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Sunday, April 06, 2008

AZTEC CALCULATION


Courtesy Der Spiegel. 1T=2,5 m (Aztec unit measurement).
Calculating the areas of a without scales agricultural fields of the year 1544, with precisions of measurements of their sides -although without measures of their corners and heights of their triangles- with the help of another agricultural field of 1540, that included exact measures of hundred of fields of Tepetlaoztoc (Aztec city at 6 km of Texcoco), Barbara J. Williams of the University of Wisconsin and María del Carmen J. and J. of the Fenomec/University National of Mexico calculated the way (congruent arithmetic, based on proportions between the standard linear Acolhua measure : 1T, and their units of shorter lenght), according to which the Aztecs calculated agricultural areas. The Acolhua of Tepetlaoztoc used hands, hearts, arrows and bones, instead of numbers. A tlaquahuitl (1T= longitudinal measure), was equal to: 2,5 m. A filled circle =20T. For borders without exact correspondence, beside the T number, fractions were embedded: an arrow = 1/2 T, a hand = 3/5 T, bones = 1/5 T. Nevertheless in the studied plane, alone whole numbers or squares of T are consigned, without decimals, neither commas.

They would have used 6 algorithms to calculate the size of a quadrilateral field: I) They usually multiplied 2 neighboring sides: a * b, like today. If the areas were not rectangular they used other formulas: II) For 1 quadrangle with sides: a, b, c, d, the approximate of the area was calculated: a + c)/2 * b + d)/2. III) A quadrangle was sometimes divided in 2 triangles by means of a diagonal: a*b/2 and c*d/2. IV) Other times, the Acolhua subtracted 1T to one of the sides and added it to another neighboring area. Having developed different formulas with whole numbers, a question remains: how they decided to use a specific formula for any or another property: experience, intuition?

MATEMATICA AZTECA

Calculando las áreas de campos de agricultura en 1 plano sin escalas del año 1544, con precisiones de las longitudes de sus lados -aunque sin medidas de sus esquinas o, alturas de sus triángulos- con la ayuda de otro plano de 1540, que incluía medidas exactas de cientos de áreas de agricultura en Tepetlaoztoc (ciudad azteca a 6km de Texcoco), Barbara J. Williams de la U. De Wisconsin y María del Carmen J. y J. de la U. Nacional de México, calcularon el modo (aritmética congruente, cercana a la exactitud), según la cual los aztecas calcularon áreas de campos de agricultura. Los Acolhua de Tepetlaoztoc emplearon manos, corazones, flechas y huesos, en lugar de números. Un tlaquahuitl (1T=medida longitudinal), equivalía a : 2,5 m. Un circulo rellenado =20T. Para bordes sin correspondencia exacta, al lado del número T, se adosaban fracciones : una flecha = 1/2 T, una mano = 3/5 T, huesos = 1/5 T. No obstante en el plano estudiado, solo se consignan números totales o cuadrados de T, sin decimales, ni comas.

Habrian empleado 6 algoritmos para calcular el tamaño de una propiedad cuadrilátera :I) Usualmente multiplicaban 2 lados vecinos : a * b, al igual que hoy. Si las áreas no eran rectangulares empleaban otras fórmulas :II) Asi, para 1 cuadrangulo con lados : a, b, c, d, el aproximado del área se calculaba : a + c)/2 * b + d)/2. III) A veces un cuadrángulo era dividido en 2 triángulos mediante una diagonal : a*b/2 y c*d/2. IV) Otras veces, los Acolhua restaban 1 T a uno de los lados y lo añadian a otra área vecina. Habiendo desarrollado diferentes fórmulas con números enteros, aún yace una incógnita : como decidían emplear tal o cual fórmula para una u otra propiedad : experiencia, intuición?

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