176-569: Spherics (sometimes spelled sphaerics or sphaerica ) is a term used in the history of mathematics for historical works on spherical geometry , exemplified by the Spherics ( Ancient Greek : τὰ σφαιρικά tá sphairiká ), a treatise by the Hellenistic mathematician Theodosius (2nd or early 1st century BC), and another treatise of the same title by Menelaus of Alexandria ( c. 100 AD ). This geometry-related article
352-528: A circle with approximately the same area as a given square , which imply several different approximations of the value of π. In addition, they compute the square root of 2 to several decimal places, list Pythagorean triples, and give a statement of the Pythagorean theorem . All of these results are present in Babylonian mathematics, indicating Mesopotamian influence. It is not known to what extent
528-524: A place value system and the first use of negative numbers . The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics through the work of Muḥammad ibn Mūsā al-Khwārizmī . Islamic mathematics, in turn, developed and expanded
704-409: A tally of the earliest known demonstration of sequences of prime numbers or a six-month lunar calendar. Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC, with prime numbers probably not being understood until about 500 BC. He also writes that "no attempt has been made to explain why
880-601: A 365-day cycle. This calendar, which contained an error of 11 minutes and 14 seconds, was later corrected by the Gregorian calendar organized by Pope Gregory XIII ( r. 1572–1585 ), virtually the same solar calendar used in modern times as the international standard calendar. At roughly the same time, the Han Chinese and the Romans both invented the wheeled odometer device for measuring distances traveled,
1056-467: A base of 60), is dated around 305 BC and is perhaps the oldest surviving mathematical text of China. Of particular note is the use in Chinese mathematics of a decimal positional notation system, the so-called "rod numerals" in which distinct ciphers were used for numbers between 1 and 10, and additional ciphers for powers of ten. Thus, the number 123 would be written using the symbol for "1", followed by
1232-496: A chain of fourteen lakes runs across the central drainage basin of Petén. To the south the plain gradually rises towards the Guatemalan Highlands. The dense Maya forest covers northern Petén and Belize, most of Quintana Roo , southern Campeche , and a portion of the south of Yucatán state. Farther north, the vegetation turns to lower forest consisting of dense scrub. The littoral zone of Soconusco lies to
1408-403: A circle, as well as the use of seconds and minutes of arc to denote fractions of a degree. It is thought the sexagesimal system was initially used by Sumerian scribes because 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30, and for scribes (doling out the aforementioned grain allotments, recording weights of silver, etc.) being able to easily calculate by hand was essential, and so
1584-485: A city were often linked by causeways . Architecturally, city buildings included palaces , pyramid-temples , ceremonial ballcourts , and structures specially aligned for astronomical observation. The Maya elite were literate, and developed a complex system of hieroglyphic writing. Theirs was the most advanced writing system in the pre-Columbian Americas. The Maya recorded their history and ritual knowledge in screenfold books , of which only three uncontested examples remain,
1760-485: A collection of 150 algebraic problems dealing with exact solutions to determinate and indeterminate equations . The Arithmetica had a significant influence on later mathematicians, such as Pierre de Fermat , who arrived at his famous Last Theorem after trying to generalize a problem he had read in the Arithmetica (that of dividing a square into two squares). Diophantus also made significant advances in notation,
1936-529: A complex web of rivalries, periods of dominance or submission, vassalage, and alliances. At times, different polities achieved regional dominance, such as Calakmul, Caracol , Mayapan, and Tikal. The first reliably evidenced polities formed in the Maya lowlands in the 9th century BC. During the Late Preclassic, the Maya political system coalesced into a theopolitical form, where elite ideology justified
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#17327756031672112-636: A device corresponding to a binary numeral system . His discussion of the combinatorics of meters corresponds to an elementary version of the binomial theorem . Pingala's work also contains the basic ideas of Fibonacci numbers (called mātrāmeru ). The next significant mathematical documents from India after the Sulba Sutras are the Siddhantas , astronomical treatises from the 4th and 5th centuries AD ( Gupta period ) showing strong Hellenistic influence. They are significant in that they contain
2288-595: A diagram of Pascal's triangle with coefficients of binomial expansions through the eighth power, though both appear in Chinese works as early as 1100. The Chinese also made use of the complex combinatorial diagram known as the magic square and magic circles , described in ancient times and perfected by Yang Hui (AD 1238–1298). Even after European mathematics began to flourish during the Renaissance , European and Chinese mathematics were separate traditions, with significant Chinese mathematical output in decline from
2464-529: A formula for obtaining Pythagorean triples bears his name. Eudoxus developed the method of exhaustion , a precursor of modern integration and a theory of ratios that avoided the problem of incommensurable magnitudes . The former allowed the calculations of areas and volumes of curvilinear figures, while the latter enabled subsequent geometers to make significant advances in geometry. Though he made no specific technical mathematical discoveries, Aristotle (384– c. 322 BC ) contributed significantly to
2640-424: A new king was a highly elaborate ceremony, involving a series of separate acts that included enthronement upon a jaguar-skin cushion, human sacrifice, and receiving the symbols of royal power, such as a headband bearing a jade representation of the so-called "jester god", an elaborate headdress adorned with quetzal feathers, and a sceptre representing the god Kʼawiil . Maya political administration, based around
2816-440: A number of Maya sites with English architect and draftsman Frederick Catherwood . Their illustrated accounts of the ruins sparked strong popular interest, and brought the Maya to world attention. The later 19th century saw the recording and recovery of ethnohistoric accounts of the Maya, and the first steps in deciphering Maya hieroglyphs. The final two decades of the 19th century saw the birth of modern scientific archaeology in
2992-578: A result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. Pythagoras established the Pythagorean School , whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The Pythagoreans are credited with
3168-533: A sexagesimal system is pragmatically easier to calculate by hand with; however, there is the possibility that using a sexagesimal system was an ethno-linguistic phenomenon (that might not ever be known), and not a mathematical/practical decision. Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a place-value system, where digits written in the left column represented larger values, much as in
3344-536: A significant Maya presence remained into the Postclassic period after the abandonment of the major Classic period cities; the population was particularly concentrated near permanent water sources. Unlike during previous cycles of contraction, abandoned lands were not quickly resettled in the Postclassic. Activity shifted to the northern lowlands and the Maya Highlands; this may have involved migration from
3520-741: A small empire covering a large part of the western Guatemalan Highlands and the neighbouring Pacific coastal plain. However, in the decades before the Spanish conquest of the Kaqchikel kingdom had been steadily eroding the kingdom of the Kʼicheʼ. In 1511, a Spanish caravel was wrecked in the Caribbean, and about a dozen survivors made landfall on the coast of Yucatán. They were seized by a Maya lord, and most were sacrificed , although two escaped. From 1517 to 1519, three separate Spanish expeditions explored
3696-544: A small number of geometrical theorems as well. It also defined the concepts of circumference , diameter , radius , and volume . In 212 BC, the Emperor Qin Shi Huang commanded all books in the Qin Empire other than officially sanctioned ones be burned. This decree was not universally obeyed, but as a consequence of this order little is known about ancient Chinese mathematics before this date. After
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#17327756031673872-403: A strategy of increasing administration, and filling administrative posts with loyal supporters rather than blood relatives. Within a polity, mid-ranking population centres would have played a key role in managing resources and internal conflict. The Maya political landscape was highly complex and Maya elites engaged in political intrigue to gain economic and social advantage over neighbours. In
4048-425: A successful military campaign could vary in its impact on the defeated polity. In some cases, entire cities were sacked, and never resettled, as at Aguateca. In other instances, the victors would seize the defeated rulers, their families, and patron gods. The captured nobles and their families could be imprisoned, or sacrificed. At the least severe end of the scale, the defeated polity would be obliged to pay tribute to
4224-530: A tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10." The Ishango bone, according to scholar Alexander Marshack , may have influenced the later development of mathematics in Egypt as, like some entries on the Ishango bone, Egyptian arithmetic also made use of multiplication by 2; this however, is disputed. Predynastic Egyptians of
4400-557: Is a stub . You can help Misplaced Pages by expanding it . History of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past . Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer , Akkad and Assyria , followed closely by Ancient Egypt and
4576-526: Is a modern term used to refer collectively to the various peoples that inhabited this area, as Maya peoples have not had a sense of a common ethnic identity or political unity for the vast majority of their history. Early Spanish and Mayan-language colonial sources in the Yucatán Peninsula used the term "Maya" to denote both the language spoken by the Yucatec Maya and the area surrounding
4752-463: Is also due the systematic use of the 360 degree circle. Heron of Alexandria ( c. 10 –70 AD) is credited with Heron's formula for finding the area of a scalene triangle and with being the first to recognize the possibility of negative numbers possessing square roots. Menelaus of Alexandria ( c. 100 AD ) pioneered spherical trigonometry through Menelaus' theorem . The most complete and influential trigonometric work of antiquity
4928-515: Is considered to be of particular importance because it gives a method for finding the volume of a frustum (truncated pyramid). Finally, the Berlin Papyrus 6619 (c. 1800 BC) shows that ancient Egyptians could solve a second-order algebraic equation . Greek mathematics refers to the mathematics written in the Greek language from the time of Thales of Miletus (~600 BC) to the closure of
5104-644: Is found on a wax tablet dated to the 1st century AD (now found in the British Museum ). The association of the Neopythagoreans with the Western invention of the multiplication table is evident in its later Medieval name: the mensa Pythagorica . Plato (428/427 BC – 348/347 BC) is important in the history of mathematics for inspiring and guiding others. His Platonic Academy , in Athens , became
5280-429: Is independent of Western mathematics; To this period belongs the mathematician Seki Takakazu , of great influence, for example, in the development of wasan (traditional Japanese mathematics), and whose discoveries (in areas such as integral calculus ), are almost simultaneous with contemporary European mathematicians such as Gottfried Leibniz . Japanese mathematics of this period is inspired by Chinese mathematics and
5456-635: Is named Babylonian mathematics due to the central role of Babylon as a place of study. Later under the Arab Empire , Mesopotamia, especially Baghdad , once again became an important center of study for Islamic mathematics . In contrast to the sparsity of sources in Egyptian mathematics , knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s. Written in Cuneiform script , tablets were inscribed whilst
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5632-525: Is not yet deciphered, but it was held only by the most powerful kings of the strongest dynasties. It indicated an overlord, or high king , and was only in use during the Classic period. By the Late Classic, the absolute power of the kʼuhul ajaw had weakened, and the political system had diversified to include a wider aristocracy, that by this time may well have expanded disproportionately. A sajal
5808-710: Is oriented towards essentially geometric problems. On wooden tablets called sangaku, "geometric enigmas" are proposed and solved; That's where, for example, Soddy's hexlet theorem comes from. The earliest civilization on the Indian subcontinent is the Indus Valley civilization (mature second phase: 2600 to 1900 BC) that flourished in the Indus river basin. Their cities were laid out with geometric regularity, but no known mathematical documents survive from this civilization. The oldest extant mathematical records from India are
5984-528: Is sometimes taken as the end of the era of the Alexandrian Greek mathematics, although work did continue in Athens for another century with figures such as Proclus , Simplicius and Eutocius . Although Proclus and Simplicius were more philosophers than mathematicians, their commentaries on earlier works are valuable sources on Greek mathematics. The closure of the neo-Platonic Academy of Athens by
6160-474: Is supported by the existence of languages which preserve the distinction between "one", "two", and "many", but not of numbers larger than two. The Ishango bone , found near the headwaters of the Nile river (northeastern Congo ), may be more than 20,000 years old and consists of a series of marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either
6336-458: Is the Almagest of Ptolemy ( c. AD 90 –168), a landmark astronomical treatise whose trigonometric tables would be used by astronomers for the next thousand years. Ptolemy is also credited with Ptolemy's theorem for deriving trigonometric quantities, and the most accurate value of π outside of China until the medieval period, 3.1416. Following a period of stagnation after Ptolemy,
6512-736: Is the Rhind papyrus (sometimes also called the Ahmes Papyrus after its author), dated to c. 1650 BC but likely a copy of an older document from the Middle Kingdom of about 2000–1800 BC. It is an instruction manual for students in arithmetic and geometry. In addition to giving area formulas and methods for multiplication, division and working with unit fractions, it also contains evidence of other mathematical knowledge, including composite and prime numbers ; arithmetic , geometric and harmonic means ; and simplistic understandings of both
6688-562: The Moscow Mathematical Papyrus (Egyptian c. 1890 BC). All of these texts mention the so-called Pythagorean triples , so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. The study of mathematics as a "demonstrative discipline" began in the 6th century BC with the Pythagoreans , who coined the term "mathematics" from
6864-625: The suan pan , or Chinese abacus. The date of the invention of the suan pan is not certain, but the earliest written mention dates from AD 190, in Xu Yue 's Supplementary Notes on the Art of Figures . The oldest extant work on geometry in China comes from the philosophical Mohist canon c. 330 BC , compiled by the followers of Mozi (470–390 BC). The Mo Jing described various aspects of many fields associated with physical science, and provided
7040-571: The Academy of Athens in 529 AD. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to North Africa, but were united by culture and language. Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. Greek mathematics was much more sophisticated than the mathematics that had been developed by earlier cultures. All surviving records of pre-Greek mathematics show
7216-564: The Antikythera mechanism , the odometer of Vitruvius featured chariot wheels measuring 4 feet (1.2 m) in diameter turning four-hundred times in one Roman mile (roughly 4590 ft/1400 m). With each revolution, a pin-and-axle device engaged a 400-tooth cogwheel that turned a second gear responsible for dropping pebbles into a box, each pebble representing one mile traversed. An analysis of early Chinese mathematics has demonstrated its unique development compared to other parts of
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7392-462: The Arithmetica being the first instance of algebraic symbolism and syncopation. Among the last great Greek mathematicians is Pappus of Alexandria (4th century AD). He is known for his hexagon theorem and centroid theorem , as well as the Pappus configuration and Pappus graph . His Collection is a major source of knowledge on Greek mathematics as most of it has survived. Pappus is considered
7568-704: The Brahmagupta theorem , Brahmagupta's identity and Brahmagupta's formula , and for the first time, in Brahma-sphuta-siddhanta , he lucidly explained the use of zero as both a placeholder and decimal digit , and explained the Hindu–Arabic numeral system . It was from a translation of this Indian text on mathematics (c. 770) that Islamic mathematicians were introduced to this numeral system, which they adapted as Arabic numerals . Islamic scholars carried knowledge of this number system to Europe by
7744-488: The Confucian -based East Asian cultural sphere . Korean and Japanese mathematics were heavily influenced by the algebraic works produced during China's Song dynasty, whereas Vietnamese mathematics was heavily indebted to popular works of China's Ming dynasty (1368–1644). For instance, although Vietnamese mathematical treatises were written in either Chinese or the native Vietnamese Chữ Nôm script, all of them followed
7920-856: The Egyptian language . From the Hellenistic period , Greek replaced Egyptian as the written language of Egyptian scholars. Mathematical study in Egypt later continued under the Arab Empire as part of Islamic mathematics , when Arabic became the written language of Egyptian scholars. Archaeological evidence has suggested that the Ancient Egyptian counting system had origins in Sub-Saharan Africa. Also, fractal geometry designs which are widespread among Sub-Saharan African cultures are also found in Egyptian architecture and cosmological signs. The most extensive Egyptian mathematical text
8096-750: The Maya Region , an area that today comprises southeastern Mexico , all of Guatemala and Belize , and the western portions of Honduras and El Salvador . It includes the northern lowlands of the Yucatán Peninsula and the Guatemalan Highlands of the Sierra Madre , the Mexican state of Chiapas , southern Guatemala , El Salvador, and the southern lowlands of the Pacific littoral plain. Today, their descendants, known collectively as
8272-529: The Middle Ages , periods of mathematical discovery were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 15th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day. This includes the groundbreaking work of both Isaac Newton and Gottfried Wilhelm Leibniz in
8448-581: The Petén Basin , and the city of Kaminaljuyu rose to prominence in the Guatemalan Highlands . Beginning around 250 AD, the Classic period is largely defined as when the Maya were raising sculpted monuments with Long Count dates . This period saw the Maya civilization develop many city-states linked by a complex trade network . In the Maya Lowlands two great rivals, the cities of Tikal and Calakmul , became powerful. The Classic period also saw
8624-468: The Pythagorean theorem , and a mathematical formula for Gaussian elimination . The treatise also provides values of π , which Chinese mathematicians originally approximated as 3 until Liu Xin (d. 23 AD) provided a figure of 3.1457 and subsequently Zhang Heng (78–139) approximated pi as 3.1724, as well as 3.162 by taking the square root of 10. Liu Hui commented on the Nine Chapters in
8800-569: The Sieve of Eratosthenes and perfect number theory (namely, that of the number 6). It also shows how to solve first order linear equations as well as arithmetic and geometric series . Another significant Egyptian mathematical text is the Moscow papyrus , also from the Middle Kingdom period, dated to c. 1890 BC. It consists of what are today called word problems or story problems , which were apparently intended as entertainment. One problem
8976-402: The Sulba Sutras (dated variously between the 8th century BC and the 2nd century AD), appendices to religious texts which give simple rules for constructing altars of various shapes, such as squares, rectangles, parallelograms, and others. As with Egypt, the preoccupation with temple functions points to an origin of mathematics in religious ritual. The Sulba Sutras give methods for constructing
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#17327756031679152-415: The ajaw was a member of the ruling class and a major city could have more than one, each ruling over different districts. Paramount rulers distinguished themselves from the extended nobility by prefixing the word kʼuhul to their ajaw title. A kʼuhul ajaw was "divine lord", originally confined to the kings of the most prestigious and ancient royal lines. Kalomte was a royal title, whose exact meaning
9328-562: The axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework. The Elements was known to all educated people in the West up through the middle of the 20th century and its contents are still taught in geometry classes today. In addition to
9504-702: The book burning of 212 BC, the Han dynasty (202 BC–220 AD) produced works of mathematics which presumably expanded on works that are now lost. The most important of these is The Nine Chapters on the Mathematical Art , the full title of which appeared by AD 179, but existed in part under other titles beforehand. It consists of 246 word problems involving agriculture, business, employment of geometry to figure height spans and dimension ratios for Chinese pagoda towers, engineering, surveying , and includes material on right triangles . It created mathematical proof for
9680-456: The decimal system. The power of the Babylonian notational system lay in that it could be used to represent fractions as easily as whole numbers; thus multiplying two numbers that contained fractions was no different from multiplying integers, similar to modern notation. The notational system of the Babylonians was the best of any civilization until the Renaissance , and its power allowed it to achieve remarkable computational accuracy; for example,
9856-538: The lunar calendar of the Republican era contained 355 days, roughly ten-and-one-fourth days shorter than the solar year , a discrepancy that was solved by adding an extra month into the calendar after the 23rd of February. This calendar was supplanted by the Julian calendar , a solar calendar organized by Julius Caesar (100–44 BC) and devised by Sosigenes of Alexandria to include a leap day every four years in
10032-519: The opus tessellatum pieces on average measuring eight millimeters square and the finer opus vermiculatum pieces having an average surface of four millimeters square. The creation of the Roman calendar also necessitated basic mathematics. The first calendar allegedly dates back to 8th century BC during the Roman Kingdom and included 356 days plus a leap year every other year. In contrast,
10208-478: The spiral bearing his name, obtained formulas for the volumes of surfaces of revolution (paraboloid, ellipsoid, hyperboloid), and an ingenious method of exponentiation for expressing very large numbers. While he is also known for his contributions to physics and several advanced mechanical devices, Archimedes himself placed far greater value on the products of his thought and general mathematical principles. He regarded as his greatest achievement his finding of
10384-579: The theoretical mathematics and geometry that were prized by the Greeks. It is unclear if the Romans first derived their numerical system directly from the Greek precedent or from Etruscan numerals used by the Etruscan civilization centered in what is now Tuscany , central Italy . Using calculation, Romans were adept at both instigating and detecting financial fraud , as well as managing taxes for
10560-774: The treasury . Siculus Flaccus , one of the Roman gromatici (i.e. land surveyor), wrote the Categories of Fields , which aided Roman surveyors in measuring the surface areas of allotted lands and territories. Aside from managing trade and taxes, the Romans also regularly applied mathematics to solve problems in engineering , including the erection of architecture such as bridges , road-building , and preparation for military campaigns . Arts and crafts such as Roman mosaics , inspired by previous Greek designs , created illusionist geometric patterns and rich, detailed scenes that required precise measurements for each tessera tile,
10736-435: The "Golden Age" of Greek mathematics, with advances in pure mathematics henceforth in relative decline. Nevertheless, in the centuries that followed significant advances were made in applied mathematics, most notably trigonometry , largely to address the needs of astronomers. Hipparchus of Nicaea ( c. 190 –120 BC) is considered the founder of trigonometry for compiling the first known trigonometric table, and to him
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#173277560316710912-614: The 12th century, and it has now displaced all older number systems throughout the world. Various symbol sets are used to represent numbers in the Hindu–Arabic numeral system, all of which evolved from the Brahmi numerals . Each of the roughly dozen major scripts of India has its own numeral glyphs. In the 10th century, Halayudha 's commentary on Pingala 's work contains a study of the Fibonacci sequence and Pascal's triangle , and describes
11088-410: The 13th century onwards. Jesuit missionaries such as Matteo Ricci carried mathematical ideas back and forth between the two cultures from the 16th to 18th centuries, though at this point far more mathematical ideas were entering China than leaving. Japanese mathematics , Korean mathematics , and Vietnamese mathematics are traditionally viewed as stemming from Chinese mathematics and belonging to
11264-500: The 1950s, the texts revealed the warlike activities of the Classic Maya kings, undermining the view of the Maya as peaceful. Unlike the Aztecs and the Inca , the Maya political system never integrated the entire Maya cultural area into a single state or empire. Rather, throughout its history, the Maya area contained a varying mix of political complexity that included both states and chiefdoms . These polities fluctuated greatly in their relationships with each other and were engaged in
11440-452: The 3rd century AD and gave a value of π accurate to 5 decimal places (i.e. 3.14159). Though more of a matter of computational stamina than theoretical insight, in the 5th century AD Zu Chongzhi computed the value of π to seven decimal places (between 3.1415926 and 3.1415927), which remained the most accurate value of π for almost the next 1000 years. He also established a method which would later be called Cavalieri's principle to find
11616-485: The 5th millennium BC pictorially represented geometric designs. It has been claimed that megalithic monuments in England and Scotland , dating from the 3rd millennium BC, incorporate geometric ideas such as circles , ellipses , and Pythagorean triples in their design. All of the above are disputed however, and the currently oldest undisputed mathematical documents are from Babylonian and dynastic Egyptian sources. Babylonian mathematics refers to any mathematics of
11792-472: The Babylonian tablet YBC 7289 gives an approximation of √ 2 accurate to five decimal places. The Babylonians lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context. By the Seleucid period, the Babylonians had developed a zero symbol as a placeholder for empty positions; however it was only used for intermediate positions. This zero sign does not appear in terminal positions, thus
11968-412: The Babylonians came close but did not develop a true place value system. Other topics covered by Babylonian mathematics include fractions, algebra, quadratic and cubic equations, and the calculation of regular numbers , and their reciprocal pairs . The tablets also include multiplication tables and methods for solving linear , quadratic equations and cubic equations , a remarkable achievement for
12144-413: The Chinese format of presenting a collection of problems with algorithms for solving them, followed by numerical answers. Mathematics in Vietnam and Korea were mostly associated with the professional court bureaucracy of mathematicians and astronomers , whereas in Japan it was more prevalent in the realm of private schools . The mathematics that developed in Japan during the Edo period (1603-1887)
12320-456: The Classic period, such trophy heads no longer appeared on the king's belt, but Classic period kings are frequently depicted standing over humiliated war captives. Right up to the end of the Postclassic period, Maya kings led as war captains. Maya inscriptions from the Classic show that a defeated king could be captured, tortured, and sacrificed. The Spanish recorded that Maya leaders kept track of troop movements in painted books. The outcome of
12496-414: The Early Classic period, the Maya cities of Tikal and Kaminaljuyu were key Maya foci in a network that extended into the highlands of central Mexico; there was a strong Maya presence at the Tetitla compound of Teotihuacan. The Maya city of Chichen Itza and the distant Toltec capital of Tula had an especially close relationship . The Petén region consists of densely forested low-lying limestone plain;
12672-598: The Early Classic. This was a 0.5-metre-long (1.6 ft) stick with a notched end to hold a dart or javelin . The stick was used to launch the missile with more force and accuracy than simply hurling it with the arm. Evidence in the form of stone blade points recovered from Aguateca indicate that darts and spears were the primary weapons of the Classic Maya warrior. Commoners used blowguns in war, which also served as their hunting weapon. The bow and arrow
12848-565: The Guatemalan Highlands. In the 16th century, the Spanish Empire colonised the Mesoamerican region, and a lengthy series of campaigns saw the fall of Nojpetén , the last Maya city, in 1697. Rule during the Classic period centred on the concept of the "divine king", who was thought to act as a mediator between mortals and the supernatural realm. Kingship was usually (but not exclusively) patrilineal , and power normally passed to
13024-451: The Late Classic, some cities established a long period of dominance over other large cities, such as the dominance of Caracol over Naranjo for half a century. In other cases, loose alliance networks were formed around a dominant city. Border settlements, usually located about halfway between neighbouring capitals, often switched allegiance over the course of their history, and at times acted independently. Dominant capitals exacted tribute in
13200-549: The Levantine state of Ebla began using arithmetic , algebra and geometry for purposes of taxation , commerce , trade and also in the field of astronomy to record time and formulate calendars . The earliest mathematical texts available are from Mesopotamia and Egypt – Plimpton 322 ( Babylonian c. 2000 – 1900 BC), the Rhind Mathematical Papyrus ( Egyptian c. 1800 BC) and
13376-491: The Maya area, trade routes particularly focused on central Mexico and the Gulf coast. In the Early Classic, Chichen Itza was at the hub of an extensive trade network that imported gold discs from Colombia and Panama , and turquoise from Los Cerrillos, New Mexico . Long-distance trade of both luxury and utilitarian goods was probably controlled by the royal family. Prestige goods obtained by trade were used both for consumption by
13552-664: The Maya lowlands, where large structures have been dated to around 750 BC. The northern lowlands of Yucatán were widely settled by the Middle Preclassic. By approximately 400 BC, early Maya rulers were raising stelae. A developed script was already being used in Petén by the 3rd century BC. In the Late Preclassic Period, the enormous city of El Mirador grew to cover approximately 16 square kilometres (6.2 sq mi). Although not as large, Tikal
13728-591: The Maya region, with the meticulous work of Alfred Maudslay and Teoberto Maler . By the early 20th century, the Peabody Museum was sponsoring excavations at Copán and in the Yucatán Peninsula. In the first two decades of the 20th century, advances were made in deciphering the Maya calendar, and identifying deities, dates, and religious concepts. Since the 1930s, archaeological exploration increased dramatically, with large-scale excavations across
13904-511: The Maya region. In the 1960s, Mayanist J. Eric S. Thompson promoted the ideas that Maya cities were essentially vacant ceremonial centres serving a dispersed population in the forest, and that the Maya civilization was governed by peaceful astronomer-priests. These ideas began to collapse with major advances in the decipherment of the script in the late 20th century, pioneered by Heinrich Berlin, Tatiana Proskouriakoff , and Yuri Knorozov . With breakthroughs in understanding of Maya script since
14080-405: The Maya, number well over 6 million individuals, speak more than twenty-eight surviving Mayan languages , and reside in nearly the same area as their ancestors. The Archaic period , before 2000 BC, saw the first developments in agriculture and the earliest villages. The Preclassic period ( c. 2000 BC to 250 AD ) saw the establishment of the first complex societies in the Maya region, and
14256-481: The Pacific coast, and the Maya were already cultivating the staple crops of maize, beans, squash, and chili pepper. This period was characterised by sedentary communities and the introduction of pottery and fired clay figurines. During the Middle Preclassic Period , small villages began to grow to form cities. Nakbe in the Petén department of Guatemala is the earliest well-documented city in
14432-479: The Roman model first described by the Roman civil engineer and architect Vitruvius ( c. 80 BC – c. 15 BC ). The device was used at least until the reign of emperor Commodus ( r. 177 – 192 AD ), but its design seems to have been lost until experiments were made during the 15th century in Western Europe. Perhaps relying on similar gear-work and technology found in
14608-589: The Spanish were invited as allies into Iximche , the capital city of the Kaqchikel Maya. Good relations did not last, due to excessive Spanish demands for gold as tribute, and the city was abandoned a few months later. This was followed by the fall of Zaculeu , the Mam Maya capital, in 1525. Francisco de Montejo and his son, Francisco de Montejo the Younger , launched a long series of campaigns against
14784-517: The Sulba Sutras influenced later Indian mathematicians. As in China, there is a lack of continuity in Indian mathematics; significant advances are separated by long periods of inactivity. Pāṇini (c. 5th century BC) formulated the rules for Sanskrit grammar . His notation was similar to modern mathematical notation, and used metarules, transformations , and recursion . Pingala (roughly 3rd–1st centuries BC) in his treatise of prosody uses
14960-463: The Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period. Babylonian mathematics were written using a sexagesimal (base-60) numeral system . From this derives the modern-day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 × 6) degrees in
15136-508: The Terminal Classic, the northern cities of Chichen Itza and Uxmal showed increased activity. Major cities in the northern Yucatán Peninsula were inhabited long after the cities of the southern lowlands ceased to raise monuments. Classic Maya social organization was based on the ritual authority of the ruler, rather than central control of trade and food distribution. This model was poorly structured to respond to changes, because
15312-524: The Tikal king Kʼinich Muwaan Jol II, was sent to found a new city at Dos Pilas , in the Petexbatún region, apparently as an outpost to extend Tikal's power beyond the reach of Calakmul. For the next two decades he fought loyally for his brother and overlord at Tikal. In 648, king Yuknoom Chʼeen II of Calakmul captured Balaj Chan Kʼawiil. Yuknoom Chʼeen II then reinstated Balaj Chan Kʼawiil upon
15488-586: The Yucatán coast, and engaged in a number of battles with the Maya inhabitants. After the Aztec capital Tenochtitlan fell to the Spanish in 1521, Hernán Cortés despatched Pedro de Alvarado to Guatemala with 180 cavalry, 300 infantry, 4 cannons, and thousands of allied warriors from central Mexico; they arrived in Soconusco in 1523. The Kʼicheʼ capital, Qʼumarkaj, fell to Alvarado in 1524. Shortly afterwards,
15664-531: The ancient Greek μάθημα ( mathema ), meaning "subject of instruction". Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs ) and expanded the subject matter of mathematics. The ancient Romans used applied mathematics in surveying , structural engineering , mechanical engineering , bookkeeping , creation of lunar and solar calendars , and even arts and crafts . Chinese mathematics made early contributions, including
15840-455: The backs of porters when going overland; if the trade route followed a river or the coast, then goods were transported in canoes. A substantial Maya trading canoe made from a large hollowed-out tree trunk was encountered off Honduras on Christopher Columbus 's fourth voyage . The canoe was 2.5 metres (8.2 ft) broad and was powered by 25 rowers. Trade goods carried included cacao, obsidian, ceramics, textiles, and copper bells and axes. Cacao
16016-549: The centers of mathematical innovation were to be found elsewhere by this time. Although ethnic Greek mathematicians continued under the rule of the late Roman Republic and subsequent Roman Empire , there were no noteworthy native Latin mathematicians in comparison. Ancient Romans such as Cicero (106–43 BC), an influential Roman statesman who studied mathematics in Greece, believed that Roman surveyors and calculators were far more interested in applied mathematics than
16192-557: The changes were catastrophic and resulted in the rapid depopulation of cities. Within a couple of generations, large swathes of the central Maya area were all but abandoned. Both the capitals and their secondary centres were generally abandoned within a period of 50 to 100 years. One by one, cities stopped sculpting dated monuments; the last Long Count date was inscribed at Toniná in 909. Stelae were no longer raised, and squatters moved into abandoned royal palaces. Mesoamerican trade routes shifted and bypassed Petén. Although much reduced,
16368-438: The city's ruler, and as luxury gifts to consolidate the loyalty of vassals and allies. Trade routes not only supplied physical goods, they facilitated the movement of people and ideas throughout Mesoamerica. Shifts in trade routes occurred with the rise and fall of important cities in the Maya region, and have been identified in every major reorganization of the Maya civilization, such as the rise of Preclassic Maya civilization,
16544-562: The clay was moist, and baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework. The earliest evidence of written mathematics dates back to the ancient Sumerians , who built the earliest civilization in Mesopotamia. They developed a complex system of metrology from 3000 BC that was chiefly concerned with administrative/financial counting, such as grain allotments, workers, weights of silver, or even liquids, among other things. From around 2500 BC onward,
16720-530: The collapse of the kingdoms of the Petexbatún region of western Petén. The rapid abandonment of Aguateca by its inhabitants has provided a rare opportunity to examine the remains of Maya weaponry in situ . Aguateca was stormed by unknown enemies around 810 AD, who overcame its formidable defences and burned the royal palace. The elite inhabitants of the city either fled or were captured, and never returned to collect their abandoned property. The inhabitants of
16896-403: The complete destruction of an enemy state. Little is known about Maya military organization, logistics, or training. Warfare is depicted in Maya art from the Classic period, and wars and victories are mentioned in hieroglyphic inscriptions. Unfortunately, the inscriptions do not provide information upon the causes of war, or the form it took. In the 8th–9th centuries, intensive warfare resulted in
17072-415: The conquest. At times, the colonial administration encouraged the traditional economy in order to extract tribute in the form of ceramics or cotton textiles, although these were usually made to European specifications. Maya beliefs and language proved resistant to change, despite vigorous efforts by Catholic missionaries. The 260-day tzolkʼin ritual calendar continues in use in modern Maya communities in
17248-474: The cultivation of the staple crops of the Maya diet , including maize , beans , squashes , and chili peppers . The first Maya cities developed around 750 BC, and by 500 BC these cities possessed monumental architecture, including large temples with elaborate stucco façades. Hieroglyphic writing was being used in the Maya region by the 3rd century BC. In the Late Preclassic, a number of large cities developed in
17424-439: The development of infinitesimal calculus during the course of the 17th century. The origins of mathematical thought lie in the concepts of number , patterns in nature , magnitude , and form . Modern studies of animal cognition have shown that these concepts are not unique to humans. Such concepts would have been part of everyday life in hunter-gatherer societies. The idea of the "number" concept evolving gradually over time
17600-515: The development of mathematics by laying the foundations of logic . In the 3rd century BC, the premier center of mathematical education and research was the Musaeum of Alexandria . It was there that Euclid ( c. 300 BC ) taught, and wrote the Elements , widely considered the most successful and influential textbook of all time. The Elements introduced mathematical rigor through
17776-479: The early Spanish explorers reported wealthy coastal cities and thriving marketplaces. During the Late Postclassic, the Yucatán Peninsula was divided into a number of independent provinces that shared a common culture but varied in internal sociopolitical organization. On the eve of the Spanish conquest, the highlands of Guatemala were dominated by several powerful Maya states. The Kʼicheʼ had carved out
17952-791: The eldest son . A prospective king was expected to be a successful war leader as well as a ruler. Closed patronage systems were the dominant force in Maya politics, although how patronage affected the political makeup of a kingdom varied from city-state to city-state. By the Late Classic period, the aristocracy had grown in size, reducing the previously exclusive power of the king. The Maya developed sophisticated art forms using both perishable and non-perishable materials, including wood , jade , obsidian , ceramics , sculpted stone monuments, stucco, and finely painted murals. Maya cities tended to expand organically. The city centers comprised ceremonial and administrative complexes, surrounded by an irregularly shaped sprawl of residential districts. Different parts of
18128-400: The elite in the form of staple goods such as maize, flour and game. It is likely that hard-working commoners who displayed exceptional skills and initiative could become influential members of Maya society. Warfare was prevalent in the Maya world. Military campaigns were launched for a variety of reasons, including the control of trade routes and tribute, raids to take captives, scaling up to
18304-702: The elite. During the Contact period, Maya nobility took part in long-distance trading expeditions. The majority of traders were middle class, but were largely engaged in local and regional trade rather than the prestigious long-distance trading that was the preserve of the elite. The travelling of merchants into dangerous foreign territory was likened to a passage through the underworld ; the patron deities of merchants were two underworld gods carrying backpacks. When merchants travelled, they painted themselves black, like their patron gods, and went heavily armed. The Maya had no pack animals, so all trade goods were carried on
18480-525: The emperor Justinian in 529 AD is traditionally held as marking the end of the era of Greek mathematics, although the Greek tradition continued unbroken in the Byzantine empire with mathematicians such as Anthemius of Tralles and Isidore of Miletus , the architects of the Hagia Sophia . Nevertheless, Byzantine mathematics consisted mostly of commentaries, with little in the way of innovation, and
18656-413: The enemy as the seizure of captives and plunder. There is some evidence from the Classic period that women provided supporting roles in war, but they did not act as military officers with the exception of those rare ruling queens. By the Postclassic, the native chronicles suggest that women occasionally fought in battle. The atlatl (spear-thrower) was introduced to the Maya region by Teotihuacan in
18832-461: The epoch were abandoned; the cause of this collapse is unknown. The Classic period is largely defined as the period during which the lowland Maya raised dated monuments using the Long Count calendar. This period marked the peak of large-scale construction and urbanism , the recording of monumental inscriptions, and demonstrated significant intellectual and artistic development, particularly in
19008-493: The extent of the influence is disputed, they were probably inspired by Egyptian and Babylonian mathematics . According to legend, Pythagoras traveled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests. Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem . As
19184-548: The familiar theorems of Euclidean geometry , the Elements was meant as an introductory textbook to all mathematical subjects of the time, such as number theory , algebra and solid geometry , including proofs that the square root of two is irrational and that there are infinitely many prime numbers. Euclid also wrote extensively on other subjects, such as conic sections , optics , spherical geometry , and mechanics, but only half of his writings survive. Archimedes ( c. 287 –212 BC) of Syracuse , widely considered
19360-581: The first instance of trigonometric relations based on the half-chord, as is the case in modern trigonometry, rather than the full chord, as was the case in Ptolemaic trigonometry. Through a series of translation errors, the words "sine" and "cosine" derive from the Sanskrit "jiya" and "kojiya". Around 500 AD, Aryabhata wrote the Aryabhatiya , a slim volume, written in verse, intended to supplement
19536-599: The first proof of the Pythagorean theorem , though the statement of the theorem has a long history, and with the proof of the existence of irrational numbers . Although he was preceded by the Babylonians , Indians and the Chinese , the Neopythagorean mathematician Nicomachus (60–120 AD) provided one of the earliest Greco-Roman multiplication tables , whereas the oldest extant Greek multiplication table
19712-498: The form of luxury items from subjugated population centres. Political power was reinforced by military power, and the capture and humiliation of enemy warriors played an important part in elite culture. An overriding sense of pride and honour among the warrior aristocracy could lead to extended feuds and vendettas, which caused political instability and the fragmentation of polities. From the Early Preclassic, Maya society
19888-402: The form of quilted cotton that had been soaked in salt water to toughen it; the resulting armour compared favourably to the steel armour worn by the Spanish when they conquered the region. Warriors bore wooden or animal hide shields decorated with feathers and animal skins. Trade was a key component of Maya society, and in the development of the Maya civilization. The cities that grew to become
20064-530: The formation of a matrix . In the 12th century, Bhāskara II , who lived in southern India, wrote extensively on all then known branches of mathematics. His work contains mathematical objects equivalent or approximately equivalent to infinitesimals, the mean value theorem and the derivative of the sine function although he did not develop the notion of a derivative. In the 14th century, Narayana Pandita completed his Ganita Kaumudi . Maya civilization The Maya civilization ( / ˈ m aɪ ə / )
20240-448: The great metropolis of Teotihuacan in the distant Valley of Mexico . In AD 378, Teotihuacan decisively intervened at Tikal and other nearby cities, deposed their rulers, and installed a new Teotihuacan-backed dynasty. This intervention was led by Siyaj Kʼakʼ ("Born of Fire"), who arrived at Tikal in early 378. The king of Tikal, Chak Tok Ichʼaak I , died on the same day, suggesting a violent takeover. A year later, Siyaj Kʼakʼ oversaw
20416-493: The greatest mathematician of antiquity, used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series , in a manner not too dissimilar from modern calculus. He also showed one could use the method of exhaustion to calculate the value of π with as much precision as desired, and obtained the most accurate value of π then known, 3+ 10 / 71 < π < 3+ 10 / 70 . He also studied
20592-687: The highlands of Guatemala and Chiapas, and millions of Mayan-language speakers inhabit the territory in which their ancestors developed their civilization. The agents of the Catholic Church wrote detailed accounts of the Maya, in support of their efforts at Christianization , and absorption of the Maya into the Spanish Empire. This was followed by various Spanish priests and colonial officials who left descriptions of ruins they visited in Yucatán and Central America. In 1839, American traveller and writer John Lloyd Stephens set out to visit
20768-470: The installation of a new king, Yax Nuun Ahiin I . This led to a period of political dominance when Tikal became the most powerful city in the central lowlands. Tikal's great rival was Calakmul, another powerful city in the Petén Basin. Tikal and Calakmul both developed extensive systems of allies and vassals; lesser cities that entered one of these networks gained prestige from their association with
20944-406: The intrusive intervention of the central Mexican city of Teotihuacan in Maya dynastic politics. In the 9th century, there was a widespread political collapse in the central Maya region, resulting in civil wars , the abandonment of cities, and a northward shift of population. The Postclassic period saw the rise of Chichen Itza in the north, and the expansion of the aggressive Kʼicheʼ kingdom in
21120-642: The last major innovator in Greek mathematics, with subsequent work consisting mostly of commentaries on earlier work. The first woman mathematician recorded by history was Hypatia of Alexandria (AD 350–415). She succeeded her father ( Theon of Alexandria ) as Librarian at the Great Library and wrote many works on applied mathematics. Because of a political dispute, the Christian community in Alexandria had her stripped publicly and executed. Her death
21296-417: The leap to coordinate geometry, Apollonius' treatment of curves is in some ways similar to the modern treatment, and some of his work seems to anticipate the development of analytical geometry by Descartes some 1800 years later. Around the same time, Eratosthenes of Cyrene ( c. 276 –194 BC) devised the Sieve of Eratosthenes for finding prime numbers . The 3rd century BC is generally regarded as
21472-407: The mathematical center of the world in the 4th century BC, and it was from this school that the leading mathematicians of the day, such as Eudoxus of Cnidus (c. 390 - c. 340 BC), came. Plato also discussed the foundations of mathematics, clarified some of the definitions (e.g. that of a line as "breadthless length"), and reorganized the assumptions. The analytic method is ascribed to Plato, while
21648-535: The mathematics known to these civilizations. Contemporaneous with but independent of these traditions were the mathematics developed by the Maya civilization of Mexico and Central America , where the concept of zero was given a standard symbol in Maya numerals . Many Greek and Arabic texts on mathematics were translated into Latin from the 12th century onward, leading to further development of mathematics in Medieval Europe . From ancient times through
21824-420: The mediator between the mortal realm and that of the gods. From very early times, kings were specifically identified with the young maize god , whose gift of maize was the basis of Mesoamerican civilization. Maya royal succession was patrilineal , and royal power only passed to queens when doing otherwise would result in the extinction of the dynasty. Typically, power was passed to the eldest son. A young prince
22000-487: The most important usually controlled access to vital trade goods, or portage routes. Cities such as Kaminaljuyu and Qʼumarkaj in the Guatemalan Highlands, and Chalchuapa in El Salvador, variously controlled access to the sources of obsidian at different points in Maya history. The Maya were major producers of cotton , which was used to make the textiles to be traded throughout Mesoamerica. The most important cities in
22176-428: The most part continued to manage their own affairs. Maya communities and the nuclear family maintained their traditional day-to-day life. The basic Mesoamerican diet of maize and beans continued, although agricultural output was improved by the introduction of steel tools. Traditional crafts such as weaving, ceramics, and basketry continued to be practised. Community markets and trade in local products continued long after
22352-482: The natural terrain. One of the most important cities in the Guatemalan Highlands at this time was Qʼumarkaj , the capital of the aggressive Kʼicheʼ kingdom . The government of Maya states, from the Yucatán to the Guatemalan highlands, was often organised as joint rule by a council. However, in practice one member of the council could act as a supreme ruler, while the other members served him as advisors. Mayapan
22528-502: The northern Yucatán Peninsula controlled access to the sources of salt. In the Postclassic, the Maya engaged in a flourishing slave trade with wider Mesoamerica. The Maya engaged in long-distance trade across the Maya region, and across greater Mesoamerica and beyond. As an illustration, an Early Classic Maya merchant quarter has been identified at the distant metropolis of Teotihuacan, in central Mexico. Within Mesoamerica beyond
22704-401: The peoples of Mesopotamia (modern Iraq ) from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity . The majority of Babylonian mathematical work comes from two widely separated periods: The first few hundred years of the second millennium BC (Old Babylonian period), and the last few centuries of the first millennium BC ( Seleucid period). It
22880-413: The period between 250 and 350 AD is sometimes referred to as the "Silver Age" of Greek mathematics. During this period, Diophantus made significant advances in algebra, particularly indeterminate analysis , which is also known as "Diophantine analysis". The study of Diophantine equations and Diophantine approximations is a significant area of research to this day. His main work was the Arithmetica ,
23056-506: The periphery abandoned the site soon after. This is an example of intensive warfare carried out by an enemy in order to eliminate a Maya state, rather than subjugate it. Research at Aguateca indicated that Classic period warriors were primarily members of the elite. From as early as the Preclassic period, the ruler of a Maya polity was expected to be a distinguished war leader, and was depicted with trophy heads hanging from his belt. In
23232-401: The permanent foundations of market stalls. A 2007 study compared soils from a modern Guatemalan market to a proposed ancient market at Chunchucmil ; unusually high levels of zinc and phosphorus at both sites indicated similar food production and vegetable sales activity. The calculated density of market stalls at Chunchucmil strongly suggests that a thriving market economy already existed in
23408-559: The polities of the Yucatán Peninsula in 1527, and finally completed the conquest of the northern portion of the peninsula in 1546. This left only the Maya kingdoms of the Petén Basin independent. In 1697, Martín de Ursúa launched an assault on the Itza capital Nojpetén and the last independent Maya city fell to the Spanish. The Spanish conquest stripped away most of the defining features of Maya civilization. However, many Maya villages remained remote from Spanish colonial authority, and for
23584-470: The population, but relatively little is known about them. Their houses were generally constructed from perishable materials, and their remains have left little trace in the archaeological record. Some commoner dwellings were raised on low platforms, and these can be identified, but an unknown quantity of commoner houses were not. Such low-status dwellings can only be detected by extensive remote-sensing surveys of apparently empty terrain. The range of commoners
23760-555: The priesthood, the warrior aristocracy, and other aristocratic courtiers. Where ruling councils existed, as at Chichen Itza and Copán, these may have formed an additional faction. Rivalry between different factions would have led to dynamic political institutions as compromises and disagreements were played out. In such a setting, public performance was vital. Such performances included ritual dances , presentation of war captives, offerings of tribute, human sacrifice, and religious ritual. Commoners are estimated to have comprised over 90% of
23936-403: The rest having been destroyed by the Spanish. In addition, a great many examples of Maya texts can be found on stelae and ceramics. The Maya developed a highly complex series of interlocking ritual calendars, and employed mathematics that included one of the earliest known instances of the explicit zero in human history. As a part of their religion, the Maya practised human sacrifice . "Maya"
24112-442: The royal court, was not bureaucratic in nature. Government was hierarchical, and official posts were sponsored by higher-ranking members of the aristocracy; officials tended to be promoted to higher levels of office over their lives. Officials are referred to as being "owned" by their sponsor, and this relationship continued even after the death of the sponsor. The Maya royal court was a vibrant and dynamic political institution. There
24288-444: The ruler's actions were limited by tradition to such activities as construction, ritual, and warfare. This only served to exacerbate systemic problems. By the 9th and 10th centuries, this resulted in collapse of this system of rulership. In the northern Yucatán, individual rule was replaced by a ruling council formed from elite lineages. In the southern Yucatán and central Petén, kingdoms declined; in western Petén and some other areas,
24464-472: The ruler's authority, and was reinforced by public display, ritual, and religion. The divine king was the centre of political power, exercising ultimate control over administrative, economic, judicial, and military functions. The divine authority invested within the ruler was such that the king was able to mobilize both the aristocracy and commoners in executing huge infrastructure projects, apparently with no police force or standing army. Some polities engaged in
24640-519: The rules of calculation used in astronomy and mathematical mensuration, though with no feeling for logic or deductive methodology. It is in the Aryabhatiya that the decimal place-value system first appears. Several centuries later, the Muslim mathematician Abu Rayhan Biruni described the Aryabhatiya as a "mix of common pebbles and costly crystals". In the 7th century, Brahmagupta identified
24816-496: The same title, and Mark Zender has suggested that the holder of this title may have been the spokesman for the ruler. Courtly titles are overwhelmingly male-oriented, and in those relatively rare occasions where they are applied to a woman, they appear to be used as honorifics for female royalty. Titled elites were often associated with particular structures in the hieroglyphic inscriptions of Classic period cities, indicating that such office holders either owned that structure, or that
24992-636: The south of the Sierra Madre de Chiapas, and consists of a narrow coastal plain and the foothills of the Sierra Madre. The Maya highlands extend eastwards from Chiapas into Guatemala, reaching their highest in the Sierra de los Cuchumatanes . Their major pre-Columbian population centres were in the largest highland valleys, such as the Valley of Guatemala and the Quetzaltenango Valley. In
25168-635: The southern highlands, a belt of volcanic cones runs parallel to the Pacific coast. The highlands extend northwards into Verapaz , and gradually descend to the east. The history of Maya civilization is divided into three principal periods: the Preclassic, Classic, and Postclassic. These were preceded by the Archaic Period, during which the first settled villages and early developments in agriculture emerged. Modern scholars regard these periods as arbitrary divisions of Maya chronology, rather than indicative of cultural evolution or decline. Definitions of
25344-407: The southern lowland regions. The Classic period Maya political landscape has been likened to that of Renaissance Italy or Classical Greece , with multiple city-states engaged in a complex network of alliances and enmities. The largest cities had 50,000 to 120,000 people and were linked to networks of subsidiary sites. During the Early Classic, cities throughout the Maya region were influenced by
25520-543: The southern lowlands, because many Postclassic Maya groups had migration myths. Chichen Itza and its Puuc neighbours declined dramatically in the 11th century, and this may represent the final episode of Classic Period collapse. After the decline of Chichen Itza, the Maya region lacked a dominant power until the rise of the city of Mayapan in the 12th century. New cities arose near the Caribbean and Gulf coasts, and new trade networks were formed. The Postclassic Period
25696-400: The start and end dates of period spans can vary by as much as a century, depending on the author. The Maya developed their first civilization in the Preclassic period. Scholars continue to discuss when this era of Maya civilization began. Maya occupation at Cuello (modern Belize) has been carbon dated to around 2600 BC. Settlements were established around 1800 BC in the Soconusco region of
25872-446: The structure was an important focus for their activities. A lakam , or standard-bearer, was possibly the only non-elite post-holder in the royal court. The lakam was only found in larger sites, and they appear to have been responsible for the taxation of local districts. Different factions may have existed in the royal court. The kʼuhul ahaw and his household would have formed the central power-base, but other important groups were
26048-412: The surface area and volume of a sphere, which he obtained by proving these are 2/3 the surface area and volume of a cylinder circumscribing the sphere. Apollonius of Perga ( c. 262 –190 BC) made significant advances to the study of conic sections , showing that one can obtain all three varieties of conic section by varying the angle of the plane that cuts a double-napped cone. He also coined
26224-417: The symbol for "100", then the symbol for "2" followed by the symbol for "10", followed by the symbol for "3". This was the most advanced number system in the world at the time, apparently in use several centuries before the common era and well before the development of the Indian numeral system. Rod numerals allowed the representation of numbers as large as desired and allowed calculations to be carried out on
26400-499: The terminology in use today for conic sections, namely parabola ("place beside" or "comparison"), "ellipse" ("deficiency"), and "hyperbola" ("a throw beyond"). His work Conics is one of the best known and preserved mathematical works from antiquity, and in it he derives many theorems concerning conic sections that would prove invaluable to later mathematicians and astronomers studying planetary motion, such as Isaac Newton. While neither Apollonius nor any other Greek mathematicians made
26576-502: The territory now in the modern countries of Guatemala and Belize, as well as the western portions of Honduras and El Salvador. Most of the peninsula is formed by a vast plain with few hills or mountains and a generally low coastline. The territory of the Maya covered a third of Mesoamerica , and the Maya were engaged in a dynamic relationship with neighbouring cultures that included the Olmecs , Mixtecs , Teotihuacan, and Aztecs . During
26752-437: The then-abandoned city of Mayapán . The term "Maya" was derived from the city of Mayapán. Some colonial Mayan-language sources also used "Maya" to refer to other Maya groups, sometimes pejoratively in reference to Maya groups more resistant to Spanish rule. The Maya civilization occupied a wide territory that included southeastern Mexico and northern Central America. This area included the entire Yucatán Peninsula and all of
26928-477: The throne of Dos Pilas as his vassal. He thereafter served as a loyal ally of Calakmul. In the southeast, Copán was the most important city. Its Classic-period dynasty was founded in 426 by Kʼinich Yax Kʼukʼ Moʼ . The new king had strong ties with central Petén and Teotihuacan. Copán reached the height of its cultural and artistic development during the rule of Uaxaclajuun Ubʼaah Kʼawiil , who ruled from 695 to 738. His reign ended catastrophically when he
27104-510: The time. Tablets from the Old Babylonian period also contain the earliest known statement of the Pythagorean theorem . However, as with Egyptian mathematics, Babylonian mathematics shows no awareness of the difference between exact and approximate solutions, or the solvability of a problem, and most importantly, no explicit statement of the need for proofs or logical principles. Egyptian mathematics refers to mathematics written in
27280-406: The top-tier city, and maintained peaceful relations with members of the network. Tikal and Calakmul engaged in the manoeuvering of their alliance networks against each other. At various points during the Classic period, one or other of these powers would gain a strategic victory over its great rival, resulting in respective periods of florescence and decline. In 629, Bʼalaj Chan Kʼawiil , a son of
27456-488: The transition to the Classic, and the Terminal Classic collapse. Even the Spanish Conquest did not immediately terminate all Maya trading activity; for example, the Contact period Manche Chʼol traded the prestige crops of cacao, annatto and vanilla into colonial Verapaz. Little is known of Maya merchants, although they are depicted on Maya ceramics in elaborate noble dress, so at least some were members of
27632-423: The use of inductive reasoning , that is, repeated observations used to establish rules of thumb. Greek mathematicians, by contrast, used deductive reasoning . The Greeks used logic to derive conclusions from definitions and axioms, and used mathematical rigor to prove them. Greek mathematics is thought to have begun with Thales of Miletus (c. 624–c.546 BC) and Pythagoras of Samos (c. 582–c. 507 BC). Although
27808-470: The victor. During the Contact period, certain military positions were held by members of the aristocracy, and were passed on by patrilineal succession. It is likely that the specialised knowledge inherent in the particular military role was taught to the successor, including strategy, ritual, and war dances. Maya armies of the Contact period were highly disciplined, and warriors participated in regular training exercises and drills; every able-bodied adult male
27984-649: The volume of a sphere . The high-water mark of Chinese mathematics occurred in the 13th century during the latter half of the Song dynasty (960–1279), with the development of Chinese algebra. The most important text from that period is the Precious Mirror of the Four Elements by Zhu Shijie (1249–1314), dealing with the solution of simultaneous higher order algebraic equations using a method similar to Horner's method . The Precious Mirror also contains
28160-642: The world, leading scholars to assume an entirely independent development. The oldest extant mathematical text from China is the Zhoubi Suanjing (周髀算經), variously dated to between 1200 BC and 100 BC, though a date of about 300 BC during the Warring States Period appears reasonable. However, the Tsinghua Bamboo Slips , containing the earliest known decimal multiplication table (although ancient Babylonians had ones with
28336-423: Was a Mesoamerican civilization that existed from antiquity to the early modern period . It is known by its ancient temples and glyphs (script). The Maya script is the most sophisticated and highly developed writing system in the pre-Columbian Americas . The civilization is also noted for its art , architecture , mathematics , calendar , and astronomical system . The Maya civilization developed in
28512-532: Was a royal scribe, usually a member of the royal family; the ah chʼul hun was the Keeper of the Holy Books, a title that is closely associated with the ajaw title, indicating that an ajaw always held the ah chʼul hun title simultaneously. Other courtly titles, the functions of which are not well understood, were yajaw kʼahk' ("Lord of Fire"), tiʼhuun and ti'sakhuun . These last two may be variations on
28688-406: Was abandoned around 1448, after a period of political, social and environmental turbulence that in many ways echoed the Classic period collapse in the southern Maya region . The abandonment of the city was followed by a period of prolonged warfare, disease and natural disasters in the Yucatán Peninsula, which ended only shortly before Spanish contact in 1511. Even without a dominant regional capital,
28864-415: Was already a significant city by around 350 BC. In the highlands, Kaminaljuyu emerged as a principal centre in the Late Preclassic. Takalik Abaj and Chocolá were two of the most important cities on the Pacific coastal plain, and Komchen grew to become an important site in northern Yucatán. The Late Preclassic cultural florescence collapsed in the 1st century AD and many of the great Maya cities of
29040-541: Was already a sprawling city by 300. In the north of the Maya area, Coba was the most important capital. During the 9th century AD, the central Maya region suffered major political collapse, marked by the abandonment of cities, the ending of dynasties, and a northward shift in activity. No universally accepted theory explains this collapse, but it likely had a combination of causes, including endemic internecine warfare, overpopulation resulting in severe environmental degradation , and drought . During this period, known as
29216-425: Was available for military service. Maya states did not maintain standing armies; warriors were mustered by local officials who reported back to appointed warleaders. There were also units of full-time mercenaries who followed permanent leaders. Most warriors were not full-time, however, and were primarily farmers; the needs of their crops usually came before warfare. Maya warfare was not so much aimed at destruction of
29392-563: Was broad; it consisted of everyone not of noble birth, and therefore included everyone from the poorest farmers to wealthy craftsmen and commoners appointed to bureaucratic positions. Commoners engaged in essential production activities, including that of products destined for use by the elite, such as cotton and cacao , as well as subsistence crops for their own use, and utilitarian items such as ceramics and stone tools. Commoners took part in warfare, and could advance socially by proving themselves as outstanding warriors. Commoners paid taxes to
29568-455: Was called a chʼok ("youth"), although this word later came to refer to nobility in general. The royal heir was called bʼaah chʼok ("head youth"). Various points in the prince's childhood were marked by ritual; the most important was a bloodletting ceremony at age five or six. Although being of the royal bloodline was of utmost importance, the heir also had to be a successful war leader, as demonstrated by taking of captives. The enthronement of
29744-471: Was captured by his vassal, king Kʼakʼ Tiliw Chan Yopaat of Quiriguá . The captured lord of Copán was taken back to Quiriguá and was decapitated in a public ritual. It is likely that this coup was backed by Calakmul, in order to weaken a powerful ally of Tikal. Palenque and Yaxchilan were the most powerful cities in the Usumacinta region. In the highlands, Kaminaljuyu in the Valley of Guatemala
29920-408: Was held communally by noble houses or clans . Such clans held that the land was the property of the ancestors, and ties between the land and the ancestors were reinforced by the burial of the dead within residential compounds. Classic Maya rule was centred in a royal culture that was displayed in all areas of Classic Maya art. The king was the supreme ruler and held a semi-divine status that made him
30096-489: Was marked by changes from the preceding Classic Period. The once-great city of Kaminaljuyu in the Valley of Guatemala was abandoned after continuous occupation of almost 2,000 years. Across the highlands and neighbouring Pacific coast, long-occupied cities in exposed locations were relocated, apparently due to a proliferation of warfare . Cities came to occupy more-easily defended hilltop locations surrounded by deep ravines, with ditch-and-wall defences sometimes supplementing
30272-404: Was no universal structure for the Maya royal court, instead each polity formed a royal court that was suited to its own individual context. A number of royal and noble titles have been identified by epigraphers translating Classic Maya inscriptions. Ajaw is usually translated as "lord" or "king". In the Early Classic, an ajaw was the ruler of a city. Later, with increasing social complexity,
30448-493: Was ranked below the ajaw , and indicated a subservient lord. A sajal would be lord of a second- or third-tier site, answering to an ajaw , who may himself have been subservient to a kalomte . A sajal would often be a war captain or regional governor, and inscriptions often link the sajal title to warfare; they are often mentioned as the holders of war captives. Sajal meant "feared one". The titles of ah tzʼihb and ah chʼul hun are both related to scribes. The ah tzʼihb
30624-603: Was sharply divided between the elite and commoners. As population increased over time, various sectors of society became increasingly specialised, and political organization increasingly complex. By the Late Classic, when populations had grown enormously and hundreds of cities were connected in a complex web of political hierarchies, the wealthy segment of society multiplied. A middle class may have developed that included artisans, low ranking priests and officials, merchants, and soldiers. Commoners included farmers, servants, labourers, and slaves. According to indigenous histories, land
30800-496: Was used as currency (although not exclusively), and its value was such that counterfeiting occurred by removing the flesh from the pod, and stuffing it with dirt or avocado rind. Marketplaces are difficult to identify archaeologically. However, the Spanish reported a thriving market economy when they arrived in the region. At some Classic period cities, archaeologists have tentatively identified formal arcade-style masonry architecture and parallel alignments of scattered stones as
30976-417: Was used by the ancient Maya for both war and hunting. Although present in the Maya region during the Classic period, its use as a weapon of war was not favoured; it did not become a common weapon until the Postclassic. The Contact period Maya also used two-handed swords crafted from strong wood with the blade fashioned from inset obsidian, similar to the Aztec macuahuitl . Maya warriors wore body armour in
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