**Tlalcuahuitl** [t͡ɬaɬˈkʷawit͡ɬ] or **land rod**^{ [1] } also known as a **cuahuitl** [ˈkʷawit͡ɬ] was an Aztec unit of measuring distance that was approximately 2.5 m (8.2 ft),^{ [2] }6 ft (1.8 m) to 8 ft (2.4 m)^{ [3] } or 7.5 ft (2.3 m) long.^{ [3] }

The abbreviation used for *tlalcuahuitl* is (T) and the unit square of a *tlalcuahuitl* is (T²).^{ [1] }

Glyph | English | Nahuatl | IPA | Fraction of Tlalcuahuitl | Metric Equivalent where 1 T = 2.5 m |
---|---|---|---|---|---|

arrow | cemmitl | [ˈsemmit͡ɬ] | 1/2 T | 1.25 m | |

arm | cemacolli | [semaˈkolːi] | 1/3 T | 0.83 m | |

bone | cemomitl | [seˈmomit͡ɬ] | 1/5 T | 0.5 m | |

heart | cenyollotli | [senjoˈlːot͡ɬi] | 2/5 T | 1.0 m | |

hand | cemmatl | [ˈsemmat͡ɬ] | 3/5 T | 1.5 m | |

Using their knowledge of *tlalcuahuitl*, Barbara J. Williams of the Department of Geology at the University of Wisconsin and María del Carmen Jorge y Jorge of the Research Institute for Applied Mathematics and FENOMEC Systems at the National Autonomous University of Mexico believe the Aztecs used a special type of arithmetic. This arithmetic (*tlapōhuallōtl* [t͡ɬapoːˈwalːoːt͡ɬ] ) the researchers called Acolhua [aˈkolwa] Congruence Arithmetic and it was used to calculate the area of Aztec people's land as demonstrated below:^{ [1] }

Field Id. | Side lengths a, b, c, d in (T) | Recorded Area (T²) | Calculated Area (T²) | |||
---|---|---|---|---|---|---|

Multiplication of two adjacent sides | ||||||

1-207-31 | 20 + ht | 19 + hd | 20 + ht | 19 + a | 380 | 20 x 19 = 380 |

3-50-7 | 17 | 23 | 16 | 24 | 391 | 17 x 23 = 391 |

Average length of one pair of opposite sides times an adjacent side | ||||||

4-27-16 | 42 | 12 | 40 | 11 | 451 | 11 x (42 +40)/2 = 11 x 41 = 451 |

5-12-2 | 52 | 21 | 56 | 13 | 884 | 52 x (21 + 13)/2 = 52 x 17 = 884 |

5-145-31 | 40 | 8 | 27 | 24 | 432 | 27 x (8 + 24)/2 = 27 x 16 = 432 |

Surveyors' Rule, A = (a + c)/2 x (b + d)/2 | ||||||

5-111-21 | 26 | 32 | 30 | 10 | 588 | (26 + 30)/2 x (32 + 10)/2 = 28 x 21 = 588 |

5-46-4 | 23 | 15 + hd | 25 + hd | 11 | 312 | (23 + 25)/2 + (15 + 11)/2 = 24 x 13 = 312 |

1-2-1 | 16 | 10 | 11 | 9 | 126 | (16 + 11)/2 = 13.5ru = 14, (10 + 9)/2 = 9.5rd = 9, 14 x 9 = 126 |

Triangle Rule, A = (a x b)/2 + (c x d)/2, or (a x d)/2 + (b x c)/2 | ||||||

2-2-1 | 41 | 11 | 35 | 8 + a | 366 | (41 + 11)/2 = 225.5ru = 226, (35 x 8)/2 = 140, 226 + 140 = 366 |

2-30-6 | 24 | 16 | 25 | 24 | 492 | (24 x 16)/2 + (24 x 25)/2 = 192 + 300 = 492 |

5-34-3 | 49 | 14 | 47 | 12 + a | 623 | (49 x 12)/2 + (14 x 47)/2 = 294 + 329 = 623 |

Plus-Minus Rule, one sidelength +1 or +2 times another sidelength -1 or -2 | ||||||

1-106-25 | 16 | 8 | 16 | 7 | 126 | (16 - 2) x (7 + 2) = 14 x 9 = 126 |

5-139-30 | 18 | 19 | 13 | 13 + a | 252 | (19 - 1) x (13 + 1) = 18 x 14 = 252 |

1-189-27 | 14 | 6 | 13 | 6 | 75 | (14 + 1) x (6 - 1) = 15 x 5 = 75 |

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- 1 2 3 4 5 Williams, B.J. & Jorge, M. (2008). Aztec Arithmetic Revisited: land-Area Algorithms and Acolhua Congruence Arithmetic. In
*Science (320)*. - ↑ Jorge, M et al. (2011).
*Mathematical accuracy of Aztec land surveys assessed from records in the Codex Vergara. PNAS:*University of Michigan. - 1 2
*Nahuatl Dictionary.*(1997). Wired Humanities Project. Retrieved September 8, 2012, from link.

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