Egyptian astrologers also played an important role in developing a rather sophisticated calendar system that was already well established by the time the pyramids were being built. A new year was defined by the first sighting of the brightest star in the sky, Sirius (Sopdet to the Egyptians), just before sunrise in midsummer. The year was divided into 12 months of 30 days each, with 5 extra days of worship or parties tacked onto the end for a 365-day year. They also knew from carefully observing and recording star positions on different dates that they needed to add an extra day every fourth year—what we call a leap day—to keep their calendar synced to the motions of the sky. The predawn rising times of a number of bright stars were tracked in order to determine times for major religious festivals, as well as to plan for the annual floods of the Nile.
The pyramid shape itself may even represent a facet of ancient Egyptian cosmology, as some myths claim that the god of creation, Atum, lived within a pyramid that, along with the land, had emerged from the primordial ocean.
SEE ALSO Birth of Cosmology (c. 5000 BCE), Sundial (c. 3000 BCE) Sun-Centered Universe (1543).
The great pyramids of Giza, burial places of the pharaohs and astronomical pointers to the presumed gateway to the heavens at the north celestial pole. These were the largest human-made structures in the world for nearly 4,000 years.
c. 1850 BCE
Arch • Clifford A. Pickover
In architecture, an arch is a curved structure that spans a space while supporting weight. The arch has also become a metaphor for extreme durability created by the interaction of simple parts. The Roman philosopher Seneca wrote, “Human society is like an arch, kept from falling by the mutual pressure of its parts.” According to an ancient Hindu proverb, “An arch never sleeps.”
The oldest existing arched city gate is the Ashkelon gate in Israel, built c. 1850 BCE of mud-brick with some calcareous limestone. Mesopotamian brick arches are even older, but the arch gained particular prominence in ancient Rome, where it was applied to a wide range of structures.
In buildings, the arch allows the heavy load from above to be channeled into horizontal and vertical forces on supporting columns. The construction of arches usually relies upon wedge-shaped blocks, called voussoirs, that precisely fit together. The surfaces of neighboring blocks conduct loads in a mostly uniform manner. The central voussoir, at the top of the arch, is called the keystone. To build an arch, a supporting wooden framework is often used until the keystone is finally inserted, locking the arch in place. Once inserted, the arch becomes self-supporting. One advantage of the arch over earlier kinds of supporting structures is its creation from easily transported voussoirs and its spanning of large openings. Another advantage is that gravitational forces are distributed throughout the arch and converted to forces that are roughly perpendicular to voussoirs’ bottom faces. However, this means that the base of the arch is subject to some lateral forces, which must be counterbalanced by materials (e.g. a brick wall) located at the bottom sides of the arch. Much of the force of the arch is converted to compressional forces on the voussoirs—forces that stones, concrete, and other materials can easily withstand. Romans mostly constructed semicircular arches, although other shapes are possible. In Roman aqueducts, the lateral forces of neighboring arches served to counteract each other.
SEE ALSO Pulley (c. 230 BCE), Gears (c. 50), Roman Concrete (c. 126).
The arch allows the heavy load from above to be channeled into horizontal and vertical forces. Arches usually rely upon wedge-shaped blocks, called voussoirs, that fit closely together as in these ancient Turkish arches.
c. 1650 BCE
Rhind Papyrus • Clifford A. Pickover
Ahmes (c. 1680 BCE–c. 1620 BCE), Alexander Henry Rhind (1833–1863)
The Rhind Papyrus is considered to be the most important known source of information concerning ancient Egyptian mathematics. This scroll, about a foot (30 centimeters) high and 18 feet (5.5 meters) long, was found in a tomb in Thebes on the east bank of the river Nile. Ahmes, the scribe, wrote it in hieratic, a script related to the hieroglyphic system. Given that the writing occurred in around 1650 BCE, this makes Ahmes the earliest-named individual in the history of mathematics! The scroll also contains the earliest-known symbols for mathematical operations—plus is denoted by a pair of legs walking toward the number to be added.
In 1858, Scottish lawyer and Egyptologist Alexander Henry Rhind had been visiting Egypt for health reasons when he bought the scroll in a market in Luxor. The British Museum in London acquired the scroll in 1864.
Ahmes wrote that the scroll gives an “accurate reckoning for inquiring into things, and the knowledge of all things, mysteries . . . all secrets.” The content of the scroll concerns mathematical problems involving fractions, arithmetic progressions, algebra, and pyramid geometry, as well as practical mathematics useful for surveying, building, and accounting. The problem that intrigues me the most is Problem 79, the interpretation of which was initially baffling.
Today, many interpret Problem 79 as a puzzle, which may be translated as “Seven houses contain seven cats. Each cat kills seven mice. Each mouse had eaten seven ears of grain. Each ear of grain would have produced seven hekats (measures) of wheat. What is the total of all of these?” Interestingly, this indestructible puzzle meme, involving the number 7 and animals, seems to have persisted through thousands of years! We observe something quite similar in Fibonacci’s Liber Abaci (Book of Calculation), published in 1202, and later in the St. Ives puzzle, an Old English children’s rhyme involving 7 cats.
SEE ALSO Ishango Bone (c. 18,000 BCE), Pythagorean Theorem and Triangles (c. 600 BCE), Fibonacci’s Liber Abaci (1202).
The Rhind Papyrus is the most important source of information concerning ancient Egyptian mathematics. The scroll, a portion of which is shown here, includes mathematical problems involving fractions, arithmetic progressions, algebra, geometry, and accounting.
c. 1300 BCE
Iron Smelting • Derek B. Lowe
The Iron Age definitively replaced the Bronze Age, so you would assume that the newly available iron must have been clearly superior. Not so—good bronze was harder and much more corrosion-resistant. However, major disturbances and population movements in the Mediterranean and Near East around 1300 BCE may have disrupted the metal trade that bronze-working depended on. Iron ore was much easier to come by, but higher-temperature furnaces were needed to smelt it, and these often depended on forced air. Iron production was thus, sometimes, a seasonal event, with furnaces built to take advantage of monsoons and other dependable winds. Objects made of iron from before 1300 BCE are known but uncommon, and many of these are not even from our own planet—produced from solid nickel-iron meteorites, they must have been very valuable objects indeed.
Given a chance, iron will react with oxygen to produce rust (iron oxide), and smelting iron ore is basically the reverse process. The early iron-smelting device, a clay or stone furnace with air inlet tubes, was called a bloomery. Charcoal and iron ore were heated, producing a lump of crude smelted iron (the bloom) in the bottom of the furnace. This was a laborious process, since the bloom needed further heating, and lumps of impurities had to be beaten out before it could be useful. Still, iron technology spread rapidly, and it seems to have been discovered independently in several locations, including India and sub-Saharan Africa. Ancient wind-driven iron furnaces evolved into the modern blast furnace—in which ore is fed in continuously from the top and has its oxygen stripped away by contact with carbon monoxide gas of ferocious temperatures—as early as the first or second century BCE in China.
Iron’s properties change dramatically depending on what is mixed into it. Careful addition of some of the charcoal’s carbon produces steel—a superior metal in every way—but this was a job for experienced craftsmen: too little carbon produced soft wrought iron, while too much carbon yielded a very hard metal that is too brittle for most uses. Now, the varieties of iron alloys and steels in modern metallurgy are
almost too many to count.
SEE ALSO Bronze (c. 3300 BCE) Bessemer Process (1855), Plastic (1856).
A modern blast furnace can produce molten iron on a scale that ancient craftsmen could only dream of. But by any route, ironworking has always been a very energy-intensive process.
c. 1000 BCE
Olmec Compass • Clifford A. Pickover
Michael D. Coe (b. 1929), John B. Carlson (b. 1945)
For many centuries, navigators have used compasses with magnetized pointers for determining the Earth’s magnetic north pole. The Olmec compass in Mesomerica may represent the earliest known compass. The Olmecs were an ancient pre-Columbian civilization situated in south-central Mexico from around 1400 BCE to 400 BCE and famous for the colossal artwork in the form of heads carved from volcanic rock.
American astronomer John B. Carlson used Radiocarbon Dating methods of the relevant layers of an excavation to determine that a flattened, polished, oblong piece of hematite (iron oxide) had its origin about 1400–1000 BCE. Carlson has speculated that the Olmecs used such objects as direction indicators for astrology and geomancy, and for orienting burial sites. The Olmec compass is part of a polished lodestone (magnetized piece of the mineral) bar with a groove at one end that was possibly used for sighting. Note that the ancient Chinese invented the compass some time before the second century, and the compass was used for navigation by the eleventh century.
Carlson writes in “Lodestone Compass: Chinese or Olmec Primacy?”
Considering the unique morphology (purposefully shaped polished bar with a groove) and composition (magnetic mineral with magnetic moment vector in the floating plane) of M-160, and acknowledging that the Olmec were a sophisticated people who possessed advanced knowledge and skill in working iron ore minerals, I would suggest for consideration that the Early Formative artifact M-160 was probably manufactured and used as what I have called a zeroth-order compass, if not a first-order compass. Whether such a pointer would have been used to point to something astronomical (zeroth-order compass) or to geomagnetic north-south (first-order compass) is entirely open to speculation.
In the late 1960s, Yale University archeologist Michael Coe found the Olmec bar at San Lorenzo in the Mexican state of Veracruz, and it was tested by Carlson in 1973. Carlson floated it on mercury or on water with a cork mat.
SEE ALSO Ampère’s Law of Electromagnetism (1825), Faraday’s Law of Induction (1831), Telegraph System (1837), Radiocarbon Dating (1949).
In the most general definition, a lodestone refers to a naturally magnetized mineral, such as those used in fragments that ancient people used to create magnetic compasses. Shown here is a lodestone in the Hall of Gems at the National Museum of Natural History, administered by The Smithsonian Institution.
c. 600 BCE
Pythagorean Theorem and Triangles • Clifford A. Pickover
Baudhayana (c. 800 BCE), Pythagoras of Samos (c. 580 BCE–c. 500 BCE)
Today, young children sometimes first hear of the famous Pythagorean theorem from the mouth of the Scarecrow, when he finally gets a brain in MGM’s 1939 film version of The Wizard of Oz. Alas, the Scarecrow’s recitation of the famous theorem is completely wrong!
The Pythagorean theorem states that for any right triangle, the square of the hypotenuse length c is equal to the sum of the squares on the two (shorter) “leg” lengths a and b—which is written as a2 + b2 = c2. The theorem has more published proofs than any other, and Elisha Scott Loomis’s book Pythagorean Proposition contains 367 proofs.
Pythagorean triangles (PTs) are right triangles with integer sides. The “3-4-5” PT— with legs of lengths 3 and 4, and a hypotenuse of length 5—is the only PT with three sides as consecutive numbers and the only triangle with integer sides, the sum of whose sides (12) is equal to double its area (6). After the 3-4-5 PT, the next triangle with consecutive leg lengths is 21-20-29. The tenth such triangle is much larger: 27304197-27304196-38613965.
In 1643, French mathematician Pierre de Fermat (1601–1665) asked for a PT, such that both the hypotenuse c and the sum (a + b) had values that were square numbers. It was startling to find that the smallest three numbers satisfying these conditions are 4,565,486,027,761, 1,061,652,293,520, and 4,687,298,610,289. It turns out that the second such triangle would be so “large” that if its numbers were represented as feet, the triangle’s legs would project from Earth to beyond the sun!
Although Pythagoras is often credited with the formulation of the Pythagorean theorem, evidence suggests that the theorem was developed by the Hindu mathematician Baudhayana centuries earlier around 800 BCE in his book Baudhayana Sulba Sutra. Pythagorean triangles were probably known even earlier to the Babylonians.
SEE ALSO Platonic Solids (c. 350 BCE), Golden Ratio (1509), Descartes’ La Geometrie (1637).
Persian mathematician Nasr al-Din al-Tusi (1201–1274) presented a version of Euclid’s proof of the Pythagorean theorem. Al-Tusi was a prolific mathematician, astronomer, biologist, chemist, philosopher, physician, and theologian.
c. 600 BCE
Sewage Systems • Clifford A. Pickover
Given the very large variety of diseases that can be caused by sewage or sewage-contaminated water, the development of effective sewage systems deserves an entry in this book. As an example, the following sewage-related diseases are possible dangers in the United States today, and many can cause severe diarrhea: campylobacteriosis (the most common diarrheal illness in the United States, caused by the bacterium Campylobacter, which can spread to the bloodstream and cause a life-threatening infection in people with weakened immune systems), cryptosporidiosis (caused by the microscopic parasite Cryptosporidium parvum), diarrheagenic E. coli (different varieties of the Escherichia coli bacteria), encephalitis (a viral disease transmitted by mosquitoes that often lay eggs in water contaminated by sewage), viral gastroenteritis (caused by many viruses, including rotavirus), giardiasis (caused by the one-celled microscopic parasite Giardia intestinalis), hepatitis A (a liver disease caused by a virus), leptospirosis (caused by bacteria), and methaemoglobinaemia (also known as blue-baby syndrome, triggered when infants drink well-water high in nitrates from septic systems).
Other sewage-related diseases include poliomyelitis (caused by a virus) and the following diseases caused by bacteria: salmonellosis, shigellosis, paratyphoid fever, typhoid fever, yersiniosis, and cholera.
This entry is dated to around 600 BCE, which is traditionally thought to be the date of the initial construction of the Cloaca Maxima, one of the world’s most famous early and large sewage systems, constructed in ancient Rome in order to drain local marshes and channel wastes to the River Tiber. However, older sewage disposal systems were built in ancient India, prehistoric Middle East, Crete, and Scotland. Today, sewage treatment often involves various filters and the biological degradation of wastes by microorganisms in a managed habitat, followed by disinfection to reduce the number of microorganisms before the water is discharged into the environment. Disinfectants may include chlorine, ultraviolet light, and ozone. Chemicals are sometimes used to reduce the levels of nitrogen and phosphorus. Prior to sewage systems, city dwellers often threw waste into the streets.
SEE ALSO Zoo Within Us (1683), Semmelweis’s Hand Washing (1847), Antiseptics (1865), Chlorination of Water (1910).
The latrines of Housesteads Roman Fort along Hadrian’s Wall in the ancient Roman province of Britannia. The flow of water from adjacent tanks flushed away waste matter.
c. 350 BCE
Aristotle’s Organon • Clifford A. Pickover
Aristotle (384 BCE–322 BCE)
Aristotle was a Greek philosopher and scientist, a pupil of Plato, and a teacher of Alexander the Great. The Organon (Instrument) refers to the collection of six of Aristotle’s works on logic: Categories, Prior Analytics, De Interpretatione, Posterior Analytics, Sophistical Refutations, and Topics. Andronicus of Rhodes determined the ordering of the six works around 40 BCE Although Plato (c. 428–348 BCE) and Socrates (c. 470�
��399 BCE) delved into logical themes, Aristotle actually systematized the study of logic, which dominated scientific reasoning in the Western world for 2,000 years.
The goal of the Organon is not to tell readers what is true, but rather to give approaches for how to investigate truth and how to make sense of the world. The primary tool in Aristotle’s tool kit is the syllogism, a three-step argument, such as “All women are mortal; Cleopatra is a woman; therefore, Cleopatra is mortal.” If the two premises are true, we know that the conclusion must be true. Aristotle also made a distinction between particulars and universals (general categories). Cleopatra is a particular term. Woman and mortal are universal terms. When universals are used, they are preceded by “all,” “some,” or “no.” Aristotle analyzed many possible kinds of syllogisms and showed which of them are valid.
Aristotle also extended his analysis to syllogisms that involved modal logic—that is, statements containing the words “possibly” or “necessarily.” Modern mathematical logic can depart from Aristotle’s methodologies or extend his work into other kinds of sentence structures, including ones that express more complex relationships or ones that involve more than one quantifier, such as “No women like all women who dislike some women.” Nevertheless, Aristotle’s systematic attempt at developing logic is considered to be one of humankind’s greatest achievements, providing an early impetus for fields of mathematics that are in close partnership with logic and even influencing theologians in their quest to understand reality.
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