On the evidence of a trigonometric function value system in Babylon

TL;DR

This paper presents evidence that ancient Babylonian mathematicians used a systematic trigonometric function value system, predating Greek and Arabic trigonometry, as demonstrated through the analysis of the Plimpton 322 tablet.

Contribution

It reconstructs the Babylonian calculation sequence, showing they employed a form of trigonometric functions and geometric methods similar to modern practices.

Findings

Babylonian calculations used trigonometric function values.

The system predates Greek and Arabic trigonometry.

The accuracy of their calculations is comparable to modern methods.

Abstract

Ratios and coefficients are used to simplify calculations. For geometric usage these values also called function values. Like in Egypt also in Babylon such a value system can be shown. The reconstructed calculation sequence, of the Plimpton 322 cuneiform tablet, presented and described here, shows in its completeness that, around 3800 years ago there already was a systematically applied exact measuring system, with the usage of trigonometric function values. With this approach one can plausibly explain that, as we still it practice today, a geometry of the circle has been used for this calculation. It is based on, but not only, of the usage of the regularities, which 1200 years later, were named after Pythagoras. During a second calculation step, for an intended scaled documentation, presentation or transfer to other locations, the dimensionless calculated function value, was extended, with a fractional part, or more exact spelled a common factor. This transformation creates a real measurable length from the ratios, always related to the smallest unit. The systematic usage by means of inclining triangles already in old Babylonian times, goes far beyond the previously known level of this age. The accuracy of the individual calculation is just as verifiable and on exactly the same level as the trigonometric functions used today. However, at least at this point in time, the Babylonians were content with dividing a quarter circle into at least 150 inclining triangles. This old Babylonian trigonometric function value system, together with the here described "Babylonian diagonal calculation", is thus the forerunner of the Greek chord calculation and the Indo-Arabic trigonometric functions of the early Middle Ages, which we still use today. In addition, the new approach to Plimpton 322 plausible explains why and how such values could have been used.