John Philoponus (490–570 CE), whose surname means lover of toil, was a Greek Christian who lived and worked as a philosopher, theologian, and scientist in the century immediately following the invasion of Roman Italy by Germanic tribes in 476 CE. While he flourished more than a century after Theodosius (347–395 CE) had established Catholic Christianity as the official religion of the empire in 380 CE, Philoponus was taught by and worked with pagan philosophers associated with the library in Alexandria. Philoponus wrote extensive commentaries on Aristotle and in several treatises argued against the Aristotelian doctrine of the eternity of the world. He believed that the heavens have the same properties as the earth and, as well a Christian might, that the heavens are not divine.
Philoponus’s analysis of motion critiqued Aristotle’s. Aristotle had argued that continuous motion requires either an internal or an external mover in continuous contact with the object moved. Accordingly, the natural downward motion of a heavy object is caused by the object’s inner nature and opposed by the air through which it falls. In contrast, the horizontal motion of a projectile is unnatural and requires external movers: at first a mover that initiates the horizontal motion and then the continued push of the air. Philoponus, quite reasonably, doubted that the air could at the same time resist a projectile’s natural downward motion and cause its unnatural horizontal motion.
Aristotle also claimed that the time required for an object to fall from rest from a given height is in inverse proportion to its weight. Thus, the heavier an object, the more quickly it should fall. But, according to Philoponus:
This view of Aristotle’s is completely erroneous, and our view may be corroborated by actual observations more effectively than by any sort of verbal argument. For if you let fall from the same height two weights, one many times as heavy as the other, you will see that the ratio of the times required for the motion does not depend [solely] on the ratio of the weights, but that the difference in time is very small. And so if the difference in the weights is not considerable, that is, if one is, let us say, double the other there will be no difference, or else an imperceptible difference, in time.
Figure 16 illustrates the situation Philoponus describes. When two objects, one several times heavier than the other, are simultaneously released, the heavier object, according to Philoponus’s observation, reaches the ground only slightly ahead of the lighter object—certainly not, as according to Aristotle, several times more quickly.
However, Aristotle’s view is not without foundation. Given the difficulty of measuring small time intervals, Aristotle may well have simulated descent in air with descent in water by, for instance, simultaneously dropping heavy and light stones in a pool of clear water. If so, Aristotle would have observed that heavier objects do, indeed, as illustrated in figure 17, fall significantly faster than lighter ones.
Figure 17
We now know that free fall, that is, fall through a near vacuum or through a relatively short distance in air, is not comparable to fall through water or oil. In a vacuum, all massive objects fall at exactly the same rate just as, for instance, a hammer and feather dropped together on the surface of our relatively airless moon do. However, in a sufficiently viscous fluid, two similarly shaped objects fall at terminal speeds that are, as Aristotle expected, proportional to the object’s weight. To observe such descent all one needs is a tall glass of water and two objects of approximately the same shape and size but with very different masses—perhaps a stony pebble and a ball bearing.
The barbarian invasions that led to the fall of the Western Empire and the subsequent breakdown of Roman institutions disrupted communication between the Latin West and the Greek East. As a result, Philoponus’s books and commentaries as well as many other Greek texts became, for centuries, physically and linguistically unavailable to the Latin scholars of the West. Nestorian and Muslim scholars of the ninth and tenth centuries translated many of these texts from Greek into Syriac and Arabic. By 1000 this work of translation was largely complete and another translation movement began—this time from the Greek, Syriac, and Arabic into Latin. Witness, for example, the indefatigable Gerard of Cremona (1114–1187) who managed to translate some seventy to eighty books, including Ptolemy, Aristotle, and Euclid, from Arabic into Latin.
Although Philoponus’s books were not translated into Latin until the fourteenth century, this was well in time for Simon Stevin (1548–1620) and Galileo Galilei (1564–1642) to make use of them. Stevin actually reproduced, in Delft in 1588, the free fall experiment Philoponus described. And, while Galileo certainly grasped and exploited the meaning of Philoponus’s and Stevin’s observations, there is scant evidence that, in similar fashion, he dropped objects from the leaning tower of Pisa. It is an irony of popular history that Galileo is given credit for an experiment he probably did not do and that in his own time was already a thousand years old.
11. The Optics of Vision (1020 CE)
Figure 18
It is said that an Egyptian caliph of the Fatimid dynasty convinced the Muslim sage Ibn al-Haytham (ca. 965–ca. 1040), also known by his Latinized name Alhazen, to leave his native Basra in Iraq and come to Egypt in order to design and build a waterway that would regulate the flow of the Nile. Upon close inspection, Alhazen found that the project was not feasible. Then, fearing the wrath of the disappointed caliph, Alhazen feigned insanity. This tactic preserved Alhazen’s life at the cost of his forced confinement. Even so, Alhazen was able to continue his scholarly work, and, when the caliph died, he recovered his freedom.
During his enforced leisure Alhazen may have occupied himself with copying two works he held in high esteem: Euclid’s Elements and Ptolemy’s Almagest. These and other Greek philosophical, mathematical, and medical texts were available to Alhazen in Arabic translation thanks to the work of ninth- and tenth-century linguists associated with the House of Wisdom in Bagdad.
Greek scholars presented Alhazen with two theories of vision: (1) an emission theory, held by Euclid and Ptolemy, according to which rays were emitted from the eye and upon reaching an object rendered it visible, and (2) an intromission theory, held by Aristotle and Galen (ca. 129–200 CE), according to which rays of light traveled in the other direction, that is, from the visible object to the eye. Alhazen knew that very bright objects could damage the eye and so it seemed unlikely to him that, as the emission theory required, the eye could harm itself. He also realized that observing the nighttime sky by virtue of rays that travel from one’s eyes to the most distant parts of the universe in the instant required to lift one’s eyelids was absurd.
Yet the emission theory had one quite useful feature: The rays emerging from the eye (considered as a point) and encompassing the visible object form a cone, the visual cone, whose apex is at the eye and whose base outlines the object seen. The visual cone explains, for instance, depth perception. After all, more distant objects of the same size form visual cones with smaller solid angles at the cone’s apex. Thus, the angle subtended by a familiar object indicates its distance. How to preserve the useful features of the visual cone and yet adopt the more plausible intromission of rays was Alhazen’s problem.
It is important to know that Alhazen labored under the incorrect assumption that the crystalline humor or lens was the visually sensitive part of the eye. Not until Johannes Kepler’s dissection of an ox’s eye in 1604 was it understood that the images responsible for vision are formed on the retina at the back of the eye. Alhazen also supposed, this time correctly, that rays of light emerge in all directions from every point of an illuminated object. For this reason, many rays emerging from a single point enter the surface of the eye at different points and from slightly different directions. For example, consider the two rays coming from the top of the crescent in figure 18. How does the surface of the eye make sense of these two rays and others like them?
Alhazen cleverly, but arbitrarily, answered this question and, in the process, created a theory of vision. He insisted that the surface of the eye is sensitive only to those rays that enter it perpendicular to its curved surface, that is, only to those rays that do not bend or refract upon entering the eye. These unrefracted rays form the visual cone. Presumably, refracted rays are, in some fashion, dissipated or rendered incapable of stimulating the lens. In this way the virtues of the visual cone were merged with those of the intromission of rays. In other words, the geometry of Euclid and Ptolemy was merged with the causation and anatomy of Aristotle and Galen.
Alhazen’s theory of vision, however flawed we now understand it to be, answered the questions of its day and, as a consequence, was immensely influential. His Optica was translated into Latin around 1200. Subsequent contributors to the science of optics, Roger Bacon (1214–1294), Johannes Kepler (1571–1630), Willebrord Snell (1580–1626), and Pierre de Fermat (1601–1665), all refer to Alhazen.
Alhazen’s theory of vision was not his only contribution to optics. He also explained the principles behind the camera obscura and understood that when a ray of light leaves one medium, say, air, and enters another, say, water or glass, the incident, reflected, and refracted rays all lie in a single plane—for instance, as represented by the plane containing figure 19.
Figure 19
Many centuries before Alhazen, Ptolemy (90–168 CE) had made an empirical study of reflection and refraction. Ptolemy correctly surmised that the angle of incidence is always equal to the angle of reflection so that . But he also incorrectly proposed that the angle of refraction is directly proportional to the angle of incidence so that where the proportionality constant k characterized the different media on either side of the interface. For example, when going from air to water (as in figure 19), Ptolemy found that and so . Alhazen showed that Ptolemy’s expression describes refraction only through relatively small angles. Not until the seventeenth century did more generally applicable and accurate theories of refraction become widely available.
12. Oresme’s Triangle (1360)
Figure 20
Oresme’s triangle, embedded in figure 20, relates two quantities, speed (in the vertical direction) and time (in the horizontal direction), graphically rather than pictorially. It is possibly the earliest such graph. As such it illustrates a proof of a theorem sometimes called the mean speed theorem or the Merton rule according to which a uniformly accelerated object starting from rest traverses the same distance in a given time as an object moving uniformly at half the accelerated object’s final speed. In expressing this proof in graphic language, Nicole Oresme (1323–1382), who later became the bishop of Lisieux in northwestern France, built upon ideas first articulated in antiquity.
The preeminent mathematics of Greek and Roman antiquity and of the Middle Ages was geometry, and the preeminent geometry text was Euclid’s Elements. While the earlier, more familiar books of the Elements are nonnumerical, Euclid used straight lines to represent numerical magnitudes in the latter books of the Elements. The longer the line the greater the magnitude, with a doubly long line having double the magnitude.
While Euclid’s lines and magnitudes are abstract quantities without physical reference, Aristotle used straight lines to represent distances before Euclid just as Archimedes and Eratosthenes did, to great effect, after Euclid. After all, that a straight line can represent the distance between two points in space follows naturally from sketching an object extended in space. By dividing a straight line into standard units, the Greeks quantified space just as, by dividing a time interval into so many drops leaving the bowl of a water clock, they quantified time.
The related concept of speed, even if composed of distance and time, was not similarly quantified until the period 1325–1350 by a group of mathematicians and logicians associated with Merton College of Oxford University: Thomas Bradwardine, William Heytesbury, John of Dumbleton, and Richard Swineshead. These Merton scholars also distinguished among different kinds of motion and investigated their relationships including that identified by the Merton rule.
Oxford University and its counterparts in Bologna and Paris had grown in the late twelfth century out of the professionally oriented guilds of masters and scholars. Their masters were expert in the arts of rhetoric, law, medicine, and theology while their scholars were in need of these arts. Contemporaneous with the growth of universities was the recovery and translation into Latin of important Greek and Arabic texts. Thus, the Merton scholars of 1325–1350 and, a few years later, Oresme at the University of Paris had access to all thirteen books of Euclid’s Elements and the entire corpus of Aristotle’s work.
Oresme’s contribution was to translate the largely verbal discussions of the Merton scholars into geometrical language. In the process he also constructed a neat proof of the Merton rule—a proof that hinges on the distinction between uniform and uniformly accelerated motion. According to a definition, widely circulated in the late Middle Ages, an object in uniform motion traverses the same distance in equal intervals of time no matter how small the interval. The Merton scholars constructed a structurally similar definition of uniformly accelerated motion: an object in uniformly accelerated motion increases its speed by equal amounts in equal intervals of time no matter how small the interval.
Oresme represented the speed of an object in uniform motion with a series of equal length, vertical lines, separated in the horizontal direction by equal intervals of time as illustrated in the left panel of figure 21. Oresme also represented the speed of an object in uniformly accelerated motion with a series of successively, equally incremented vertical lines separated in the horizontal direction by equal intervals as illustrated in the center panel. In both cases, the lines representing speed are perpendicular to a single, horizontal line representing the passage of time. The dashed lines outline the area occupied by the vertical speed lines. The right panel similarly illustrates an arbitrary case of non-uniformly accelerated motion.
Figure 21
Clearly, the speed lines in the left panel of figure 21, representing an object in uniform motion, fill out an area that is proportional to the distance traversed. After all, a car traveling for 2 hours at 90 kilometers per hour has traversed 180 kilometers, that is, the product of the base times the height of the rectangle occupied by its representative speed lines. Oresme assumed, correctly but without justification, that the areas occupied by the speed lines of any kind of motion, non-uniform as well as uniform, also represent the distance traversed—a general theorem that requires the calculus for proof.
Given Oresme’s assumption and his use of speed lines, a proof of the Merton rule follows from closely inspecting the triangle ABC in figure 20. Because triangle ABC outlines the speed lines of a uniformly accelerated object, its area represents the distance traversed by that object. By construction, the horizontal line FD bisects the vertical line BC at D so that ABDF is a rectangle with a height BD that represents half the final speed BC of the uniformly accelerated object. Therefore, the area of the rectangle ABDF represents the distance traversed by an object whose uniform speed equals half the final speed of the uniformly accelerated object. In order to prove the Merton rule, Oresme needed only to prove that rectangle ABDF and triangle ABC have the same area.
Today we would simply observe that since the area of any triangle is equal to its base times one-half its height, the area of the triangle ABC in figure 20 is equal to its base AB times one-half its height BD. But AB times BD is also the area of the rectangle ABDF. Therefore, the rectangle ABDF and the triangle ABC have the same area.
However, Oresme’s proof was closely based on Euclidean propositions. Accordingly, note that the vertical angles CED and AEF are equal (Proposition 15 of Book I of Euclid’s Elements) as are the right angles AFE and CDE (Postulate 4). Therefore, the remaining angles FAE and DCE of the two triangles, CDE and AFE, are also equal. Furthermore, the sides CD and FA are equal because, by construction, the horizontal line FD bisects the vertical line BC at D. Therefore, the triangles CDE and AFE are equal in area (Proposition 26). Adding these equal area triangles to the same quadrilateral ABDE produces two differently shaped but equal area figures: the triangle ABC and the rectangle ABDF (Common Notion 2). Since these figures represent the distances traversed, respectively, by a uniformly accelerated object and by an object with uniform speed equal to half the final speed of the accelerated object, the Merton rule is proved. The right panel of figure 20 merely extends the proof to allow for objects with non-zero initial speed.
While this analysis may appear wordy and inefficient, even obscure, to those habituated to the algebraic methods of today, our aim here is to understand the medieval science of motion rather than to judge it. And to understand Oresme’s analysis we need to reproduce Oresme’s pattern of thought.
The Merton scholars’ distinction between uniform and uniformly accelerated motion, their discovery of the Merton rule, and Oresme’s geometrical proof of it, are exercises in kinematics, that is, exercises in the description of motion rather than an exploration of the dynamics or the causes of motion. While the dynamics of the fourteenth century remained enmeshed in Aristotelian concepts, the kinematics of the Merton scholars and of Oresme was a real advance on Aristotle. In due course, Galileo restated Oresme’s proof of the mean speed theorem under the heading “Naturally Accelerated Motion” in the “Third Day” of his text Two New Sciences, and René Descartes’s (1596–1650) invention of what we now call Cartesian coordinates made explicit what Oresme’s graphical analysis merely suggested.
13. Leonardo and Earthshine (1510)
Figure 22
If this text had a section called “Renaissance Science,” Leonardo da Vinci (1452–1519) would be its exemplar. Yet, while Leonardo was of the Renaissance, he was not the kind of scholar glorified by the humanists of his time: a scholar nurtured in classical history and literature, having perfect Latin, skilled at rhetoric, and able to speak with confidence at public gatherings. Rather, Leonardo’s education was incomplete, his Latin was poor, and he had little interest in public affairs. But Leonardo was a keen observer of nature, an avid experimentalist, and a man drawn to practical applications. While the classically educated scholars of the Italian Renaissance quoted authors, Leonardo cited experience.
Leonardo poured much of his experience into 13,000 notebook pages of drawings and text—pages that have enriched the language of visualization. He originated the aerial view so helpful in topography and mapmaking and the idea of presenting different sides of the same object, for instance, of the aorta of an ox. He pioneered the use of anatomical cross-sections, and observed that, at the same distance, a bright object appears larger than a less bright object of the same size.
Leonardo may have intended his notebooks to comprise a profusely illustrated encyclopedia of all technical knowledge. But, as they come to us, their pages have no order other than that imposed by the vicissitudes of Leonardo’s life. He usually wrote from right to left with characters slanting leftward: so-called mirror-image cursive. We do not know whether this practice was meant to preserve the privacy of his entries or was simply more convenient for the left-handed Leonardo.
The notebooks do, however, help us understand how Leonardo could be a prolifically inventive genius and yet have so little influence on the development of science. Like Archimedes, he focused on isolated problems. But, unlike Archimedes, Leonardo failed to develop collections of coherent ideas that explain more than the subject at hand. It is as if the very fertility of his mind and the concreteness of his artistic vision fragmented his scientific efforts and, in this way, kept him from developing powerful, abstract theoretical explanations. Even so, the fragments of his thought are often intriguing. Figure 22 illustrates one of them, on earthshine.
When the moon is a waxing or waning crescent, the shaded, relatively dark surface between the horns of its crescent glows with a faint, ghostly light—as suggested in the left panel of figure 22. Leonardo’s explanation of earthshine, illustrated in the right panel, is the earliest documented explanation of this phenomenon. According to Leonardo, a significant part of the sunlight striking the earth is reflected from its surface. The fraction of sunlight that reflects from the earth’s surface, known as its albedo, is close to 30 percent. Some of this reflected light strikes the dark side of the moon and some of that light is reflected back to the earth and observed as earthshine.
Leonardo got one detail of his explanation wrong. He believed that sunlight reflects primarily from the earth’s oceans, in particular from the tops of ocean waves. In fact, the earth’s clouds reflect much more sunlight than do its oceans. Photos taken from orbiting spacecraft confirm that the brightest parts of the earth are its cloud-covered areas. And when the earth’s cloud cover changes, the albedo of the earth also changes. In contrast, the moon has virtually no atmosphere and its albedo, about 12 percent, remains constant in time. Therefore, measuring changes in the intensity of the earthshine is equivalent to measuring changes in the earth’s albedo. The latter has become an important input to climate change models.
When walking the streets of Florence and Milan, Leonardo carried a notebook in which he sketched whatever caught his attention: people, buildings, and landscapes. On occasion, he would follow a stranger for hours until he could rough out their visage on paper. Leonardo also drew what he could only imagine: flying machines, cannons that shot exploding shells, and shoes that allowed one to walk on water. He designed a car powered by two sets of springs. While one set unwound and propelled the car forward, the car’s passenger would wind the other set. He envisioned a rotating spit powered by the same fire that cooked the flesh impaled upon it and a house of prostitution with an unusually large number of doors. Many of his designs are of practical devices that would, in time, be built. But no one has yet constructed Leonardo’s wake-up device contrived of mechanical relays that, when triggered by a water clock, jerked a sleeper’s feet into the air.
Leonardo was also keenly interested in mathematics and prepared illustrations for a mathematical text De Divina Proportione (1509) written by his friend Luca Pacioli. But, of course, Leonardo is most famous for his paintings—above all, The Last Supper and The Mona Lisa—paintings with animated postures; expressive faces; modeled, pointing hands; and pyramidal compositions. Leonardo painted in oils with a broader range of light and dark shades than is usually seen with the eye—a technique art historians call chiaroscuro. It may have been that the artist in Leonardo was drawn to the chiaroscuro of earthshine while the scientist in him sought its explanation.
. But he also incorrectly proposed that the angle of refraction is directly proportional to the angle of incidence so that 
where the proportionality constant k characterized the different media on either side of the interface. For example, when going from air to water (as in figure 19), Ptolemy found that 
and so 
. Alhazen showed that Ptolemy’s expression describes refraction only through relatively small angles. Not until the seventeenth century did more generally applicable and accurate theories of refraction become widely available.
