SEE ALSO Euclid’s Elements (c. 300 BCE), Bayes’ Theorem (1761), Gödel’s Theorem (1931).
Italian Renaissance artist Raphael depicts Aristotle (right), holding his Ethics, next to Plato. This Vatican fresco, The School of Athens, was painted between 1510 and 1511.
c. 350 BCE
Platonic Solids • Clifford A. Pickover
Plato (c. 428 BCE–c. 348 BCE)
A Platonic solid is a convex multifaceted 3-D object whose faces are all identical polygons, with sides of equal length and angles of equal degrees. A Platonic solid also has the same number of faces meeting at every vertex. The best-known example of a Platonic solid is the cube, whose faces are six identical squares.
The ancient Greeks recognized and proved that only five Platonic solids can be constructed: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. For example, the icosahedron has 20 faces, all in the shape of equilateral triangles.
Plato described the five Platonic solids in Timaeus in around 350 BCE. He was not only awestruck by their beauty and symmetry, but he also believed that the shapes described the structures of the four basic elements thought to compose the cosmos. In particular, the tetrahedron was the shape that represented fire, perhaps because of the polyhedron’s sharp edges. The octahedron was air. Water was made up of icosahedra, which are smoother than the other Platonic solids. Earth consisted of cubes, which look sturdy and solid. Plato decided that God used the dodecahedron for arranging the constellations in the heavens.
Pythagoras of Samos—the famous mathematician and mystic who lived in the time of Buddha and Confucius, around 550 BCE—probably knew of three of the five Platonic solids (the cube, tetrahedron, and dodecahedron). Slightly rounded versions of the Platonic solids made of stone have been discovered in areas inhabited by the late Neolithic people of Scotland at least 1,000 years before Plato. The German astronomer Johannes Kepler (1571–1630) constructed models of Platonic solids nested within one another in an attempt to describe the orbits of the planets about the sun. Although Kepler’s theories were wrong, he was one of the first scientists to insist on a geometrical explanation for celestial phenomena.
SEE ALSO Pythagorean Theorem and Triangles (c. 600 BCE), Euclid’s Elements (c. 300 BCE), Tesseract (1888).
A traditional dodecahedron is a polyhedron with 12 pentagonal faces. Shown here is Paul Nylander’s graphical approximation of a hyperbolic dodecahedron, which uses a portion of a sphere for each face.
c. 300 BCE
Euclid’s Elements • Clifford A. Pickover
Euclid of Alexandria (c. 325 BCE–c. 270 BCE)
The geometer Euclid of Alexandria lived in Hellenistic Egypt, and his book Elements is one of the most successful textbooks in the history of mathematics. His presentation of plane geometry is based on theorems that can all be derived from just five simple axioms, or postulates, one of which is that only one straight line can be drawn between any two points. Given a point and a line, another famous postulate suggests that only one line through the point is parallel to the first line. In the 1800s, mathematicians finally explored Non-Euclidean Geometries, in which the parallel postulate was no longer always required. Euclid’s methodical approach of proving mathematical theorems by logical reasoning not only laid the foundations of geometry but also shaped countless other areas concerning logic and mathematical proofs.
Elements consists of 13 books that cover two- and three-dimensional geometries, proportions, and the theory of numbers. Elements was one of the first books to be printed after the invention of the printing press and was used for centuries as part of university curricula. More than 1,000 editions of Elements have been published since its original printing in 1482. Although Euclid was probably not the first to prove the various results in Elements, his clear organization and style made the work of lasting significance. Mathematical historian Thomas Heath called Elements “the greatest mathematical textbook of all time.” Scientists like Galileo Galilei and Isaac Newton were strongly influenced by Elements. Philosopher and logician Bertrand Russell wrote, “At the age of eleven, I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined that there was anything so delicious in the world.” The poet Edna St. Vincent Millay wrote, “Euclid alone has looked on Beauty bare.”
SEE ALSO Pythagorean Theorem and Triangles (c. 600 BCE), Aristotle’s Organon (c. 350 BCE), Descartes’ La Géométrie (1637), Non-Euclidean Geometry (1829).
This is the frontispiece of Adelard of Bath’s translation of Euclid’s Elements, c. 1310. This translation from Arabic to Latin is the oldest surviving Latin translation of Elements.
c. 250 BCE
Archimedes’ Principle of Buoyancy • Clifford A. Pickover
Archimedes (c. 287 BCE–c. 212 BCE)
Imagine that you are weighing an object—like a fresh, uncooked egg—that is submerged in a kitchen sink. If you weigh the egg by hanging it from a scale, the egg would weigh less while in the water, according to the scale, than when the egg is lifted out of the sink and weighed. The water exerts an upward force that partially supports the weight of the egg. This force is more obvious if we perform the same experiment with an object of lower density, such as a cube made out of cork, which floats while being partially submerged in the water.
The force exerted by the water on the cork is called a buoyant force, and for a cork held under water, the upward force is greater than its weight. This buoyant force depends on the density of the liquid and the volume of the object, but not on the shape of the object or the material of which the object is composed. Thus, in our experiment, it doesn’t matter if the egg is shaped like a sphere or a cube. An equal volume of egg or wood would experience the same buoyant force in water.
According to Archimedes’ Principle of Buoyancy, named after the Greek mathematician and inventor famous for his geometric and hydrostatic studies, a body wholly or partially submerged in liquid is buoyed up by a force equal to the weight of displaced liquid.
As another example, consider a small pellet of lead placed in a bathtub. The pellet weighs more than the tiny weight of water it displaces, so the pellet sinks. A wooden rowboat is buoyed up by the large weight of water that it displaces, and hence the rowboat floats. A submarine floating underwater displaces a volume of water that has a weight that is precisely equal to the submarine’s weight. In other words, the total weight of the submarine—which includes the people, the metal hull, and the enclosed air—equals the weight of displaced seawater.
SEE ALSO Acceleration of Falling Objects (1638), Newton’s Laws of Motion and Gravitation (1687), Bernoulli’s Law of Fluid Dynamics (1738).
When plesiosaurs (extinct reptiles) floated within the sea, their total weights equaled the weights of the water they displaced. Gastrolith stones discovered in the stomach region of plesiosaur skeletons may have helped in controlling buoyancy and flotation.
c. 250 BCE
π • Clifford A. Pickover
Archimedes of Syracuse (c. 287 BCE–c. 212 BCE)
Pi, symbolized by the Greek letter π, is the ratio of a circle’s circumference to its diameter and is approximately equal to 3.14159. Perhaps ancient peoples observed that for every revolution of a cartwheel, a cart moves forward about three times the diameter of the wheel—an early recognition that the circumference is about three times the diameter. An ancient Babylonian tablet states that the ratio of the circumference of a circle to the perimeter of an inscribed hexagon is 1 to 0.96, implying a value of pi of 3.125. Greek mathematician Archimedes (c. 250 BCE) was the first to give us a mathematically rigorous range for π—a value between 223/71 and 22/7. The Welsh mathematician William Jones (1675–1749) introduced the symbol π in 1706, most likely after the Greek word for periphery, which starts with the letter π.
The most famous ratio in mathematics is π, on Earth and probably for any advanced civilization in the universe. The digits of π never end, nor has anyone detected an orderly
pattern in their arrangement. The speed with which a computer can compute π is an interesting measure of a computer’s computational ability, and today we know more than a trillion digits of π.
We usually associate π with a circle, and so did pre-seventeenth-century humanity. However, in the seventeenth century, π was freed from the circle. Many curves were invented and studied (for example, various arches, hypocycloids, and curves known as witches), and it was found that their areas could be expressed in terms of π. Finally, π appeared to flee geometry altogether, and today π relates to unaccountably many areas in number theory, probability, complex numbers, and series of simple fractions, such as π/4 = 1 − 1/3 + 1/5 − 1/7. . . . In 2006, Akira Haraguchi, a retired Japanese engineer, set a world record for memorizing and reciting 100,000 digits of π.
SEE ALSO Golden Ratio (1509), Euler’s Number, e (1727), Transcendental Numbers (1844).
Pi is approximately equal to 3.14 and is the ratio of a circle’s circumference to its diameter. Ancient peoples may have noticed that for every revolution of a cart wheel, the cart moves forward about three times the diameter of the wheel.
c. 240 BCE
Eratosthenes Measures the Earth • Clifford A. Pickover
Eratosthenes of Cyrene (c. 276 BCE–c. 194 BCE)
According to author Douglas Hubbard, “Our first mentor of measurement did something that was probably thought by many in his day to be impossible. An ancient Greek named Eratosthenes made the first recorded measurement of the circumference of Earth. . . . [He] didn’t use accurate survey equipment, and he certainly didn’t have lasers and satellites. . . .” However, Eratosthenes knew of a particular deep well in Syene, a city in southern Egypt. The bottom of this well was entirely lit by the noon Sun one day out of the year, and thus the sun was directly overhead. He also was aware that, at the same time in the city of Alexandria, objects cast a shadow, which suggested to Eratosthenes that the Earth was spherical, not flat. He assumed that the Sun’s rays were essentially parallel, and he knew that the shadow made an angle that was 1/50th of a circle. Thus, he determined that the circumference of the Earth must be approximately 50 times the known distance between Alexandria and Syene. Assessments of Eratosthenes’ accuracy vary, due to the conversion of his ancient units of measure to modern units, along with other factors, but his measurements are usually deemed to be within a few percent of the actual circumference. Certainly, his estimation was more accurate than other estimates of his day. Today, we know that the circumference of the Earth at the equator is about 24,900 miles (40,075 kilometers). Curiously, if Columbus had not ignored the results of Eratosthenes, thereby underestimating the circumference of the Earth, the goal of reaching Asia by sailing west might have been considered to be an impossible task.
Eratosthenes was born in Cyrene (now in Libya) and later was a director of the great Library of Alexandria. He is also famous for founding scientific chronology (a system that endeavors to fix dates of events at correctly proportioned intervals), along with developing a simple algorithm for identifying prime numbers (numbers such as 13, divisible only by themselves and 1). In old age, Eratosthenes became blind and starved himself to death.
SEE ALSO Sundial (c. 3000 BCE) Telescope (1608), Measuring the Solar System (1672).
Eratosthenes’ map of the world (1895 reconstruction). Eratosthenes measured the circumference of the Earth without leaving Egypt. Ancient and medieval European scholars often believed that the world was spherical, although they were not aware of the Americas.
c. 240 BCE
Sieve of Eratosthenes • Clifford A. Pickover
Eratosthenes (c. 276 BCE–c. 194 BCE)
A prime number is a number larger than 1, such as 5 or 13, that is divisible only by itself or 1. The number 14 is not prime because 14 = 7 × 2. Prime numbers have fascinated mathematicians for more than two thousand years. Around 300 BCE, Euclid showed that there is no “largest prime” and that an infinitude of prime numbers exists. But how can we determine if a number is prime? Around 240 BCE, the Greek mathematician Eratosthenes developed the first-known test for primality, which we today call the Sieve of Eratosthenes. In particular, the Sieve can be used to find all prime numbers up to a specified integer. (The ever-versatile Eratosthenes served as the director of the famous library in Alexandria and was also the first person to provide a reasonable estimation of the diameter of the Earth.)
The French theologian and mathematician Marin Mersenne (1588–1648) was also fascinated by prime numbers, and he tried to find a formula that he could use to find all primes. Although he did not find such a formula, his work on Mersenne numbers of the form 2p − 1, where p is an integer, continues to be of interest to us today. Mersenne numbers, with p a prime number, are the easiest type of number to prove prime, so they are usually the largest primes of which humanity is aware. The forty-fifth known Mersenne prime (243,112,609 − 1) was discovered in 2008, and it contains 12,978,189 digits!
Today, prime numbers play an important role in public-key cryptography algorithms that may be used for sending secure messages. More important, for pure mathematicians, prime numbers have been at the heart of many intriguing unsolved conjectures through history, including the Riemann Hypothesis, which concerns the distribution of prime numbers, and the strong Goldbach Conjecture, which states that every even integer greater than 2 can be written as a sum of two primes.
SEE ALSO Ishango Bone (c. 18,000 BCE) Riemann Hypothesis (1859), Proof of the Prime Number Theorem (1896), Public-Key Cryptography (1977).
Polish artist Andreas Guskos creates contemporary art by concatenating thousands of prime numbers and using them as textures on various surfaces. This work is called Eratosthenes, after the Greek mathematician who developed the first-known test for primality.
c. 230 BCE
Pulley • Clifford A. Pickover
Archimedes (c. 287 BCE–c. 212 BCE)
A pulley is a mechanism that usually consists of a wheel on an axle. A rope runs over the wheel so that the pulley can change the direction of an applied force, for example, when helping a human or a machine lift or pull heavy loads. The pulley also makes it easier to move a load because it decreases the applied force needed.
The pulley probably had its birth in prehistoric times when someone tossed a rope over a horizontal tree branch and used it to lift a heavy object. Author Kendall Haven writes, “By 3000 BCE, such pulleys with grooved wheels (so that the rope wouldn’t slip off) existed in Egypt and Syria. The Greek mathematician and inventor Archimedes gets credit for inventing the compound pulley in about 230 BCE . . . in which a number of wheels and ropes combine to lift a single object . . . to multiply the lifting power of a person. Modern block and tackle systems are examples of compound pulleys.”
Pulleys almost seem magical in the way they can decrease the width and strength of the rope required, and of the force needed, to lift heavy objects. In fact, according to legends and the writings of the Greek historian Plutarch, Archimedes may have used a compound pulley to help move heavy ships with minimal effort. Of course, no laws of nature are violated. Work, which is defined as force times the distance moved, remains the same—pulleys allow one to pull with less force but over a longer distance. In practice, more pulleys increase the sliding friction, and, thus, a system of pulleys may become decreasingly efficient after a certain number are employed. When performing computations to estimate the effort needed to use a pulley system, engineers often assume that the pulley and rope weigh very little compared to the weight that is being moved. Through history, block-and-tackle systems were particularly common on sailing ships, where motorized aids were not always available.
SEE ALSO Gears (c. 50), Acceleration of Falling Objects (1638), Newton’s Laws of Motion and Gravitation (1687).
Close-up of a pulley system on a vintage yacht. Ropes in pulleys travel over wheels so that the pulley can change the direction of applied forces and make it easier to move a load.
c. 125 BCE
Antikythera
Mechanism • Clifford A. Pickover
Valerios Stais (1857–1923)
The Antikythera mechanism is an ancient geared computing device that was used to calculate astronomical positions and that mystified scientists for over a century. Discovered around 1902 by archaeologist Valerios Stais in a shipwreck off the coast of the Greek island Antikythera, the device is thought to have been built about 150–100 BCE. Journalist Jo Marchant writes, “Among the salvaged hoard subsequently shipped to Athens was a piece of formless rock that no one noticed at first, until it cracked open, revealing bronze gearwheels, pointers, and tiny Greek inscriptions. . . . A sophisticated piece of machinery consisting of precisely cut dials, pointers and at least thirty interlocking gear wheels, nothing close to its complexity appears again in the historical record for more than a thousand years, until the development of astronomical clocks in medieval Europe.”
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