Tlalcuahuitl
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Tlalcuahuitl [t͡ɬaɬˈkʷawit͡ɬ] or land rod also known as a cuahuitl [ˈkʷawit͡ɬ] 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.
The abbreviation used for tlalcuahuitl is (T) and the unit square of a tlalcuahuitl is (T²).
Subdivisions of tlalcuahuitl
| 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:
| 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 |