The article provides new arguments in favor of old disputes (squares of diagonals or widths mistakes in previous analysis of errors in P322 ). It has been shown that the presumed original table with its 7 columns and 39 rows represented: a table of square roots of numbers from 0 to 2 for mathematicians an earliest rudiments of a trigonometric table for builders and surveyors where angles are not measured as an arc in a unit circle but as a side of a unit right-angled triangle a list of the 39 exercises on reciprocal pairs, unit and integer-side right triangles (rectangles), factorization and square numbers for teachers. This article deals with the damaged and incomplete Old Babylonian tablet Plimpton 322 which contains 4 columns and 15 rows of a cuneiform mathematical text. įurther explanations about the calculation and the possible usage of Plimpton 322 are in preparation. It can fill the gaps between the given angles within Plimpton 322 and it shows us the wide mathematical and geometrical knowing and their usage during this time. In Conclusion, this table shows the wider range of usable angles according the original procedure.
This seems to be far from another, but for instance at a right-angled triangle with a base leg (longer one) of a Babylonian cubit (around 0,5m) in length, this made just a difference of a less than a half of an old Babylonian Digitus, (the smallest unit for normal use), for the other triangle leg opposite of the angle. In average, the decrease is a third of a degree, but some angle-differences are also more than half of a degree. The angles decreasing steadily but according to the given procedure, there is no completely uniform distribution between them, because of the limitation that the solution (ib-si) should be just a right angled triangle with 3 integer sides. This now given table here was made just with the same content per line, like the original. Additional, the found angles and the computing procedure enlighten us, why the tablet was possibly written. However, in this state of research it is possible to say, how the clay-tablet was calculated, which steps came at first and where the calculation errors came from. Triples was much more than 1000 but I’m sure that the Chaldeans didn´t used them all. According to the same proceeding, a second table with further 100 or more triples between forty-five and ninety Degree is in preparation. This table presents beside of the already known triangles with their included angles, a total number of 140 Pythagorean triples within a range between zero and forty-five degree. In addition, the new approach to Plimpton 322 plausible explains why and how such values could have been used.ĭuring my recent research, I was able to reveal the original procedure to calculate the rows of the so called Pythagorean triples within Plimpton 322. 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. However, at least at this point in time, the Babylonians were content with dividing a quarter circle into at least 150 inclining triangles. The accuracy of the individual calculation is just as verifiable and on exactly the same level as the trigonometric functions used today. The systematic usage by means of inclining triangles already in old Babylonian times, goes far beyond the previously known level of this age. This transformation creates a real measurable length from the ratios, always related to the smallest unit. 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. It is based on, but not only, of the usage of the regularities, which 1200 years later, were named after Pythagoras.
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. 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. Like in Egypt also in Babylon such a value system can be shown. For geometric usage these values also called function values. Ratios and coefficients are used to simplify calculations.