Mathematicians of ancient Greece knew that there were five, and only five, regular solids. These are three-dimensional geometrical objects whose faces are all identical to one another. The solids that satisfy this requirement are the cube, the tetrahedron (triangle-based pyramid), the octahedron, the icosahedron, and the dodecahedron. Plato admired the discovery and ascribed much importance to these solids; so they became known as the Platonic solids. The Greeks associated the five solids with elements of nature: earth, water, air, and fire, as in the figure.
This illustration of the five Platonic solids appeared in Johannes Kepler’s Mysterium Cosmographicum, published in 1596.
Of the five solids, Plato considered the dodecahedron so important that he named it the “fifth essence,” or quintessence, from which we get the word quintessential.
EUDOXUS OF KNIDUS
Eudoxus was born in 408 BCE to an impoverished family in Knidus, Asia Minor. Because of his family’s low socioeconomic status, he would have had no chance at a successful life if it weren’t for his powerful mathematical skills. As a young adult, Eudoxus heard about Plato’s Academy and borrowed money to travel there. Many of the philosophers at the Academy ignored the young man, but Plato recognized his genius and supported him in his mathematical pursuits.
There was no remuneration for membership in the esteemed Academy, and Eudoxus had so little money that he could not afford to live with the other members in Athens. He was forced to rent a small room in the nearby city of Piraeus, where rents were low and basic food could be obtained inexpensively. He commuted daily to Athens to attend the discussions at the Academy. Eventually, after having proved several major theorems in geometry that no one had been able to tackle, Eudoxus earned the respect of the other philosophers. Thanks in part to the constant encouragement he received from Plato, Exodus surpassed all the mathematicians who came before him by devising the basic ideas of integral calculus 2,100 years prior to its formal and complete introduction by Newton and Leibniz.
Eudoxus was able to calculate volumes and areas using essentially the calculus ideas we use today. In fact, in modern mathematical analysis we use “Eudoxus sums” as part of the derivation of the integral. Unfortunately, the resulting envy of lesser mathematicians in the academy finally drove Eudoxus to leave Athens and settle in Cyzicus, where he learned and then practiced medicine. Exodus became very wealthy and was even elected a legislator.
ARCHIMEDES OF SYRACUSE
Archimedes (ca. 287–212 BCE) was a relative of the ruler of Syracuse, Hieron II, and his father was the astronomer Phidias. Belonging to one of Syracuse’s most aristocratic families, Archimedes didn’t have to work, and mathematics became his passion. In fact, he is believed to have cared so little about daily life that he left meals uneaten when a mathematical problem occupied his mind.4
Archimedes is famous for a great discovery he made while taking a bath. The episode with the bath actually begins with a request by Hieron to his bright relative to help him determine whether a goldsmith had stolen some of his gold. The king had apparently asked the goldsmith to make him a new crown, providing him with the needed gold. When the king got his new crown from the goldsmith, he became suspicious that the goldsmith had replaced some of the gold inside the crown with silver or another cheaper metal and then pocketed the missing gold. The king needed Archimedes to find a way to compare the density of the crown with the density of gold without damaging the crown.
Archimedes pondered this dilemma while taking a bath, absently noting that the tub water rose when he got in. Suddenly it hit him: the water displaced was equal in volume to that of his body. Famously, he jumped out of the bath and ran naked through the streets of Syracuse, shouting, “Eureka, eureka!” (I found it, I found it!)
The Greek mathematician Archimedes discovered that a sphere circumscribed within a cylinder of the same radius has a volume and area equal to two-thirds of the volume and area of the circumscribing cylinder.
Archimedes now had a method for measuring the volume of an irregularly shaped object, such as a crown or a human body, by immersing it in water and measuring the volume of the water it replaces. Then, by weighing the object—the crown, in this case—one could find its density (weight divided by volume). And by comparing this density with that of gold, one could determine whether the goldsmith had cheated the king and stolen his gold! We don’t know whether the goldsmith was found guilty or not, however.5
Beyond this discovery, Archimedes helped his king and the people of Syracuse fight against the invading Romans by inventing many machines useful in warfare, including various kinds of catapults and (supposedly) a set of mirrors that focused the sun’s rays on attacking ships, causing them to burn, although there’s no strong proof that this last invention was viable. Many of the weapons he invented enabled the Syracusans to defend themselves—at least for a while—against Roman attack.
Archimedes also applied Eudoxus’s amazingly powerful analysis to the study of the areas and volumes of solids. As such, his work also anticipated the development of calculus. He is also known for discovering the Archimedean spiral, which is the locus of all points traced through time as a point moves away from the origin at constant speed and as the coordinate system rotates at a constant angular speed. Another result, appealing in its simplicity, was Archimedes’ discovery that a sphere circumscribed within a cylinder of the same radius has a volume and area equal to 2/3 of the volume and area of the circumscribing cylinder. Archimedes was so pleased with this result that he requested that a bronze sphere inside a cylinder be placed on his grave.
According to historical record, Archimedes’ life was cut short while he was trying to prove a geometrical theorem by drawing in the sand. When a Roman soldier approached him, Archimedes cried, “Don’t disturb my circles!” With that, he was promptly killed by a swing of the sword.
Despite orders from the Roman general Marcellus that Archimedes’ life should be spared, the great scientist was killed by a Roman soldier during the Siege of Syracuse around 212 BCE. Syracuse had been protected by weapons designed by Archimedes.
Many centuries later Europe revived the work of Eudoxus and Archimedes, culminating with the development of calculus. In the meantime, however, the mathematical treasures of the Greek world traveled to Arabia through Euclid’s famous books, the Elements, and Europe descended into the darkness of the Middle Ages.
Alexandria, Egypt, was home to many of history’s most influential and important mathematicians, including Apollonius, Eratosthenes, and Diophantus. This undated illustration depicts scholars poring over scrolls in the ancient city’s famed library.
THREE
ALEXANDRIA
Greece declined as Rome gained influence and forced the infusion of the harsher and more pragmatic Roman lifestyle, which valued engineering, military goals, discipline, killing animals for entertainment, and law over the Greek values of democracy, philosophy, art, and theater. Yet the Greek art of mathematics continued to be pursued by Greek mathematicians, most notably in Egypt. Alexandria, which boasted the largest and most famous library of the time until fire destroyed it in 48 BCE, had a thriving mathematical community. In fact, Alexandria, which had been founded in 331 BCE by Alexander the Great, became the undisputed center for mathematics in the Western world and would remain so until the early Middle Ages.
This fragment from Euclid’s Elements—one of the Oxyrhynchus Papyri, discovered in Egypt in 1896–97—contains a diagram illustrating proposition 5 of Book II. It is one of the oldest and most complete extant diagrams from Euclid’s seminal work.
In the third century BCE, Alexandria had been home to Euclid, whose Elements include in their thirteen volumes all the great theorems of Thales, Pythagoras, and other Greek mathematicians, theorems that laid the foundation for geometry based on straight lines. Euclid also put forward five postulates summarizing the geometry of the ancient Greeks:
1. A straight-line segment can be drawn joining any two points.
2
. Any straight-line segment can be extended indefinitely in a straight line.
3. Given any straight-line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.
Greek mathematicians took these postulates for granted as obvious starting points for the study of geometry. No one would ever argue with the first four postulates, but the fifth was very troublesome … and would be debated for millennia.
APOLLONIUS OF PERGA
Apollonius of Perga, Asia Minor, lived roughly between 260 and 190 BCE. Although not born in Alexandria, he spent considerable time there studying and working. Just thirty years younger than Archimedes, he pursued the kind of mathematics that had been developed by the cel-ebrated mathematician and came up with what is famously known as the Problem of Apollonius: given three things—where “thing” may be a point, a line, or a circle—find a circle that is tangent to (i.e., touching but not intersecting) each of the three things. For points, the problem is easily solved as the circumscribing circle. For three circles, however, the problem is difficult. Almost two millennia later Newton finally came up with a solution using only a straightedge and compass, as had been required for the three Classical Problems of Antiquity.
Apollonius studied conic sections—geometric objects formed by making various cuts through a cone—and he named them the hyperbola, the parabola, and the ellipse. He also deduced their mathematical properties. These geometrical figures would hold the attention of mathematicians for millennia, reappearing in the work of Descartes, Fermat, Newton, and Leibniz in the seventeenth and eighteenth centuries, leading up to the invention of calculus. The intense interest in conic sections can be attributed to the fact that they are not just cuts made through a cone; they are also two-dimensional figures that, as Descartes and Fermat would later show, can be described by purely algebraic methods. The simplest parabola, for example, can be described as the curve written algebraically as y = x2, where x is measured along the horizontal axis and y along the vertical. In making this connection between conic sections and algebra, Descartes would launch a powerful link between algebra and geometry.
Ptolemy’s model of the solar system, showing a planet rotating on an epicycle, which itself is rotating on a circular orbit (deferent) around X—a point halfway between Earth and the equant.
Apollonius also invented the concept of epicycles—circles centered on larger circles—which are mathematically valid constructions but which received a bad reputation from our modern vantage point because of their famous misapplication in astronomy by Ptolemy of Egypt in his second-century treatise the Almagest (Great Compilation). Ptolemy’s model of the solar system placed Earth at the center of creation and represented the motion of the planets as epicycles on a circular orbit centered on a point halfway between Earth and the equant. Ptolemy’s model persistently stood in the way of progress in cosmology, giving fodder to the Catholic Church in its opposition to science and the Copernican system for centuries.
ERATOSTHENES OF CYRENE
Eratosthenes of Cyrene (ca. 276–194 BCE) was a young contemporary of Archimedes. While living in Athens, where he wrote poetry, studied history and mathematics, engaged in athletics, and observed the stars and planets, he showed great promise in many areas of art and science. His fame reached the ear of the ruler of Alexandria, Ptolemy III, who invited Eratosthenes to his city and offered him a position as librarian and tutor to his son. While Eratosthenes was in Alexandria, Archimedes sent him his treatise The Method of Mechanical Theorems. As the young mathematician studied the work and learned more about astronomy, he became interested in what appeared to be an extremely difficult problem: measuring Earth’s circumference.
In this illustration of Eratosthenes’ measurement of Earth’s circumference, the sun appears directly over Syene (S) on the summer solstice. At the same time, the angle of the sun to a perpendicular to the surface of the earth (φ) is approximately 7°12′, or 1/50 of of a circle, at Alexandria (A). Multiplying the distance between Syene and Alexandria (δ)—roughly 500 miles—by 50 gives us an estimate of Earth’s circumference: 25,000 miles.
Perhaps aware of the work of the “first mathematician,” Thales, Eratosthenes thought up an ingenious method for measuring Earth’s circumference using the sun and shadows. Eratosthenes found that on the day of the summer solstice, June 21, the sun shone directly down into a deep well at Syene (present-day Aswan) in Upper Egypt. This meant to him that the sun was directly overhead—it was perpendicular to Earth’s surface at that point. At the same time, an associate at Alexandria, which Eratosthenes took to be roughly on the same meridian (it is slightly west, in reality), measured the angle of the sun to a perpendicular to Earth’s surface 5,000 stades—roughly five hundred miles—to the north. At Alexandria the angle of the sun to the perpendicular was measured as 1/50 of a circle.
As seen in the picture, the angle of 1/50 of a circle is identical to the angle measured from the center of Earth. Eratosthenes deduced that since 5,000 stades equal 1/50 of Earth’s circumference, the full circumference must be 50 × 5,000 stades, or 250,000 stades (about 25,000 miles—very close to our modern value for the circumference of Earth!).
Eratosthenes’ calculation was undoubtedly one of the greatest achievements of the time. But Eratosthenes did much more. Among his achievements in pure mathematics is an advance in number theory. He invented the Sieve of Eratosthenes, a device that enables us to determine prime numbers less than or equal to any given number (although it works well only for relatively small numbers, because the computation becomes tedious and inefficient for large sets). To see how it works, let’s use an example: find all the prime numbers between 2 and 100.
Begin by listing all the numbers between 2 and 100, as shown below:
Then cross out all multiples of 2 (all even numbers, here appearing against a black background), after having identified 2 as the first prime number (1 is not considered a prime number for various technical reasons). Next, proceed to the next prime number, 3, mark it as prime, and cross out all further multiples of 3 that have not been already eliminated (here shown against a gray background). Then proceed to 5, mark it as the next prime number, and cross out all remaining multiples of it (here shown within a gray circle) until you reach the end of the list. The process continues with 7 (multiples of which are underlined here), and when we reach 11 there are no new numbers to cross out, so the remaining set of numbers represents all prime numbers between 2 and 100. Today, with computers and better technology, we can go quite far in our search for prime numbers, but Eratosthenes’ systematic search laid the groundwork and constitutes the first-ever algorithm for finding prime numbers.
DIOPHANTUS
With the influence of the Roman Empire, Greek mathematics of later centuries became increasingly practical, and so-called pure mathematics—mathematics regarded simply as an intellectual exercise and a philosophy—declined. Among the great Greek mathematicians of Alexandria during the first few centuries CE were Pappus (ca. 290–350), who achieved important strides in geometry; Proclus (410–85), who wrote an influential commentary on Euclid’s Elements; Diophantus, the Father of Algebra, who lived in the third century CE; Theon (ca. 335–405), who published an authoritative edition of Euclid’s aforementioned monumental work; and his daughter Hypatia (d. 415).
Diophantus of Alexandria made key advances in algebra, and Diophantine equations—for example, x2 + 2y3 = 25—are studied in number theory today. These equations have several variables appearing in various powers. Diophantus was the first mathematician in history to study in detail such equations and to recognize that rational numbers (ratios of integers) can be used as both coefficients and solutions.
All we know about
Diophantus is that he probably lived around the year 250 CE. A mathematical riddle that appeared in a book of puzzles by an unknown author, written sometime in the fifth or sixth century, offers a few clues about his life:
God granted him boyhood for the sixth part of his life, and adding a twelfth part to this, He clothed his cheeks with down; He lit him the light of wedlock after a seventh part, and five years after his marriage He granted him a son. Alas! Late-born wretched child; after attaining the measure of half his father’s life, chill Fate took him. After consoling his grief by this science of numbers for four years, he ended his life.1
Diophantus’s mathematics was far more complicated than this simple linear equation implies. Solving this riddle, the solution to which is 84, tells us a few things about Diophantus, such as the age he was when he married, his age when his son was born, the age he was when he died (eighty-four), and so on.
Diophantus wrote his mathematical results in a series of books called the Arithmetica, only six of which survive. His work was largely lost to history until Pierre de Fermat consulted a Latin translation of the Arithmetica, and in its margin he wrote his famous Last Theorem—a generalization of a result by Diophantus that was proved by Andrew Wiles of Princeton University in the 1990s. The centuries-old interest in solving Fermat’s problem refocused the attention of mathematicians from the seventeenth century onward on Diophantine equations—i.e., the analysis and solution of algebraic equations with several variables and the attendant study of polynomials (sums of multiples of powers of a particular variable, or of several variables)—and brought Diophantus fame as the Father of Algebra.
HYPATIA
The daughter and pupil of Theon, a notable scholar and mathematician of the era, Hypatia (ca. 360–415) was the first important female mathematician in history. Fittingly, her name is derived from the Greek word for “highest.” Hypatia studied with her father and at an early age became interested in philosophy and mathematics.
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