# Tlalcuahuitl

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Tlalcuahuitl or land rod  also known as a cuahuitl was an Aztec unit of measuring distance that was approximately 2.5 m (8.2 ft),  6 ft (1.8 m) to 8 ft (2.4 m)  or 7.5 ft (2.3 m) long. 

## Contents

The abbreviation used for tlalcuahuitl is (T) and the unit square of a tlalcuahuitl is (T²). 

## Acolhua Congruence Arithmetic

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) the researchers called Acolhua Congruence Arithmetic and it was used to calculate the area of Aztec people's land as demonstrated below: 

Calculation Examples Yielding Acolhua Recorded Area 
Field Id.Side lengths a, b, c, d in (T)Recorded Area (T²)Calculated Area (T²)
1-207-3120 + ht19 + hd20 + ht19 + a38020 x 19 = 380
3-50-71723162439117 x 23 = 391
Average length of one pair of opposite sides times an adjacent side
4-27-164212401145111 x (42 +40)/2 = 11 x 41 = 451
5-12-25221561388452 x (21 + 13)/2 = 52 x 17 = 884
5-145-31408272443227 x (8 + 24)/2 = 27 x 16 = 432
Surveyors' Rule, A = (a + c)/2 x (b + d)/2
5-111-2126323010588(26 + 30)/2 x (32 + 10)/2 = 28 x 21 = 588
5-46-42315 + hd25 + hd11312(23 + 25)/2 + (15 + 11)/2 = 24 x 13 = 312
1-2-11610119126(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
2-2-14111358 + a366(41 + 11)/2 = 225.5ru = 226, (35 x 8)/2 = 140,
226 + 140 = 366
2-30-624162524492(24 x 16)/2 + (24 x 25)/2 = 192 + 300 = 492
5-34-349144712 + a623(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
1-106-25168167126(16 - 2) x (7 + 2) = 14 x 9 = 126
5-139-3018191313 + a252(19 - 1) x (13 + 1) = 18 x 14 = 252
1-189-2714613675(14 + 1) x (6 - 1) = 15 x 5 = 75

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