# Tlalcuahuitl

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]

## Subdivisions of tlalcuahuitl

Subdivisions of Tlalcuahuitl[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

## 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 [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]

Calculation Examples Yielding Acolhua Recorded Area[1]
Field Id. Side lengths a, b, c, d in (T) Recorded Area (T²) Calculated Area (T²)
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