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Inca Empire Tawantinsuyu (Quechua)
← 1438–1533 →   →
Banner of the Tawantinsuyu
The Inca Empire at its greatest extent
Capital	Cusco (1438–1533)
Language(s)	Quechua (official), Aymara, Puquina, Jaqi family, Muchik and scores of smaller languages.
Religion	Inca religion
Government	Monarchy
Sapa Inca
- 1438–1471	Pachacuti
- 1471–1493	Túpac Inca Yupanqui
- 1493–1525	Huayna Capac
- 1525–1532	Huáscar
- 1532–1533	Atahualpa
Historical era	Pre-Columbian
- Pachacuti created the Tawantinsuyu	1438
- Civil war between Huáscar and Atahualpa	1529–1532
- Spanish conquest led by Francisco Pizarro	1533
- End of the last Inca resistance	1572
Area
- 1438[1]	800,000 km2 (308,882 sq mi)
- 1527	2,000,000 km2 (772,204 sq mi)
Population
- 1438[1] est.	12,000,000
Density	15 /km2  (38.8 /sq mi)
- 1527 est.	20,000,000
Density	10 /km2  (25.9 /sq mi)
Today part of	Argentina  Bolivia  Chile  Colombia  Ecuador  Peru

Inca Empire
Inca society
Education · Religion · Mythology
Architecture · Road system · Army
Agriculture · Andean cuisine
Inca history
Spanish conquest of the Inca Empire
v t e

The Inca Empire, or Inka Empire^[2] (Quechua: Tawantinsuyu), was the largest empire in pre-Columbian America.^[3] The administrative, political and military center of the empire was located in Cusco in modern-day Peru. The Inca civilization arose from the highlands of Peru sometime in the early 13th century.

From 1438 to 1533, the Incas used a variety of methods, from conquest to peaceful assimilation, to incorporate a large portion of western South America, centered on the Andean mountain ranges, including, besides Peru, large parts of modern Ecuador, western and south central Bolivia, northwest Argentina, north and central Chile, and southern Colombia into a state comparable to the historical empires of Eurasia.

The official language of the empire was Quechua, although hundreds of local languages and dialects of Quechua were spoken. The Inca referred to their empire as Tawantinsuyu^[4] which can be translated as The Four Regions or The Four United Provinces.

There were many local forms of worship, most of them concerning local sacred "Huacas", but the Inca leadership encouraged the worship of Inti—the sun god—and imposed its sovereignty above other cults such as that of Pachamama.^[5] The Incas considered their King, the Sapa Inca, to be the "child of the sun."^[6]

Name[link]

The Inca referred to their empire as Tawantinsuyu, "four parts together."^[7] In Quechua the term Tawantin is a group of four things (tawa "four" with the suffix -ntin which names a group). Suyu means "region" or "province". The empire was divided into four suyus, whose corners met at the capital, Cusco (Qosqo). The name Tawantinsuyu was, therefore, a descriptive term indicating a union of provinces. The Spanish transliterated the name as Tahuatinsuyo or Tahuatinsuyu which is often still used today.

The term Inka means ruler, or lord, in Quechua, and was used to refer to the ruling class or the ruling family in the empire.^[8] The Spanish adopted the term (transliterated as Inca in Spanish) as an ethnic term referring to all subjects of the empire rather than simply the ruling class. As such the name Imperio inca (Inca Empire) referred to the nation that they encountered, and subsequently conquered.

History[link]

Origin[link]

Incan oral history mentions three possible places of origin of their people. Three caves near Pacariqtambo (a place 33 km (21 mi) away from Cuzco), Lake Titicaca, or a place known as Tambo.^[9]

Pacariqtambo, which means "the dawn tavern" or "the place of origin", was a place of three caves. The center cave, Tambo Tocco, was named for Capac Tocco. The other caves were Maras Tocco and Sutic Tocco.^[9] Four brothers and four sisters stepped out of the middle cave. They were: Ayar Manco, Ayar Cachi, Ayar Auca, Ayar Uchu, and Mama Ocllo, Mama Raua, Mama Huaca, Mama Cora. Out of the side caves came the people who were to be the ancestors of all the clans of the Inca people.

Ayar Manco carried a staff made of the finest gold. Where this staff landed, the people would all live there. They travelled for a very, very long time. On the way, Ayar Cachi went too far boasting about his great strength and power, and his siblings tricked him into returning to the cave to get a sacred llama. When he went into the cave, they trapped him inside.

Ayar Uchu decided to stay somewhere on the top to look over the Incan people. The minute he proclaimed that, he turned to stone. They built a shrine around the stone and it became a sacred object. Ayar Auca also grew tired of all these and decide to travel alone. Only Ayar Manco and his four sisters remained.

Finally, they reached Cuzco. The staff sank into the ground. Before they reached here, Mama Ocllo had already bore Ayar Manco a child, Sinchi Roca. The people who were already living in the valley fought hard to keep their land, but Mama Huaca was a good fighter. When the enemy attacked, she threw her bolas—several stones tied together that spun through the air when thrown—at a soldier, or gualla, and killed him instantly. Mama Huaca ripped out his lungs and squeezed them until they exploded. The other people were so scared, they ran away.

After that, Ayar Manco became known as Manco Cápac, the founder of the Inca. It is said that he and his sisters built the first Inca homes in the valley with their own hands. When the time came, Manco Cápac turned to stone like his brothers before him. His son, Sinchi Roca, became the second emperor of the Inca.^[10]

Kingdom of Cuzco[link]

The Inca people were a pastoral tribe in the Cuzco area around the 12th century. Under the leadership of Manco Cápac, they formed the small city-state Kingdom of Cuzco (Quechua Qusqu'Qosqo). In 1438, they began a far-reaching expansion under the command of Sapa Inca (paramount leader) Pachacuti-Cusi Yupanqui, whose name literally meant "earth-shaker". The name of Pachacuti was given to him after conquering over the Tribe of Chancas (modern Apurímac). During his reign, he and his son Tupac Yupanqui brought much of the Andes mountains (roughly modern Peru and Ecuador) under Inca control.^[11]

Reorganization and formation[link]

The first image of the Inca in Europe. Pedro Cieza de Leon. Cronica del Peru, 1553.

Pachacuti reorganized the kingdom of Cusco into the Tahuantinsuyu, which consisted of a central government with the Inca at its head and four provincial governments with strong leaders: Chinchasuyu (NW), Antisuyu (NE), Kuntisuyu (SW), and Qullasuyu (SE).^[12] Pachacuti is also thought to have built Machu Picchu, either as a family home or as a summer retreat, although there is speculation that Machu Picchu was constructed as an agricultural station.^[13]

Pachacuti sent spies to regions he wanted in his empire; they brought reports on the political organization, military might and wealth. He would then send messages to the leaders of these lands extolling the benefits of joining his empire, offering them presents of luxury goods such as high quality textiles, and promising that they would be materially richer as subject rulers of the Inca.

Most accepted the rule of the Inca as a fait accompli and acquiesced peacefully. The ruler's children would then be brought to Cusco to be taught about Inca administration systems, then return to rule their native lands. This allowed the Inca to indoctrinate the former ruler's children into the Inca nobility, and, with luck, marry their daughters into families at various corners of the empire.

Expansion and consolidation[link]

Manco Cápac and Mama Ocllo, sons of the Inti.

It was traditional for the Inca's son to lead the army; Pachacuti's son Túpac Inca Yupanqui began conquests to the north in 1463, and continued them as Inca after Pachucuti's death in 1471. His most important conquest was the Kingdom of Chimor, the Inca's only serious rival for the coast of Peru. Túpac Inca's empire stretched north into modern day Ecuador and Colombia.

Túpac Inca's son Huayna Cápac added a small portion of land to the north in modern day Ecuador and in parts of Peru. At its height, the Inca Empire included Peru and Bolivia, most of what is now Ecuador, a large portion of what is today Chile north of the Maule River in central Chile. The advance south halted after the Battle of the Maule where they met determined resistance by the Mapuche tribes. The empire also extended into corners of Argentina and Colombia. However, most of the southern portion of the Inca empire, the portion denominated as Qullasuyu, was located in the Altiplano.

The Inca Empire was a patchwork of languages, cultures and peoples. The components of the empire were not all uniformly loyal, nor were the local cultures all fully integrated. The Inca empire as a whole had an economy based on exchange and taxation of luxury goods and labour. The following quote reflects a method of taxation:

"For as is well known to all, not a single village of the highlands or the plains failed to pay the tribute levied on it by those who were in charge of these matters. There were even provinces where, when the natives alleged that they were unable to pay their tribute, the Inca ordered that each inhabitant should be obliged to turn in every four months a large quill full of live lice, which was the Inca's way of teaching and accustoming them to pay tribute".^[14]

Inca civil war and Spanish conquest[link]

One of the main events in the conquest of the Incan Empire was the death of Atahualpa, the last Sapa Inca on 29 August 1533

Spanish conquistadors led by Francisco Pizarro and his brothers explored south from what is today Panama, reaching Inca territory by 1526.^[15] It was clear that they had reached a wealthy land with prospects of great treasure, and after one more expedition in 1529, Pizarro traveled to Spain and received royal approval to conquer the region and be its viceroy. This approval was received as detailed in the following quote: "In July 1529 the queen of Spain signed a charter allowing Pizarro to conquer the Incas. Pizarro was named governor and captain of all conquests in Peru, or New Castile, as the Spanish now called the land."^[16]

When they returned to Peru in 1532, a war of the two brothers between Huayna Capac's sons Huáscar and Atahualpa and unrest among newly conquered territories—and perhaps more importantly, smallpox, which had spread from Central America—had considerably weakened the empire. Pizarro did not have a formidable force; with just 168 men, 1 cannon and 27 horses, he often needed to talk his way out of potential confrontations that could have easily wiped out his party.

The Spanish horsemen, fully armored, had great technological superiority over the Inca forces. The traditional mode of battle in the Andes was a kind of siege warfare where large numbers of usually reluctant draftees were sent to overwhelm opponents. The Spaniards had developed one of the finest military machines in the premodern world, tactics learned in their centuries' long fight against Moorish kingdoms in Iberia. Along with this tactical and material superiority, the Spaniards also had acquired tens of thousands of native allies who sought to end the Inca control of their territories.

Inca expansion (1438–1533)

Their first engagement was the Battle of Puná, near present-day Guayaquil, Ecuador, on the Pacific Coast; Pizarro then founded the city of Piura in July 1532. Hernando de Soto was sent inland to explore the interior and returned with an invitation to meet the Inca, Atahualpa, who had defeated his brother in the civil war and was resting at Cajamarca with his army of 80,000 troops.

Pizarro and some of his men, most notably a friar named Vincente de Valverde, met with the Inca, who had brought only a small retinue. Through an interpreter Friar Vincente read the "Requerimiento" that demanded that he and his empire accept the yoke of King Charles I of Spain and convert to Christianity. Because of the language barrier and perhaps poor interpretation, Atahualpa became somewhat puzzled by the friar's description of Christian faith and was said to have not fully understood the envoy's intentions. After Atahualpa attempted further enquiry into the doctrines of the Christian faith under which Pizarro's envoy served, the Spanish became frustrated and impatient, attacking the Inca's retinue and capturing Atahualpa as hostage.

Atahualpa offered the Spaniards enough gold to fill the room he was imprisoned in, and twice that amount of silver. The Inca fulfilled this ransom, but Pizarro deceived them, refusing to release the Inca afterwards. During Atahualpa's imprisonment Huáscar was assassinated elsewhere. The Spaniards maintained that this was at Atahualpa's orders; this was used as one of the charges against Atahualpa when the Spaniards finally decided to put him to death, in August 1533.^[17]

Last Incas[link]

A view of Machu Picchu.

The Spanish installed Atahualpa's brother Manco Inca Yupanqui in power; for some time Manco cooperated with the Spanish, while the Spanish fought to put down resistance in the north. Meanwhile an associate of Pizarro's, Diego de Almagro, attempted to claim Cusco for himself. Manco tried to use this intra-Spanish feud to his advantage, recapturing Cusco in 1536, but the Spanish retook the city afterwards. Manco Inca then retreated to the mountains of Vilcabamba, Peru, where he and his successors ruled for another 36 years, sometimes raiding the Spanish or inciting revolts against them. In 1572 the last Inca stronghold was conquered, and the last ruler, Túpac Amaru, Manco's son, was captured and executed.^[18] This ended resistance to the Spanish conquest under the political authority of the Inca state.

After the fall of the Inca Empire many aspects of Inca culture were systematically destroyed, including their sophisticated farming system, known as the vertical archipelago model of agriculture.^[19] Spanish colonial officials used the Inca mita corvée labor system for colonial aims, sometimes brutally. One member of each family was forced to work in the gold and silver mines, the foremost of which was the titanic silver mine at Potosí. When a family member died, which would usually happen within a year or two, the family would be required to send a replacement.^{[citation needed]}

The effects of smallpox on the Inca empire were even more devastating. Beginning in Colombia, smallpox spread rapidly before the Spanish invaders first arrived in the empire. The spread was probably aided by the efficient Inca road system. Within a few years smallpox claimed between 60% and 94% of the Inca population,^{[citation needed]} with other waves of European disease weakening them further. Smallpox was only the first epidemic.^[20] Typhus (probably) in 1546, influenza and smallpox together in 1558, smallpox again in 1589, diphtheria in 1614, measles in 1618 – all ravaged the remains of Inca culture.

Society[link]

Population[link]

There is some debate about the number of people inhabiting Tawantinsuyu at its peak, with estimates ranging from as few as 4 million people, to more than 37 million. The reason for these various estimates is that in spite of the fact that the Inca kept excellent census records using their quipu, knowledge of how to read them has been lost, and almost all of them had been destroyed by the Spaniards in the course of their conquest.^[21]

Language[link]

Since the Inca Empire lacked a written language, the empire's main form of communication and recording came from quipus, ceramics and spoken Quechua, the language the Incas imposed upon the peoples within the empire. The plethora of civilizations in the Andean region provided for a general disunity that the Incas needed to subdue in order to maintain control of the empire. While Quechua had been spoken in the Andean region, like central Peru, for several years prior to the expansion of the Inca civilization, the type of Quechua the Incas imposed was an adaptation from the Kingdom of Cusco (an early form of "Southern Quechua" originally named Qhapaq Runasimi = The great language of the people) of what some historians define as "Proto-Quechua" or Cusco dialect (the original Quechua dialect).^[22][23]

The language imposed by the Incas further diverted from its original phonetic tone as some societies formed their own regional varieties, or slang. The diversity of Quechua at that point and even today does not come as a direct result from the Incas, who are just a part of the reason for Quechua's diversity. The civilizations within the empire that had previously spoken Quechua kept their own variety distinct to the Quechua the Incas spread. Although these dialects of Quechua have a similar linguistic structure, they differ according to the region in which they are spoken.^[23]

Although most of the societies within the empire implemented Quechua into their lives, the Incas allowed several societies to keep their old languages such as Aymara, which still remains a spoken language in contemporary Bolivia where it is the primary indigenous language and various regions of South America surrounding Bolivia. The linguistic body of the Inca Empire was thus largely varied, but it still remains quite an achievement for the Incas that went beyond their time as the Spanish continued the use of Quechua.^[23]

Religion[link]

Incan myths were an oral tradition until early Spanish colonists recorded them; however, some scholars believe that they may have been recorded on quipus, Andean knotted string records.^[24]

The Inca believed in reincarnation.^[25] Death was a passage to the next world that was full of difficulties. The spirit of the dead (camaquen) would need to follow a long dark road and during the trip the assistance of black dog that was able to see in the dark was required. Most Incas imagined the after world to be very similar to ours with flower covered fields and snow capped mountains. It was important for the Inca to ensure they did not die as a result of burning or that the body of the deceased did not become incinerated. This is because of the underlying belief that a vital force would disappear and threaten their passage to the after world. Those who obeyed the Incan moral code—ama suwa, ama llulla, ama quella (do not steal, do not lie, do not be lazy) —"went to live in the Sun's warmth while others spent their eternal days in the cold earth"^{[citation needed]}. The Inca also practiced cranial deformation.^[26] They achieved this by wrapping tight cloth straps around the heads of newborns in order to alter the shape of their soft skulls into a more conical form; this cranial deformation was made to distinguish social classes of the communities, with only the nobility having cranial deformation.

Deities[link]

• Viracocha (also Pachacamac) – Created all living things
• Apu Illapu – Rain God, prayed to when they need rain
• Ayar Cachi – Hot-tempered God, causes earthquakes
• Illapa – Goddess of lightning and thunder (also Yakumama water goddess)
• Inti – sun god and patron deity of the holy city of Cuzco (home of the sun)
• Kuychi – Rainbow God, connected with fertility
• Mama Kilya – Wife of Inti, called Moon Mother
• Mama Occlo – Wisdom to civilize the people, taught women to weave cloth, and build houses
• Manco Cápac – known for his courage and sent to earth to become first king of the Incas, taught people how to grow plants, make weapons, work together, share resources, and worship the Gods
• Pachamama – The Goddess of earth and wife of Viracocha, people give her offerings of coca leafs and beer and pray to her for major agricultural occasions
• Qochamama – Goddess of the sea
• Sachamama – Means Mother Tree, goddess in the shape of a snake with two heads
• Yakumama – Means mother Water, represented as a snake, when she came to earth she transformed into a great river (also Illapa)

Economy[link]

The economy of the Inca Empire has been characterized as involving a high degree of central planning. While evidence of trade between the Inca Empire and outside regions has been uncovered, there is no evidence that the Incas had an internal market economy or currency. Individuals were required to pay taxes to the state in the form of a certain amount of labor. In return, the state provided basic necessities.^[27]

Social structure[link]

The social structure of the Inca Empire varied from area to area, but still had the same basic structure, On top was the Sapa Inca, or the emperor. Then came the nobles, these were often the priests and relatives of past emperors or the current ones. After, there were craftsmen and architects, they were very high on the social ladder because of the skill that they had was required by the Empire for such buildings. then came the working class, often just farmers that were kept in their social groupings. After this, were the slaves and peasants of the society.^{[citation needed]}

The Inca State[link]

Philosophy and Ideology[link]

The Sapa Inca was conceptualized as divine and was effectively head of the state religion. Only the Willaq-Umu (or Chief Priest) was second the emperor. Local religious traditions were allowed to continue, and in some cases such as the Oracle at Pachacamac on the Peruvian coast, were officially venerated. Following Pachacuti, the Sapa Inca claimed decent from Inti, which placed a high value on imperial blood; by the end of the empire, it was common to wed brother and sister. He was “son of the sun,” and his people the intip churin, or “children of the sun,” and both his right to rule and mission to conquer derived from his holy ancestor. The Sapa Inca also presided over ideologically-important festivals, notably during the Inti Raymi, or “warriors’ cultivation,” attended by soldiers, mummified rulers, nobles, clerics, and the general population of Cuzco beginning on the auspicious June solstice and culminating nine days later with the ritual breaking of the earth using a foot plow by the Inca himself. Moreover, Cuzco itself was considered cosmologically central, loaded as it was with huacas and radiating ceque lines, and geographic center of the Four Quarters; Inca Garcilaso de la Vega himself called it “the navel of the universe.”^{[28][29][30][31]}

Organization of the empire[link]

The four suyus of the empire.

The Inca Empire was a federalist system which consisted of a central government with the Inca at its head and four quarters, or suyu: Chinchay Suyu (NW), Anti Suyu (NE), Kunti Suyu (SW), and Qulla Suyu (SE). The four corners of these quarters met at the center, Cusco. These suyu were likely created around 1460 during the reign of Pachacuti before the empire assumed it largest territorial extent. It is probably the case that at the time the suyu were established they were roughly of equal size and only later changing their proportions as the empire expanded north and south along the Andes. ^[32]

The capital area, Cusco, was likely not organized as a wamani, or province. Rather, it was probably somewhat akin a modern federal district, like Washington, D.C. or Mexico City. The city sat at the center of the four suyu and served as the preeminent center of politics and religion. While Cuzco was essentially governed by the Sapa Inca, his relatives, and the royal panaqa lineages, each suyu was governed by an Apu, a term of great esteem used for men of very high status and for venerated mountains. Just as with so much of Andean society and Inca administration, both Cuzco as a district and the four suyu as administrative regions were grouped into upper hanan and lower hurin divisions. As the Inca did not have written records, it is impossible to exhaustively list the constituent wamani. However, records created during the Spanish colonial period allow us to reconstruct a partial list. There were likely more than 86 wamani, with more than 48 in the highlands and more than 38 on the coast.^[33]

The most populous suyu, Chinchaysuyu encompassed the former lands of the Chimu empire and much of the northern Andes. At its largest extent, the suyu extended through much of modern Ecuador and just into modern Colombia. The second smallest of the suyu, Antisuyu was located northwest of Cuzco in high Andes. Indeed, it is the root of the word “Andes.”^[34] Collasuyu or Qollasuyu was named after the Aymara-speaking Qolla people and was the largest of the quarters in terms of area. This suyu encompassed the Bolivian Altiplano and much of the southern Andes, running down into Argentina and as far south as the Maule river near modern Santiago, Chile. Cuntisuyu or Kuntisuyu was the smallest suyu of all was located along the southern coast of modern Peru, extending into the highlands towards Cuzco.^[35]

Laws[link]

Inti as represented by José Bernardo de Tagle of Peru.

The Inca state had no separate judiciary or codified set of laws. While customs, expectations, and traditional local power holders did much in the way of governing behavior, the state, too, had legal force, such as through tokoyrikoq (lit. "he who sees all"), or inspectors. The highest such inspector, typically a blood relation to the Sapa Inca, acted independently of the conventional hierarchy, providing a point of view for the Sapa Inca free of bureaucratic influence.^[36]

Administration[link]

The colonial-era sources are not entirely clear or in agreement about the nature of the structure of the Inca government. However, there are basic structure can be spoken of broadly, even if the exactly duties and functions of government positions cannot be told. At the top of the chain of administration sat the Sapa Inca. Next to the Sapa Inca in terms of power may have been the Willaq Umu, literally the "priest who recounts", who was the High Priest of the Sun.^[37] However, it has been noted that beneath the Sapa Inca also sat the Inkap rantin, who was at the very least a confidant and assistant to the Sapa Inca, perhaps along the lines of a Prime Minister.^[38] From the time of Topa Inca Yupanqui on, there existed a "Council of the Realm" composed of sixteen nobles: two from hanan Cuzco; two from hurin Cuzco; four from Chinchaysuyu; two from Cuntisuyu; four from Collasuyu; and two from Antisuyu. This weighting of representation balanced the hanan and hurin divisions of the empire, both within Cuzco and within the Quarters (hanan suyukuna and hurin suyukuna).^[39]

While there was a great deal of variation in the form that Inca bureaucracy and government took at the provincial level, the basic (perhaps, ideal) form of organization was decimal. In this system of organization, taxpayers-- male heads of household of a certain age range-- were organized into corvée labor units (which often doubled as military units) that formed the muscle of the state as part of mit'a service. Each level of jurisdiction above one hundred tax-payers was headed by a kuraka, while those heading smaller units were kamayuq, a lower, non-hereditary status. However, while kuraka status was hereditary, one's actual position within the hierarchy (which was typically served for life) was subject to change based upon the privileges of those above them in the hierarchy; a pachaka kuraka (see below) could be appointed to their position by a waranqa kuraka. Furthermore, it has been suggested that one kuraka in each decimal level also served as the head of one of the nine groups at a lower level, so that one pachaka kuraka might also be a waranqa kuraka, in effect directly responsible for one unit of 100 tax-payers and less directly responsible for nine other such units.^[40]

^[41]

Arts and technology[link]

Monumental architecture[link]

Architecture was by far the most important of the Inca arts, with textiles reflecting motifs that were at their height in architecture. The main example is the capital city of Cusco. The site of Machu Picchu was constructed by Inca engineers. The stone temples constructed by the Inca used a mortarless construction that fit together so well that a knife could not be fitted through the stonework.

This was a process first used on a large scale by the Pucara (ca. 300 BC–AD 300) peoples to the south in Lake Titicaca, and later in the great city of Tiwanaku (ca. AD 400–1100) in present day Bolivia. The rocks used in construction were sculpted to fit together exactly by repeatedly lowering a rock onto another and carving away any sections on the lower rock where the dust was compressed. The tight fit and the concavity on the lower rocks made them extraordinarily stable.

Measures, Calendrics, and Mathematics[link]

Inca tunic

Tokapu. Textiles worn by the Inca elite consisting of geometric figures enclosed by rectangles or squares. There is evidence that the designs were an ideographic language

Physical measures employed by the Inca were based upon human body parts. Fingers, the distance between thumb to forginger, palms, cubits, and wingspans were among those units used. The most basic unit of distance was thatkiy or thatki, or one pace. The next largest unit was reported by Cobo to be the topo or tupu, measuring 6,000 thatkiys, or about 4.8 miles (7.7 km); careful study has shown that a range of 2.5–3.9 miles (4.0–6.3 km) is likely. Next was the wamani, composed of 30 topos (roughly 144 miles (232 km)). To measure area, 25 by 50 wingspans were used, reckoned in topos (roughly 1,266 square miles (3,280 km²)). It seems likely as well that distance was often conceptualized as being one day's walk; the distance between tambo way-stations varies widely in terms of distance, but in far less so in terms of time to walk that distance.^[42][43]

Inca calendrics were strongly tied to astronomy. Inca astronomers understood equinoxes, solstices, and likely zenith passages, not to mention the Venus cycle. They could not, however, predict eclipses. The Inca calendar was essentially lunisolar, as two calendars were maintained in parallel, one solar and one lunar. As twelve lunar months fall 11-days short of a full 365-day solar year, those in charge of the calendar had to adjust every winter solstice. The twelve lunar months were each marked with specific festivals and rituals.^[44] There apparently were no names for days of the week, and it may be the case that there were no subdivisions of time into weeks at all. Similarly, months were not grouped into seasons. Time during a given day was not reckoned in hours or minutes, but rather in terms of how far the sun had traveled or in how long it takes to perform a task.^[45]

The sophistication of Inca administration, calendrics, and engineering necessitated a certain facility with numbers. Numerical information itself was stored in the knots of quipu strings, allowing for large numbers to be stored in a small amount of space.^[46] These numbers were stored in base-10 digits, the same base as used by the Quechua language^[47] and used in administrative and military units.^[48] These numbers, stored in quipu, could be calculated on yupanas, grids with squares of positionally varying mathematical values perhaps functioning along the lines of an abacus.^[49] Moving piles of tokens, seeds, or pebbles between the different compartments of the yupana allowed for calculations to take place. It is likely that, "at minimum", Inca mathematics were capable of division of integers into integers or fractions and multiplication of integers and fractions.^[50]