The reinterpretation of Aristotle in the thirteenth century led to the emergence of European science, particularly through the researches of Robert Grosseteste and his followers in Paris and Oxford.
One of the pioneers of this new European science was Jordanus Nemorarius (fl. ca. 1220), a contemporary of Grosseteste's. Virtually nothing can be said about the life of Jordanus, who is known only through the inclusion of his works in the Bibliomania, a catalog of the library of Richard de Fournival done sometime between 1246 and 1260, in which twelve treatises are ascribed to him.
Jordanus made his greatest contribution in the medieval “science of weights” (scientia de ponderibus), now known as “statics,” the study of forces in equilibrium. One of the concepts he introduced was that of “positional gravity” (gravitas secundum situm),which he expressed in the statement that “weight is heavier positionally, when, at a given position, its path of descent is less oblique.” An example would be a block on an inclined plane, whose apparent weight, the force with which it presses against the surface, is greater if the angle of inclination is less. This is equivalent to resolving the weight into two components, one perpendicular to the plane, which is the apparent weight or “positional gravity,” and the other parallel to the surface.
The concept of positional gravity was used by Jordanus in his study of the most basic problem in statics, that of the beam balance, where two weights are suspended on either side of a fulcrum. According to Jor-danus, “if the arms of the balance are unequal, then if equal weights are suspended from their extremities, the balance will be depressed on the side of the longer arm.” His proof of this statement again uses the concept of positional gravity, which in this case is equal to the weight of the object times its lever arm, the perpendicular distance from the fulcrum to the line of action of the weight. This is now known as the “moment of a force,” or “torque,” a measure of its effectiveness in rotating the balance, equilibrium resulting if the two torques are equal and opposite.
Jordanus then proceeded to “prove” the law of the lever, which is that two objects will balance each other if their weights are inversely proportional to their lever arms. Here he used the concept of “work,” the product of the weight of an object times the distance through which it is lifted or otherwise moved, the first clear definition of this fundamental concept in physics. He also introduced the concept of “virtual velocity,” that is, one that is vanishingly small, since a real movement cannot take place in a system under equilibrium. He used this concept in examining two objects balanced on a lever, where in a virtual displacement the positive work done in lifting one weight is equal to the negative work in lowering the other, leading to the conclusion that the system is in equilibrium. His proof involves what is known as the “axiom of Jordanus,” which says that the motive power that can lift a given weight a certain height can lift a weight k times heavier to 1/k times that height, where k is any number.
The same concepts were used by Jordanus in studying the equilibrium of two different connected weights on inclined planes of different inclinations, which he treated as a generalized case of the law of the lever. The proof refers to a triangle ABC which has BCas its base and a right angle at A, where a pulley connects two weights, with w(1) on side AB and w(2) on side AC He showed that the two weights will be in equilibrium if their positional gravities are equal—that is, if the components of each weight down its plane are equal to that of the other in the opposite direction. This can be reduced to the equation w(1)/w(2) = AB/AC, which is equivalent to the law of the lever.
Jordanus also made contributions in mathematics, where he shows no trace of Islamic influence, following rather in the Greco-Roman tradition of Nichomachus and Boethius. He was the first to use the letters of the alphabet in arithmetical problems for greater generality, and he presented algebraic problems leading to linear and quadratic equations. He also worked in geometry and, following Archimedes, solved problems involving the determination of the center of gravity of triangles and other plane figures, as well as doing pioneering research in stereo-graphic projection.
The scholar known as Gerard of Brussels, who appears to have been associated with Jordanus, may have been the first European to deal with kinematics, a purely mathematical description of motion. His treatise on kinematics, De Motu, written sometime between 1187 and 1260, was strongly influenced by Euclid and Archimedes.
During the second quarter of the fourteenth century a group of scholars at Merton College, Oxford, developed the conceptual framework and technical vocabulary of the new science of motion. These were Thomas Bradwardine, William Heytesbury John of Dumbleton, and Richard Swineshead, who continued the Oxford tradition in science initiated by Robert Grosseteste.
Thomas Bradwardine (ca. 1290-1349) received his bachelor's, master's, and doctoral degrees at Oxford in the years 1321-48, and from 1323 until 1335 he was a fellow of Merton College. In 1339 he was chaplain and perhaps confessor to King Edward III, and he accompanied the king to France in the campaign of 1346. He was elected archbishop of Canterbury on 4 June 1349, but he died of the plague on 26 August of that same year.
Bradwardine's principal work is the Tractatus Proportionum, completed in 1328. The problem that Bradwardine tried to solve in this work was to find a suitable mathematical function for the Aristotelian law of motion, which states that the velocity (v) of an object is proportional to the power (p) of the mover divided by the resistance (r) of the medium. Bradwardine focused on the change in velocity rather than the velocity itself and tried to show how this was related to power and resistance. Stated mathematically, he looked for a functional relation between the dependent variable v and the two independent variables p and r; that is, given values for p and r, to find the corresponding value of v After trying and rejecting a number of equations, he finally settled on a law of motion that, in modern terminology, states that the velocity is proportional to the logarithm of p/r Bradwardine never tested his law of motion, which would have shown him that it was not correct. Nevertheless, his formulation of the problem in terms of a mathematical functional relationship was an important step forward in the science of dynamics, one that was followed by his successors at both Oxford and Paris. Their researches established the foundations of the late medieval tradition of the calcula-tores,those who studied the quantitative variation of motion, power, and qualities in space and time.
William Heytesbury's name, variously spelled, appears in the records of Merton College for 1330 and 1338-39, and he may be the William Heighterbury or Hetisbury who was chancellor of the university in 1371. His most influential work is his Regulae Solvendi Sophismata, dating from 1335.
Heytesbury in his Regulae, defined uniform acceleration as motion in which the velocity is changing at a constant rate, either increasing or decreasing. For such motion he defined acceleration as the change in velocity in a given time, which would be negative in the case of deceleration. He also introduced the notion of instantaneous velocity—that is, the speed at a particular moment—defining it as the distance traveled by a body in a given time if it continued to move with the speed that it had at that moment. He showed that, for uniformly accelerated motion, the average velocity during a time interval is equal to the instantaneous velocity at the midpoint of that interval. This was known as the mean speed rule of Merton College; it was adopted by Heytesbury's successors at both Oxford and Paris.
John of Dumbleton (fl. 1331-49) is mentioned as a fellow of Merton College in the years 1338-48. His best-known writing is the Summa Logi-cae et Phibsophiae Naturalis, a vast work that critically discusses most of the topics in physics and philosophy of his time. Here he covered rates of change, including motion, change of quality, and growth, in relation to a fixed scale such as distance or time. A change was said to be uniform when there were equal variations in equal intervals of time, and “dif-form” when the variations increased or decreased in time. Thus in uniform motion equal distances are covered in equal intervals of time, while in difform motion, the intervals traversed in successive time intervals increase or decrease.
Richard Swineshead, who was known as the “Calculator,” is best remembered as the author of the Liber Calculationum (ca. 1340-50), a work that became famous for its extensive use of mathematics in physics. The Liber Calculationum concentrates on calculating the values of physical variables and solving problems about their changes. The work is divided into sixteen treatises, of which the last three are devoted to “Local Motion;” there he elaborates at exhausting length on Bradwar-dine's law of dynamics in every conceivable type of motion, many of which have no known parallel in nature. The Liber Calculationum was disseminated widely in Europe, and it was printed at Padua (ca. 1477), Pavia (1498), and Venice (1520). The Venetian edition was later transcribed for the great German mathematician and philosopher Leibniz (1646-1716), who praised Swineshead for having introduced mathematics into scholastic philosophy.
Advances in the theory of motion were also being made in Paris, beginning with the work of Jean Buridan (ca. 1295-ca. 1358). Little is known of Buridan's origins other than the fact that he was born in the diocese of Arras. Soon after 1320 he obtained his master's degree at the University of Paris, where he was twice elected rector, first in 1328 and again in 1340.
Buridan's extant writings consist of the lectures he gave at the University of Paris, where the curriculum was based largely on the study of Aristotle, along with textbooks on logic, grammar, mathematics, and astronomy. Buridan wrote his own textbook on logic as well as two advanced treatises on the subject. All of his other writings are commentaries and books about Aristotle's principal works.
Buridan's philosophy of science is enunciated in his Questions on Aristotle's Physics and Metaphysics There he makes the distinction between premises whose necessity is determined through logic and those based on empirical evidence, whose necessity is conditional “on the supposition of the common cause of nature.” He held that the principles of natural science are of the second type, noting that they “are not immediately evident… but they are accepted because they have been observed to be true in many instances and to be false in none.”
Buridan's most important contribution to science is his so-called impetus theory, the revival of a concept first proposed in the sixth century by John Philoponus. He explains the continued motion of a projectile as being due to the impetus it received from the force of projection, and says it “would endure forever if it were not diminished and corrupted by an opposing resistance or something tending to an opposed motion.” Buridan defines impetus as a function of the body's “quantity of matter” and its velocity which is equivalent to the modern concept of momentum, or mass times velocity, where mass is the inertial property of matter, its resistance to a change in its state of motion. As applied to the case of free fall, Buridan explains that gravity not only is the primary cause of the motion but also imparts additional increments of impetus to the body as it falls, thus accelerating it—that is, increasing its velocity.
Buridan extended his concept of impetus to explain the motion of the celestial spheres, which in Aristotelian cosmology rotated with constant velocity. He argued that there was no need to have immaterial “intelligences” as the unmoved movers of the celestial spheres, as supposed by Aristotle, because their motion was inertial after the initial impetus they received from the Creator. “For it could be said that God, in creating the world, set each celestial orb in motion … and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more.” He added that this was why God could rest on the seventh day after the Creation, for the inertia of the celestial spheres would keep them in motion without the need for any additional divine effort.
In one of the questions in De Caelo et Mundo, Buridan asks if a proof can be given for Aristotle's geocentric model, in which the earth is at rest at the center of the cosmos with the stars and other celestial bodies rotating around it. He notes that many in his time believed the contrary, that the earth is rotating on its axis and that the stellar sphere is at rest, adding that it is “indisputably true that if the facts were as this theory supposes, everything in the heavens would appear to us just as it now appears.” In support of the earth's rotation, he says that it is better to account for appearances by the simplest theory, and it is more reasonable to think that the vastly greater stellar sphere is at rest and the earth is moving, rather than the other way around. But, after refuting the usual arguments against the earth's rotation, Buridan says that he himself believes the contrary, using the argument that a projectile fired directly upward will fall back to its starting point, which is true, at least approximately, whether or not the earth is rotating.
The impetus theory was adopted by Buridan's students and became known throughout Europe, though in a corrupted form that restored some Aristotelian notions. Aside from that, Buridan is credited with eliminating explanations involving Aristotelian final causes from physics. His books were required reading at universities until the seventeenth century and would have been read by both Copernicus and Galileo. Copernicus used some of Buridan's arguments in discussing the earth's motions, and Galileo revived the theory of impetus in formulating his own laws of kinematics and dynamics.
Like other famous medieval scholars, Buridan was the subject of apocryphal stories. One of these tales, perpetuated by the poet François Villon, tells of how Buridan had an affair with the wife of Charles V of France, who had him tied up in a sack and thrown into the Seine.
The most distinguished of his students was Nicole Oresme (ca. 1320-1382), who studied under Buridan at the University of Paris in the 1340s. He was chosen grand master of the university's College of Navarre in 1356 and three years later became secretary of the dauphin of France, the future King Charles V. From about 1369 he was employed by Charles to translate some of Aristotle's Latin texts into French, for which he was rewarded in 1377 when, at the behest of the king, he was made bishop of Lisieux, a post he held until his death in 1382.
Oresme gave a graphical demonstration of the Merton mean speed rule in his Tractus de Conjigurationibus Qualitatum et Motuum, written in the 1350s while he was at the College of Navarre in Paris. The graph plots the velocity (v) on the vertical axis as a function of the time (t) on the horizontal axis, as, for example, in the case of a body starting from rest and accelerating so that its velocity increases by 2 feet per second every second. Graphing the motion for 4 seconds, we see that the velocity increases each second, from 0 to 2 then 4 then 6 then 8 feet per second at the end of 4 seconds. This graphs in the form of a straight line rising from 0 to 8 feet per second over a time interval of 4 seconds, forming a right triangle with a height of 8 and a base of 4. The acceleration (a) is equal to the slope of the straight line, which is 8/4;, or 2, in units of feet per second per second. The average velocity is half of the final velocity, which is 8/2;, or 4 feet per second. The mean speed rule then gives the distance s traveled in 4 seconds as 4 times 4, or 16 feet. The rule can be applied over each one-second interval, so that the average velocity for each second increases from 1 to 3 to 5 to 7 feet per second. The distance in feet traveled in the first second is then 1, in the second 3, in the third 5, and in the fourth 7. These results can be generalized by the equations v = a × t, and s = a/2 × t2. These are the kinematic equations formulated by Galileo in his Dialogue Concerning the Two New Sciences (1638), where he used Oresme's demonstration of the Merton mean speed rule in his proof.
Oresme also had original ideas in astronomy which he presented in his Livre du Ciel et du Monde d'Aristote, written in 1377 for Charles V. One of these was his comparison of the eternal motion of the celestial spheres to a perpetual mechanical clock, set in motion by God at the moment of their creation. He writes that “it is not impossible that the heavens are moved by a power or corporeal quality in it, without violence and without work, because the resistance in the heavens does not incline them to any other movement nor to rest but only that they are not moved more quickly.”
Orseme objected to the Aristotelian notion that the earth was the stationary center of the finite cosmos and the reference point for all motion and gravitation. He argued that motion, gravity, and the directions in space must be regarded as relative, saying that God, through his omnipotence, could create an infinite space and as many universes as he chose. Oresme was thus able to reject the idea that the earth was the fixed center of the cosmos to which all gravitational motions were directed. Instead he proposed the idea that gravity was simply the tendency of bodies to move toward the center of spherical mass distributions. Gravitational motion was relative only to a particular universe; there was no absolute direction of gravity applying to all of space.
Oresme proposed, “subject to correction, that the earth is moving with daily motion and the heavens not. And first I will declare that it is impossible to show the contrary by any observation; secondly from reason; and thirdly I will give reasons in favor of the opinion.”
Oresme's arguments in favor of the earth's motion would later be used by both Copernicus and Galileo. Despite all of these arguments Orseme, who at the time had just been appointed bishop of Lisieux, in the end rejected the idea of the diurnal rotation of the earth as being contrary to Christian doctrine, saying, “For God fixed the earth, so that it does not move, notwithstanding the reasons to the contrary.” Orseme's attitude was not uncommon among his clerical contemporaries, for in his position as bishop he was sworn to uphold the doctrines of the Catholic Church, even when they conflicted with his own philosophical ideas.
Meanwhile, advances were being made in other areas of science, namely, natural philosophy, cosmology, magnetism, astronomy, and optics, as well as in mathematics and its application to astronomy and other fields of science.
Giles of Rome (ca. 1247-1316) was a student of Thomas Aquinas's in Paris. When Etienne Tempier delivered his second condemnation of Aristotelianism and Averroism in 1277, Giles's writings were censured and he was forced to leave Paris. He returned to Paris in 1285 at the request of Pope Honorius IV, after having retracted several of his theses. He was appointed archbishop of Bourges in 1295 by Pope Boniface VIII, and died in 1316 during a stay at the papal court in Avignon.
One of the original ideas proposed by Giles was that there are natural minima below which physical substances cannot exist, thus implying an atomic theory of matter. He investigated the nature of the vacuum through experiments with a clepsydra, a cupping glass, and a siphon, showing that the void exerted a force of suction. He disagreed with Aristotle and his own contemporaries in holding that celestial matter is identical to that of the terrestrial world. He rejected Aristotle's model of homocentric spheres in favor of Ptolemy's theory of eccentrics and epicycles, saying that observational evidence must settle the controversy between the Aristotelian “physicists” and the Ptolemaic “mathematicians.” He also admitted the possibility of a plurality of worlds.
The earliest extant treatise on magnetism is by Peter Peregrinus, of whom virtually nothing is known except what appears in his work and in possible references to him by Roger Bacon. Peter's treatise—actually, a letter—is the Epístola Petri Peregrini de Maricourt ad Sygerum de Foucau-court, Miltetn, De Magnete (Letter on the Magnet of Peter Peregrinus of Maricourt to Sygerus of Foucaucourt, Soldier). Peter concluded the letter with the note that it had been “Completed in camp, at the siege of Lucera, in the year of our Lord 1269, eighth day of August.” This would indicate that Peter was at the time in the army of Charles of Anjou, king of Sicily, who was then besieging the city of Lucera in southern Italy.
The Epístola is in two parts, of which the first, in ten chapters, describes the properties of the lodestone, or magnetic rock; the second is devoted to the construction of three instruments using magnets. Peter's observations led him to make the distinction between the north and south magnetic poles; to establish the rules for the attraction and repulsion of magnetic poles; to show the magnetization of iron by bringing it in contact with a magnet; and to demonstrate that a magnetic needle when broken in half forms two separate magnets. He showed that a magnetic needle oriented itself in the north-south direction, thereby inventing the compass, which he said could be used to map the meridians of the earth's magnetic field. He believed, mistakenly, that the poles of a magnetic needle point to the poles of the celestial sphere, the points about which the stars appear to be rotating, which are actually projections of the earth's axis of rotation. He attempted to construct a perpetual motion machine using magnets, and he blamed his failure on his lack of skill rather than the impossibility of creating an eternal source of energy. The Epistola was very popular in the late medieval era, as evidenced by the fact that there are at least thirty-one extant manuscript copies. It had a great impact on William Gilbert, who in his famous De Magnete (1600), paid tribute to Peter Peregrinus and acknowledged his debt to his predecessor.
Campanus of Novara, who flourished in the second half of the thirteenth century, is best known for his translation of Euclid's Elements, but he was also a distinguished astronomer. Little is known of his life other than that he was chaplain to Popes Urban IV, Nicholas IV, and Boniface VIII, and that he spent his last years at the Augustine monastery at Viterbo, in Italy, where he died in 1296.
Campanus's principal astronomical work is his Theorica Planetarium, a description of the structure and dimensions of the universe according to Ptolemaic theory, together with instructions for the construction of an instrument, later called an equatorium, designed to find the position of any of the celestial bodies at a given time. Campanus based his calculations on the work of the ninth-century Arabic astronomer Alfraganus (al-Farghani), who had in turn derived them from the Planetary Hypotheses of Ptolemy. Campanus probably also learned about the equatorium from an Arabic source, for descriptions of equatoria were written nearly two centuries earlier in Al-Andalus by Ibn al-Samh and Ibn al-Zarqali.
One of the most notable of the early European astronomers was William of St. Cloud, who flourished in France during the late thirteenth century. The earliest date of his activity is 28 December 1285, when he observed a conjunction of Saturn and Jupiter, to which he refers in his Almanack, completed in 1292. His other major work is his Calendrier de la Reine, also completed in 1292, dedicated to Queen Marie of Brabant, widow of Philip III, “the Bold,” and which he translated into French at the request of Jeanne of Navarre, wife of Philip IV, “the Fair.”
Queen Marie's Calendrier represents William's effort to put the calendar on a purely astronomical basis. This led him to contradict the computations in the ecclesiastical calendar, which he found full of errors, indicating the need for calendar reform. The purpose of his Almanack was to provide listings in which the positions of the celestial bodies were given directly, as contrasted to earlier tables, which only gave the elements by which those positions could be calculated. He points out the errors in the earlier planetary tables and shows how he corrected them. These tables were those of Toledo, used in the Muslim calendar, and of Toulouse, the adaptation of the Toledan Tables to the Christian calendar. William's Almanack makes no mention of the Alphonsine Tables, which were not used in Paris before 1320.
William's observations were remarkably precise, evidence of the high level that had been achieved in European astronomy by his time. By comparing his astronomical observations with ancient Greek values he was able to measure the change in the spring equinox, which he interpreted as a steady precession rather than the trepidation theory introduced by Theon of Alexandria.
A new set of astronomical tables was completed in 1327 by John of Ligneres and his students, whose work was heavily dependent on the Toledan Tables This work, known as the Large Tables, included a catalog of the positions of the forty-seven brightest stars and was easier to use than any of the earlier tables. Thus it became very popular, though it was eventually supplanted in Paris by the Alphonsine Tables
Levi ben Gerson (1288-1344) was a polymath who wrote books on astronomy, physics, mathematics, and philosophy, as well as commentaries on the Bible and the Talmud. He lived in Orange and Avignon, which were not affected by the expulsion of the Jews from France in 1306 by King Philip the Fair. He was also on good terms with the papal court in Avignon, as evidenced by his dedication of one of his works to Pope Clement IV in 1342.
Levi's greatest work is his Milhamot Adonai (The Wars of the Lord), a philosophical treatise in six books, the fifth of which is devoted to astronomy. Here Levi presents his model of the universe, based on several Arabic sources, principally al-Battani, Jabir ibn Aflah, and Ibn Rushd. His model differed in important respects from that of Ptolemy, whose theories did not always agree with observations made by Levi. This was particularly so in the case of Mars, where Ptolemy's theory had the apparent size of the planet varying by a factor of six, while Levi's observation found that it only doubled. The instruments used by Levi included one of his own invention, the “Jacob's staff,” a device to measure angles in astronomical observations. He also employed the camera obscura—invented by Alhazen (Ibn al-Haytham)—for observing eclipses and determining the eccentricity of the sun's orbit. Levi's astronomical work was influential in Europe for five centuries, and his Jacob's staff was also used for maritime navigation until the mid-eighteenth century.
During the first half of the fourteenth century an important school of astronomy was active at Oxford. The most notable of the Oxford astronomers was Richard of Wallingford (ca. 1292-1336). Richard, the son of a blacksmith, studied at Oxford from about 1308 until 1315, when he joined the Benedictine order at St. Albans abbey. Two years later he was sent back to Oxford for further studies, remaining there until 1327, when he was appointed abbot of St. Albans. After his appointment he visited Avignon for the papal confirmation, and when he returned to St. Albans he found that he had contracted leprosy; he died from the disease in 1336.
Richard's works were all written during his years at Oxford. One of these was his Quadripartitium, the first comprehensive treatise on spherical trigonometry written in Latin Europe. Richard's most important work was his Tractus Albionis, which dealt with the theory, construction, and use of an instrument that he had invented called the Albion, a form of equatorium used to perform all sorts of astronomical measurements and calculations.
Richard also built an enormous mechanical clock, ten feet in diameter, which he installed on the wall of the south transept in the abbey church. Besides the time of day, it also showed the motion of the celestial bodies, the phases of the moon, and the tides. The clock was destroyed in the sixteenth century, but several drafts of Richard's design have survived, the oldest extant plans of any mechanical clock, the most sophisticated chronometer of the medieval era. (The earliest mechanical clocks in Europe appear to date from the end of the thirteenth century.) The mechanical clock led to the notion of time as a physical quantity that could be expressed numerically in terms of units on a scale and used in scientific theories. Since Richard's clock was also a planetarium, it lent credence to the notion that the universe was a divinely designed clockwork.
The earliest appearance of a mechanical clock in literature seems to be in a passage of Dante's Paradiso, in the last lines of Canto X, written between 1316 and 1321, a decade or so before Richard built his clock.
Forthwith
As clock, that calleth up the spouse of God
To win her Bridegroom's love at matin's hour.
Each part of other fitly drawn and urged,
Sends out a tinkling sound, of note so sweet,
Affection springs in well-disposed breast;
Thus saw I move the glorious wheel; thus heard
Voice answering voice, so musical and soft,
It can be known but where day endless shines.
Another area in which the new European science developed was optics, the study of light, which had begun at Oxford with the work of Robert Grosseteste and his disciple Roger Bacon. The first significant advance beyond what they had done was by the Polish scholar Witelo (b. ca. 1230-35; d. after ca. 1275).
Witelo's best-known work is the Perspectiva, which is based on the works of Robert Grosseteste and Roger Bacon as well as those of Alhazen, Ptolemy, and Hero of Alexandria. It would seem that the Perspectiva was not written before 1270, since it makes use of Hero's Catop-trica, the translation of which was completed by William of Moerbeke (b. ca. 1220-35; d. before 1286) on 31 December 1269.
Witelo adopted the “metaphysics of light” directly from Grosseteste and Bacon, and in the preface to the Perspectiva he says that visible light is simply an example of the propagation of the power that is the basis of all natural causes. But he disagrees with Grosseteste and Bacon where they say that light rays travel from the observer's eye to the visible object, and instead follows Alhazen in holding that the rays emanate from the object to interact with the eye.
The Perspectiva describes experiments performed by Witelo in his study of refraction. Here his method is similar to that of Ptolemy; he measures the angle of refraction for light in passing from air into glass and also into water, for angles of incidence ranging from 10 to 80 degrees. He tried to explain the results by a number of mathematical generalizations, attempting to relate the differences in refraction to the difference in the densities of the two media. In addition, he produced the colors of the spectrum by passing light through a hexagonal crystal, observing that the blue rays were refracted more than the red.
Witelo also studied refraction in lenses, making use of the concept later known as the principle of minimum path. He justified this by the metaphysical notion of economy, saying, “It would be futile for anything to take place by longer lines, when it could better and more certainly take place by shorter lines.”
Witelo followed Grossteste in holding that the “multiplication of species” could be used to explain the propagation of any effect, including the divine emanation and astrological influences. In the preface to the Perspectiva, which he addresses to William of Moerbeke, he writes “of corporeal influences sensible light is the medium.” He also writes that “there is something wonderful in the way in which the influence of divine power flows in to things of the lower world passing through the powers of the higher world.”
The next advances in optics were made by Dietrich of Freiburg (ca. 1250-ca. 1311), who is sometimes called Theodoric. Dietrich, who is thought to be from Freiburg in Saxony, entered the Dominican order and probably taught in Germany before studying at the University of Paris, circa 1275-77.
Dietrich's principal work is his treatise On the Rainbow and Radiant Impressions, the latter term meaning phenomena produced in the upper atmosphere by radiation from the sun or any other celestial body. He was the first to realize that the rainbow is due to the individual drops of rain rather than the cloud as a whole. This led him to make observations with a glass bowl filled with water, which he used as a model raindrop, for, he writes, “a globe of water can be thought of, not as a diminutive spherical cloud, but as a magnified raindrop.” His observations and geometrical analysis led him to conclude that light is refracted when it enters and leaves each raindrop, and that it is internally reflected once in creating the primary bow and twice for the secondary arc, the second reflection reversing the order of the colors in the spectrum. Although he made a number of errors in his analysis, his theory was far superior to those of any of his predecessors, and it paved the way for researches by his successors.
The formation of primary and secondary rainbows, from Newton's Opticks.
Dietrich's theory of the rainbow is very similar to that of his Persian contemporary Kamal al-Din al-Farisi. Dietrich does not cite the work of al-Farisi, but since it was never translated from Arabic into Latin he was probably not aware of it. In any event, it seems that the emerging European science had by the beginning of the fourteenth century reached a level comparable to that of Arabic scientific research, at least in optics. But whereas the work of al-Farisi was the last great achievement of Arabic optics, Dietrich's researches would be an important stage in the further development of European studies in the science of light, culminating in the first correct theories of the rainbow and other optical phenomena in the seventeenth century.