section_path: Science and technology/Measures, calendrics and mathematics
section_name: Measures, calendrics and mathematics
paragraph_text: Measures, calendrics and mathematics
  
  Physical measures used by the Inca were based on human body parts. Units included fingers, the distance from thumb to forefinger, palms, cubits and wingspans. The most basic distance unit was thatkiy or thatki, or one pace. The next largest unit was reported by Cobo to be the topo or tupu, measuring 6,000 thatkiy s, or about 7.7 km (4.8 mi); careful study has shown that a range of 4.0 to 6.3 km (2.5 to 3.9 mi) is likely. Next was the wamani, composed of 30 topo s (roughly 232 km or 144 mi). To measure area, 25 by 50 wingspans were used, reckoned in topo s (roughly 3,280 km or 1,270 sq mi). It seems likely that distance was often interpreted as one day's walk; the distance between tambo way-stations varies widely in terms of distance, but far less in terms of time to walk that distance.
  
  Inca calendars were strongly tied to astronomy. Inca astronomers understood equinoxes, solstices and zenith passages, along with 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 12 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. Each lunar month was marked with festivals and rituals. Apparently, the days of the week were not named and days were not grouped into weeks. Similarly, months were not grouped into seasons. Time during a day was not measured in hours or minutes, but in terms of how far the sun had travelled or in how long it had taken to perform a task.
  
  The sophistication of Inca administration, calendrics and engineering required facility with numbers. Numerical information was stored in the knots of quipu strings, allowing for compact storage of large numbers. These numbers were stored in base-10 digits, the same base used by the Quechua language and in administrative and military units. These numbers, stored in quipu, could be calculated on yupanas, grids with squares of positionally varying mathematical values, perhaps functioning as an abacus. Calculation was facilitated by moving piles of tokens, seeds or pebbles between compartments of the yupana. It is likely that Inca mathematics at least allowed division of integers into integers or fractions and multiplication of integers and fractions.
  
  According to mid-17th-century Jesuit chronicler Bernabé Cobo, the Inca designated officials to perform accounting-related tasks. These officials were called quipo camayos. Study of khipu sample VA 42527 (Museum für Völkerkunde, Berlin) revealed that the numbers arranged in calendrically significant patterns were used for agricultural purposes in the "farm account books" kept by the khipukamayuq (accountant or warehouse keeper) to facilitate the closing of accounting books.