumference. This was a repudiation of Aristotle's doctrine of natural place, in which the natural places of the four elements were unique; now each world had its own set of natural places.

Many abandoned Aristotle's explanation of projectile motion in favor of the impetus theory (its possible connections with the Arab mail theory have yet to be established). Francis of Marchia was the first Latin scholastic to propose the theory of impetus in his explanation of sacramental causality when commenting on the Sentences at Paris (131920). The most important exposition of the theory was given at the University of Paris by john buridan, who measured the impetus of a body by its quantity of matter (analogous to mass) and the velocity imparted to it. From this standpoint impetus is analogous to Newton's momentum (mv ). However, unlike momentum, impetus also acted as a force that maintains a body's motion. Like the Arabs earlier, some (e.g., Buridan) thought of the impressed force as a permanent quality that is always destroyed by the external resistance of the medium, while others (e.g., Oresme; Galileo in his youth) held that it was self-expending. Buridan even suggested that the continuous circular motion of celestial bodies was a result of the action of impetus that God had implanted in them at the creation. The impetus remained constant since no form of resistance to motion existed in the heavens.

Acceleration of freely falling bodies also was explained in terms of impetus. Buridan assumed that the natural gravity of a body, which moves it downward, also produces impetus during the fall that causes the acceleration.

The Merton Theorem. In the 1300s at Merton College, Oxford, the mean speed theorem was formulated by william of heytesbury, richard of swyneshed, and john of dumbleton. Here, apparently for the first time, a definition of uniform acceleration was enunciated: the acquisition of equal increments of velocity in equal time periods (i.e., v t ). In the mean speed theorem this definition was utilized to demonstrate that a distance traversed by a uniformly accelerated motion was equivalent to the distance traversed in the same time by a body moving with a velocity equal to the velocity at the middle instant of the time of acceleration. The Merton theorem can be expressed as (1) S = 1/2 Vf t, for acceleration from rest, where S is the distance traversed, V f is the final velocity, and t is the time of acceleration; or as (2) S = [(Vo + 1/2) (Vf - Vo )] t, for acceleration from some velocity Vo. Since, in the first case, Vf = at (a being the acceleration),(1) can be reduced to the familiar formula S = 1/2 at 2. In the second case Vf - Vo = at. And thus (2) reduces to S = Vo t + 1/2 at 2. The context from which this theorem developed was the "intension and remission of forms," or "latitude of forms," a subject that considered how forms, including qualities, increased or decreased in intensity. Without operational justification, most qualities were treated in a purely quantitative manner involving addition and subtraction of quantitative parts of qualities. Motion was treated as just another quality capable of such treatment. At Oxford proofs of the Merton theorem were arithmetical, while later at Paris they were geometrical. The high point of this development was Oresme's De configurationibus qualitatum, in which uniformly and nonuniformly varying qualities were represented by geometric figures. In his demonstration of the mean speed theorem, Oresme assumes that the altitude of rectangle AFGB is equal to line DE, the mean velocity of the uniformly accelerated motion represented by triangle ABC. He then shows that the area of rectangle AFGB equals the area of triangle ABC, so that the distances traversed are equal since the two figures represent the total distances traversed. In his Two New Sciences (1638) Galileo proves the mean speed theorem (Third Day, theorem 1, prop. 1) in a manner similar to Oresme's geometric method; although Galileo inverts the coordinates, his diagram is like Oresme's. There is little doubt that Galileo is following the medieval tradition. Application of the theorem to falling bodies was made by Domingo de soto in 1555. Thus the widely accepted medieval view that the velocity of a falling body is proportional to distance (i.e., v s ) was now corrected, since the Merton mean speed theorem made velocity directly proportional to time (i.e., v t ). Galileo, who also believed that bodies fell in a manner described by this theorem, made very significant use of it in helping to formulate modern mechanics.

Statics. Acquainted with the Greek and Arabic statical treatises, Jordanus de Nemore composed a series of original works that made notable contributions to statics. In proofs of the laws of the lever, inclined plane (this had eluded the Greeks), and bent lever (proved correctly for the first time), he utilized the principle of virtual work. M. Clagett summarizes the proof of the law of the lever as follows: "(1) Either a lever with weights inversely proportional to the lever arms is in equilibrium or it is not. (2) If it is assumed that it is not, then one weight or the other descends. (3) But if one of the weights descends, it would perform the same action (i.e. work) as if it lifted a weight equal to itself and placed at an equal distance from the fulcrum through a distance equal to the distance it descended. But equal weights at equal distances are in equilibrium. Thus a weight does not have the force sufficient to lift its equal weight the same distance it descends. It, therefore, does not have the force to perform the same actionnamely, to lift a proportionally smaller weight a proportionally longer distance" (Science of Mechanics 7778). Jordan also utilized the concept of a component of force in what he called positional gravity (gravitas secundum situm ). This is equivalent to the assertion that F = W sin a, where F is the force along the oblique path, W is the weight, and a the angle of inclination of the oblique path. Medieval statics influenced such later authors as Leonardo da Vinci, N. Tartaglia, and perhaps Galileo.

Optics. Optics dealt with the anatomy of the eye and the nature of light. Greek and Arabic sources (especially the Optics of Alhazen) were fundamental. The behavior of light (reflection and refraction) was considered geometrically, but consideration was given also to the physical nature of light. Under Neoplatonic and Augustinian influence, light was taken as a fundamental universal entity (it was the basis of matter and corporeality for robert grosseteste), and its study was expected to reveal basic knowledge about the world. The transmission of light was held to be a special case of the transmission of effects through a medium (known as the "multiplication of species").

The law of reflection was well known, but no law of refraction was enunciated until Grosseteste proclaimed a quantitative, though false, law in which the angle of refraction equaled half the angle of incidence. The greatest triumph of medieval optics was theodoric of freiberg's (d. c. 1311) correct qualitative explanation of the formation of primary and secondary rainbows. Previous accounts relied on either reflection or refraction in clouds until witelo (b. c. 1230) emphasized both reflection and refraction in individual raindrops. roger bacon gave the correct maximum altitude of the primary bow as 42° and properly noted that each observer sees a different bow that moves with him. But it was Theodoric who explained that the primary bow is formed when light falling on raindrops is refracted upon entering, then reflected at the inner concave surface and refracted out again. For the secondary bow, whose colors are in reverse order, a second internal reflection occurs before the second refraction. Centuries later, descartes gave essentially the same explanation buttressed by a superior geometrical description. Of the colors of the rainbow, Theodoric could offer no adequate explanation, and none appeared until Newton's classic experiments.

The first systematic description of magnetism is found in the Letter on the Magnet (1269) by Peter of Maricourt (Peregrinus). Although some of its properties were known in antiquity (e.g., attraction and repulsion), Peter advances far beyond by distinguishing the north and south poles and explaining how to locate them on a spherical lodestone (this may be the first extant account of an artificial magnet). He also records the repulsion of like poles and the attraction of unlike, as well as the fact that each part of a broken magnet is itself a complete magnet with north and south poles. Peter includes what may be the first description of a dry pivoted magnetic compass with a graduated circle on which were marked the cardinal points.

Astronomy and Mathematics. Although no striking innovations or improvements were made in medieval technical astronomy, a conflict developed between Aristotelian cosmology and Ptolemaic astronomy. The former, with the earth in the exact center and only one motion permitted for each sphere, could not save the astronomical phenomena, while the latter could do so only by violating these generally accepted cosmological principles. In the end most accepted bothPtolemaic astronomy for computational purposes and Aristotelian cosmology as representative of physical reality. Some, however, such as Bernard of Verdun (fl. c. 1300), acting on the belief that the system that best saved the phenomena must be physically true, tried to construct a universe in terms of physical epicycles and eccentrics.

The possible diurnal rotation of the earth was considered long before Copernicus asserted its reality. Buridan and Oresme, and Aquinas less explicitly, argued that if the earth had a diurnal rotation while the sphere of the fixed stars remained immobile, the same astronomical phenomena would be saved equally well. On other grounds, however, both acquiesced in the traditional Aristotelian view of an immobile earth. But some of their arguments were to appear later in Copernicus's revolutionary treatise.

Except for the work of Leonardo Fibonacci (c. 1170c. 1240), mathematics did not attain the high level reached by the Greeks and Arabs. This may be explained partly by the absence (except for fragments) of the Conics of Apollonius and because the works of Archimedes, translated by william of moerbeke in 1269, were but little used. Fibonacci, however, had studied the best Greek and Arab authors. In a series of important treatises he systematically expounded the Hindu (or Arabic) numerals, used the Fibonacci series (named after him) where each term is equal to the sum of the two preceding terms, and handled difficult algebraic problems, as well as extracting approximative square and cubic roots. He included indeterminate problems of the first and second degree. He was also a capable geometer. No subsequent mathematician before 1500 equaled him in ability. There was, however, a great interest in mathematics; and scholastics argued about its foundations, as, for example, whether lines, planes, and solids were composed of indivisibles or of continuously divisible magnitudes. Infinite convergent and divergent series were widely discussed, and many such series were formulated in both physical and mathematical contexts. For example, Oresme shows the convergence of the series a/n + a/n (1 - 1/n ) + a/n (1 - 1/n)2 + ··· + a/n (1 - 1/n )m + ··· = a; and the divergence of the harmonic series 1 + 1/2 + 1/3 + 1/4 + ··· + 1/n + 1/n + 1 ···.

Medicine and Biology. Medicine received its academic formalization in the medieval universities of western Europe. Galen, Hippocrates, Avicenna, and Rhazès formed the foundation of the medical curriculum. Physicians from the first Western medical school at Salerno helped spread medical learning and influenced developments. A measure of advance is seen in the fact that, while animals only were dissected at Salerno, human dissection, an outgrowth of postmortems, was a regular feature at the University of Bologna from the late 13th century. Human dissection, which had been prohibited since the days of the Roman Empire, focused attention once again on the human body. Bologna became the leading medical school, where, in contrast to Paris, surgery was taught along with medicine. The greatest medieval anatomist was Mondino de' Luzzi, who personally performed dissections and wrote the most widely used anatomical text, the Anathomia. In the 15th century the Branca family of Sicily performed plastic surgery, replacing and repairing noses, lips, and ears. The flesh was taken at first from the face and then from the arm to prevent facial deformity. Of clinical interest were the consilia begun by Taddeo Alderotti (1223-c. 1300) and continued into the 17th century. A consilium was a prominent physician's case history, usually incomplete because it preceded rather than followed treatment. It had a formal basic structure, listing, in order, the ailment and its symptoms, regimen, and prescriptions for drugs and medicines. The many plagues that swept Europe in the 14th and 15th centuries compelled many cities to establish quarantines and other sanitary measures. A great body of literature emerged concerning the plague and its effects.

In biology some excellent descriptive works appeared in zoology and botany. Emperor frederick ii composed a remarkable treatise on falconry, De arte venandi cum avibus, with brilliant anatomical descriptions of birds. albert the great described animals and plants with considerable accuracy and rejected fanciful stories about certain animals (e.g., the beaver and salamander). Albert also repeated Aristotle's experiment on hen's eggs, tracing the development of the embryo by opening a collection of eggs at various intervals. Albert was perhaps the greatest natural scientist of the 13th century. He is remarkable for the reliance he placed on observation and experiment, and this alone is enough to set him above his contemporaries and many who followed him. More naturalistic representation of plants and animals are found in medieval manuscripts, although the difficulty of reproducing naturalistic drawings prior to the invention of printing prevented widespread standardization of illustrations and hindered the advance of biology. Biological classification schemes tended to follow the general principles laid down by Aristotle.

Bibliography: g. sarton, Introduction to the History of Science (Baltimore, Md. 192748). l. thorndike. A History of Magic and Experimental Science (New York 192358). p. m. m. duhem, Le Système du monde, 10 v. (Paris 191317; repr. 195459); Études sur Léonard de Vinci, 3 v. (Paris 190613; repr. 1955). m. clagett, The Science of Mechanics in the Middle Ages (Madison, Wis.1959); Archimedes in the Middle Ages (Madison, Wis. 1964). a. c. crombie, Medieval and Early Modern Science, 2 v. (2d ed. Cambridge, Mass. 1961); Robert Grosseteste and the Origins of Experimental Science (Oxford 1953). e. j. dijksterhuis, The Mechanization of the World Picture, tr. c. dikshoorn (Oxford 1961). j. a. weisheipl, The Development of Physical Theory in the Middle Ages (New York 1960). a. maier, Studien zur Naturphilo-sophie der Spätscholastik, 5 v. (Rome 194958). c. h. haskins, Studies in the History of Medieval Science (2d ed. Cambridge, Mass. 1927). r. taton, ed., A History of Science, v. 1 Ancient and Medieval Science, tr. a. j. pomerans (New York 1963). l. leclerc, Histoire de la médicine arabe, 2 v. (Paris 1876; New York 1961). a. castiglioni, History of Medicine, tr. and ed. e. b. krumbhaar (2d ed. New York 1947). k. sudhoff, Kurzes Handbuch der Geschichte der Medizin (Berlin 1922), 3d and 4th ed. of j. l. pagel, Einführung in die Geschichte der Medizin, 3 v. (Jena 190105). f. sturnz, Geschichte der Naturwissenschaften im Mittelalter (Stuttgart 1910). j. needham, A History of Embryology (2d ed. New York 1959).

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