A dial on the front of the device probably carried at least three hands, one indicating the date and the other two indicating the positions of the Sun and the Moon. The device was also probably used to track dates of ancient Olympic games, predict solar eclipses, and indicate other planetary motions.
Of special delight to physicists, the Moon mechanism uses a special train of bronze gears, two of them linked with a slightly offset axis, to indicate the position and phase of the moon. As is known today from Kepler’s Laws of Planetary Motion, the moon travels at different speeds as it orbits the Earth (e.g. faster when it is closer to the Earth), and this speed differential is modeled by the Antikythera mechanism, even though the ancient Greeks were not aware of the actual elliptical shape of the orbit. Additionally, the Earth travels faster when it is closer to the Sun than when it is far away.
Marchant writes, “By turning the handle on the box you could make time pass forwards or backwards, to see the state of the cosmos today, tomorrow, last Tuesday or a hundred years in the future. Whoever owned this device must have felt like master of the heavens.”
SEE ALSO Kepler’s Laws of Planetary Motion (1609), Slide Rule (1621), ENIAC (1946)
The Antikythera mechanism is an ancient geared computing device that was used to calculate astronomical positions. X-ray radiographs of the mechanism have revealed information about the device’s internal configuration. (Photo courtesy of Rien van de Weijgaert.)
c. 50
Gears • Clifford A. Pickover
Hero (or Heron) of Alexandria (c. 10–c. 70 A.D.)
Rotating gears, with their intermeshed teeth, have played a crucial role in the history of technology. Not only are gear mechanisms important for increasing the applied twisting force, or torque, but gears are also useful for changing the speed and direction of force. One of the oldest machines is a potter’s wheel, and primitive gears associated with these kinds of wheels probably existed for thousands of years. In the fourth century BCE, Aristotle wrote about wheels using friction between smooth surfaces to convey motions. Built around 125 BCE, the Antikythera Mechanism employed toothed gears for calculating astronomical positions. One of the earliest written references to toothed gears was made by Hero of Alexandria, c. 50 A.D. Through time, gears have played a crucial role in mills, clocks, bicycles, cars, washing machines, and drills. Because they are so useful in amplifying forces, early engineers used them for lifting heavy construction loads. The speed-changing properties of gear assemblies were put to use when ancient textile machines were powered by the movement of horses or water. The rotational speed of these power supplies was often insufficient, so a set of wooden gears was used to increase the speed for textile production.
When two gears are intermeshed, the rotational speed ratio s1/s2 is simply the reciprocal ratio of the number n of teeth on the two gears: s1/s2 = n2/n1. Thus, a small gear turns faster than its larger partner. The torque ratio has an opposite relationship. The larger gear experiences greater torque, and the higher torque implies lower velocity. This is useful, for example, for electric screwdrivers, in which the motor can produce a small amount of torque at high speed, but we wish to have a slow output speed with increased torque.
Among the simplest gears are spur gears, with their straight-cut teeth. Helical gears in which the teeth are set at an angle have the advantage of running more smoothly and quietly and usually being able to handle greater torques.
SEE ALSO Pulley (c. 230 BCE) Conservation of Energy (1843), Steam Turbine (1890).
Gears have played important roles in history. Gear mechanisms can increase the applied force or torque and are also useful for changing the speed and direction of a force.
c. 126
Roman Concrete • Derek B. Lowe
Pliny the Elder (23−79)
Concrete is everywhere in our civilization; modern construction wouldn’t be possible without it. But its chemistry is surprisingly complex, depending on two elements (aluminum and silicon) that form strong bonding networks with oxygen atoms. These species, which are abundant in Earth’s crust, form the basis for a huge variety of minerals and man-made ceramics. Concrete also requires calcium ions and a reaction with water to help hold everything together, but the technical name, hydrated calcium aluminosilicate, although an accurate description of concrete’s chemical composition, doesn’t roll off the tongue very easily.
The Romans had the finest concrete of the ancient world, and some of it can still be seen today in such magnificent structures as the famous Pantheon—completed around the year 126 and still the largest unreinforced concrete dome in the world. Roman civilization, though, was actually “science deficient”; considering their power and longevity, surprisingly little basic research was done. They didn’t have much patience for mathematics, blue-sky experimentation, or abstract theories, but practical improvements in civil and military engineering were always welcome. As such, the Romans developed a variety of concrete mixtures for different applications. Their water-resistant mix was of very high quality, and according to the natural philosopher Pliny the Elder, a key ingredient in the mortar was the ashy volcanic deposits (now known as pozzolan) from the area of Mount Vesuvius. Pliny knew the area well—too well, in the end, since he was killed in the famous 79 eruption that destroyed Pompeii.
Just in the last few years, analytical chemists have been able to work out how this recipe for Roman maritime concrete must have been made. The process requires quite a bit less energy than modern Portland cement, which was developed in nineteenth-century Britain. In terms of the fuel needed to bake the starting limestone mix, the time needed to cure the finished product, and its durability in salt water, the Roman recipe has many advantages. After almost two thousand years, it may be making a comeback.
SEE ALSO Bronze (c. 3300 BCE), Arch (c. 1850 BCE) Polyethylene (1933), Rubber (1839).
The two-thousand-year-old Pantheon in Rome still has the largest unreinforced concrete dome in the world—a solid testament to Roman engineering.
c. 650
Zero • Clifford A. Pickover
Brahmagupta (c. 598–c. 668), Bhaskara (c. 600–c. 680), Mahavira (c. 800–c. 870)
The ancient Babylonians originally had no symbol for zero, which caused uncertainty in their notation, just as today we would be confused if numbers like 12, 102, and 1,002 had no zero to distinguish them. The Babylonian scribes only left a space where a zero should be, and it was not easy to distinguish the number of spaces in the middle or at the ends of numbers. Eventually, the Babylonians did invent a symbol to mark the gap between their digits, but they probably had no concept of zero as an actual number.
Around A.D. 650, the use of the number was common in Indian mathematics, and a stone tablet was found in Gwalior, south of Delhi, with the numbers 270 and 50. The numbers on the tablet, dated to A.D. 876, look very similar to modern numbers, except that the zeros are smaller and raised. Indian mathematicians such as Brahmagupta, Mahavira, and Bhaskara used zero in mathematical operations. For example, Brahmagupta explained that a number subtracted from itself gives zero, and he noted that any number when multiplied by zero is zero. The Bakhshali Manuscript may be the first documented evidence of zero used for mathematical purposes, but its date is unclear.
Around A.D. 665, the Mayan civilization in Central America also developed the number zero, but its achievement did not seem to influence other peoples. On the other hand, the Indian concept of zero spread to the Arabs, Europeans, and Chinese, and changed the world.
Mathematician Hossein Arsham writes, “The introduction of zero into the decimal system in the thirteenth century was the most significant achievement in the development of a number system, in which calculation with large numbers became feasible. Without the notion of zero, the . . . modeling processes in commerce, astronomy, physics, chemistry, and industry would have been unthinkable. The lack of such a symbol is one of the serious drawbacks in the Roman numeral system.”
SEE ALSO Rhind Papyrus (c. 1650 BCE), al-Khwar
izmi’s Algebra (830), Fibonacci’s Liber Abaci (1202).
The notion of zero ignited a fire that eventually allowed humanity to more easily work with large numbers and to become efficient in calculations in fields ranging from commerce to physics.
830
Al-Khwarizmi’s Algebra • Clifford A. Pickover
Abu Ja’far Muhammad ibn Musa al-Khwarizmi (c. 780–c. 850)
Al-Khwarizmi was a Persian mathematician and astronomer who spent most of his life in Baghdad. His book on algebra, Kitab al-mukhtasar fi hisab al-jabr wa’l-muqabala (The Compendious Book on Calculation by Completion and Balancing), was the first book on the systematic solution of linear and quadratic equations and is sometimes referred to by the shortened title Algebra. Along with Diophantus, he is considered the “father of algebra.” The Latin translation of his works introduced the decimal positional number system to Europe. Interestingly, the word algebra comes from al-jabr, one of the two operations used in his book to solve quadratic equations.
For al-Khwarizmi, al-jabr is a method in which we can eliminate negative quantities in an equation by adding the same quantity to each side. For example, we can reduce x2 = 50x − 5x2 to 6x2 = 50x by adding 5x2 to both sides. Al-muqabala is a method whereby we gather quantities of the same type to the same side of the equation. For example, x2 + 15 = x + 5 is reduced to x2 + 10 = x.
The book helped readers to solve equations such as those of the forms x2 + 10x = 39, x2 + 21 = 10x, and 3x + 4 = x2, but more generally, al-Khwarizmi believed that the difficult mathematical problems could be solved if broken down into a series of smaller steps. Al-Khwarizmi intended his book to be practical, helping people to make calculations that deal with money, property inheritance, lawsuits, trade, and the digging of canals. His book also contained example problems and solutions.
Al-Khwarizmi worked most of his life in the Baghdad House of Wisdom, a library, translation institute, and place of learning that was a major intellectual center of the Islamic Golden Age. Alas, the Mongols destroyed the House of Wisdom in 1258, and legend says that the waters of the Tigris ran black with ink from the books tossed into its waters.
SEE ALSO Fibonacci’s Liber Abaci (1202), Development of Modern Calculus (1665), Fundamental Theorem of Algebra (1797).
A stamp from the Soviet Union, issued in 1983 in honor of al-Khwarizmi, the Persian mathematician and astronomer whose book on algebra offered a systematic solution to a wide variety of equations.
c. 850
Gunpowder • Derek B. Lowe
Gunpowder probably was discovered by alchemists trying to transmute metals or extend life rather than by weapons engineers seeking an explosive. A Chinese military compendium from 1044 listed a number of different recipes for gunpowder, showing that it had been the subject of a lot of action-packed research and development by the middle of the Song dynasty, but the first known mention comes from a mid-ninth-century Taoist text, which stressed its dangerously flammable nature. Sulfur was of great importance in alchemy, and any lab of the time would have had charcoal around for fuel. The third key ingredient—the oxidizer potassium nitrate—was available as the naturally occurring mineral niter (also known as saltpeter) or as crystals around deposits of bat guano in caves. Whoever first combined these powders and exposed the resulting mixture to a flame must have immediately realized that they were onto something big. Extending human life, though, turned out not to be gunpowder’s strong point.
Knowledge of the new weapon diffused through China and past its borders, and the Mongol invasions of the thirteenth century spread the news even farther, from India to Europe. The Chinese kept raising the amount of potassium nitrate in their gunpowder as time went on, producing bigger explosions all the time. Early artillery shells, exploding arrows, and a variety of alarming bomb designs show up in several Chinese military manuscripts. In his Treatise on Horsemanship and Stratagems of War (c. 1280), detailing 107 different explosive mixtures, Syrian chemist Hasan al-Rammah referred to potassium nitrate as “Chinese snow.” European militaries adopted gunpowder quickly: the first illustration of a firearm—a primitive metal cannon known as a pot-de-fer (French for “iron pot”) with an enormous arrow emerging from its barrel—appeared in a 1326 manuscript by the English scholar Walter de Milemete. For better or worse, it has been with us ever since.
SEE ALSO Iron Smelting (c. 1300 BCE) Internal Combustion Engine (1908), Little Boy Atomic Bomb (1945).
An explosion of gunpowder shells during the 1274 Mongol invasion of Japan, illustrated in a scroll commissioned some twenty years after the battle.
1202
Fibonacci’s Liber Abaci • Clifford A. Pickover
Leonardo of Pisa (also known as Fibonacci, c. 1175–c. 1250)
Carl Boyer refers to Leonardo of Pisa, also known as Fibonacci, as “without a doubt, the most original and most capable mathematician of the medieval Christian world.” Fibonacci, a wealthy Italian merchant, traveled through Egypt, Syria, and Barbary (Algeria), and in 1202 published the book Liber Abaci (The Book of the Abacus), which introduced the Hindu-Arabic numerals and decimal number system to Western Europe. This system is now used throughout the world, having overcome the terribly cumbersome Roman numerals common in Fibonacci’s time. In Liber Abaci, Fibonacci notes, “These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0, which in Arabic is called zephirum, any number can be represented, as will be demonstrated.”
Although Liber Abaci was not the first European book to describe the Hindu-Arabic numerals—and even though decimal numerals did not gain widespread use in Europe directly after its publication—the book is nevertheless considered to have had a strong impact on European thought because it was directed to both academicians and businesspeople.
Liber Abaci also introduced Western Europe to the famous number sequence 1, 1, 2, 3, 5, 8, 13. . ., which today is called the Fibonacci sequence. Notice that except for the first two numbers, every successive number in the sequence equals the sum of the previous two. These numbers appear in an amazing number of mathematical disciplines and in nature.
Is God a mathematician? Certainly, the universe seems to be reliably understood using mathematics. Nature is mathematics. The arrangement of seeds in a sunflower can be understood using Fibonacci numbers. Sunflower heads, like those of other flowers, contain families of interlaced spirals of seeds—one spiral winding clockwise, the other counterclockwise. The number of spirals in such heads, as well as the number of petals in flowers, is very often a Fibonacci number.
SEE ALSO Zero (c. 650), Golden Ratio (1509), Pascal’s Triangle (1654)
Sunflower heads contain families of interlaced spirals of seeds—one spiral winding clockwise, the other counterclockwise. The number of spirals in such heads, as well as the number of petals in flowers, is very often a Fibonacci number.
1284
Eyeglasses • Clifford A. Pickover
Salvino D’Armate of Florence (1258–1312), Giambattista della Porta (1535–1615), Edward Scarlett (1677–1743)
Historian Lois N. Magner writes, “The use of spectacles must have occasioned a profound effect on attitudes towards human limitations and liabilities. Spectacles not only made it possible for scholars and copyists to continue their work, they accustomed people to the idea that certain physical limitations could be transcended by the use of human inventions.”
Today, the terms eyeglasses and spectacles usually refer to lenses attached to a frame to correct vision problems. Various forms have existed through history, including the pince-nez (supported only by pinching the bridge of the nose, with no earpieces), monocle (a circular lens over one eye), and lorgnette (spectacles with a handle).
By 1000 A.D., “reading stones”—crystals or segments of a glass sphere placed on reading material to magnify the text—were common. Eyeglasses were in use in China by the time of Marco Polo’s journey, around 1270, and they may have been used in Arabia even earlier. In 1284, the Italian Salvino D’Armate be
came perhaps the most famous inventor of eyeglasses in Europe. The earliest eyeglasses made use of convex glasses for the correction of hyperopia (farsightedness) and presbyopia (age-related farsightedness). One early reference to concave lenses for nearsightedness (also called myopia, in which distant objects appear blurred and near objects are clear) occurred in Natural Magick (1558), by Italian scholar Giambattista della Porta. Convex lenses were used to see text that was close to the eye.
Spectacles were once so expensive that they were listed in wills as valuable property. Around 1727, British optician Edward Scarlett developed the modern style of glasses, held by rigid arms that hook over the ears. The American scientist Benjamin Franklin invented bifocals in 1784 to address his combination of myopia and presbyopia.
Today, many eyeglasses are made of the plastic CR-39 due to its favorable optical properties and durability. Lenses are generally used to change the focus location of light rays so that they properly intersect the retina, the light-sensitive tissue at the back of the eye.
SEE ALSO Telescope (1608), Micrographia (1665), Laser (1960).
A lorgnette is a pair of spectacles with a handle. It was invented in the 1700s by English optical designer George Adams. Some owners did not need glasses to see better, but carried ornate lorgnettes to be fashionable.
c. 1500
Early Calculus • Jim Bell
The Science Book Page 6
