Musée d’Art et d’Histoire, Geneva: 8 (2024-04-05)

Created by: Clevenstine, Emmert

An exercise in geometry

This Old Babylonian tablet (MAH 16055) demonstrates how to divide triangles into three pieces such that the uppermost (triangular) and lowest (trapezoidal) portions are equal in area. It shows ten triangles with upper and lower areas of 5 to 50 units. Regardless of the absolute dimensions of the different triangles, the desired division is obtained when the lengths of the base of a triangle and the two transverse lines dividing it are in the ratio 5:4:3. For example, the sixth triangle (upper and lower areas of 30) has a base of 2x601+46x600+40x60-1, or 166 2/3 in decimal notation. The two transversals are 2x601+13x600+20x60-1 (133 1/3) and 1x601+40x600 (100). This is the famous Pythagorean relationship c2=a2+b2 over a thousand years before Pythagoras. Mesopotamian mathematics was based on the number 60, not the 10 that we are familiar with today. The system survives in our division of hours into minutes and seconds, and of a circle into 360 degrees. As the example above shows, 60 has the advantage of being divisible by 3 as well as 2 and 5, so some calculations that produce continuing fractions in base 10 yield round numbers in base 60. Computation was always intimately associated with practical concerns, such as dividing a field among heirs or estimating the materials and labor required for construction projects. Introductions can be found online in English at a site maintained by Duncan Melville at St. Lawrence University (Canton, NY) and in French at CultureMATH, maintained by Eric Vandendriessche of the Université Paris Diderot. CDLI entry: P254721

credit: Clevenstine, Emmert