Another Derivation of Triangular and Pyramidal Numbers

The methods previously indicated are not the only way that one could arrive at the formula for the triangular and pyramidal numbers. If one guesses that the triangular numbers are proportional to n then one finds, on dividing the triangular numbers by the corresponding value of n, that the result is linear as seen below and the unknown factor is easily guessed at.

Similarly, one might guess that the pyramidal numbers are proportional to the triangular numbers and divide a pyramidal number by its corresponding triangular number. One again gets a linear sequence of numbers and the factor can be easily determined.

So it is not always obvious how a particular formula was arrived at. The formulas found in the ancient papyri were probably intended for use by scribes functioning as clerks and probably do not comprise a mathematical treatise. The ultimate source of the procedures may have been lost with the passage of time and what we now have may have been copied and recopied over thousands of years.

The procedures given in the mathematical papyri are in the most general form. One finds both the method for finding the area of a truncated triangle (with the upper portion cut off parallel to the bottom) and the volume of a truncated pyramid.

For more information see,

The Rhind Mathematical Papyrus by Gay Robins and Charles Shute

Mathematics and Measurement by O. A. W. Dilke

Mathematics in the Time of the Pharaohs by Richard J. Gillings