The rows of Plimpton could have been generated by simply proceeding through such a table and computing. So Bruins advocates a standard method of generation with an advanced table of reciprocals; while Neugebauer advocates an advanced method of generation with the standard table of reciprocals. Neither the advanced method nor the advanced table of reciprocals are otherwise known to have existed at the time. There has been much discussion about the method of generation and what it can tell us about the possible contents of the missing columns.
However, Robson believes that the missing fragment of the tablet contained only the generation parameters Robson, , p. Many scholars believe that Mesopotamian scribes performed SPVN calculations using some kind of computational device similar to an abacus or counting board Woods , Proust , Proust , Middeke-Conlin The device could potentially be in the form of an auxiliary clay tablet or wax writing board Robson, , p.
This device, whatever it may be, is susceptible to two kinds of error. For emphasis, we underline those digits that are considered erroneous. The first kind is copy error which occurs when SPVN numbers are carelessly transferred between the device and tablet Proust, , p. Such errors can occur whenever numbers are copied, either as they enter the device before computation or as they are copied from the device after computation. Friberg calls the second category of error telescoping error Friberg, , p.
Telescoping errors occur during computation, as opposed to copy errors that occur before or after. According to Proust there are three possible types of computational error: the merging of two consecutive digits, the insertion of an extraneous null digit, and the omission of a digit. These are known as types 1, 2 and 3 respectively Proust, , p. The digits 45 : 15 were accidentally combined into the single digit 59, and so we say this is a type 1 error. The digits 50 : 06 were accidentally combined into the single digit This appears to be a type 1 error, although it could also be considered a copy error.
Finally, the loss of the middle digit 36 from the number 19 : 36 : 15 would be considered a type 3 error. See Table 9 for a list of the errors in Plimpton and their types.
We argue that the error in row 2 resulted from a type 2 computational error during factorization and that the error in row 13 resulted from a copy error. However, the four entries listed in Table 10 are inconsistent with this hypothesis. The single entries in rows 2 and 15 are usually dismissed as uncategorized computational errors. But the two entries in row 11 are much harder to dismiss this way. Britton et. Friberg believes this is not an error at all, and instead suggests this row was already sufficiently reduced for the purpose of computing squares Friberg, , p.
Here we argue that the operation is simply factorization; not common factorization as previously supposed. Rows 11 and 15 are correct under this hypothesis and yield valuable information, and row 2 contains a single type 2 computational error. Moreover, the rows which retain regular factors show that the scribe deliberately chose to cease factorization early, and this gives us a clue as to what they were looking for. Those numbers which retain regular factors were sufficient for something.
We begin with the generation procedure for row 1. The value in column II is obtained by removing factors from the short side, and the value in column III is obtained by removing factors from the diagonal. We can only distinguish synchronous from independent factorization through analysis of the rows where additional regular factors are present.
This occurs in rows 2, 5, 11, and 15, which warrant careful attention. Row 2 makes it clear that factorization is independent, simply because more factors are removed from the diagonal than from the short side Friberg, , p. The steps of the factorization procedure are given in Table This error can be easily explained by the insertion of an intermediate null during the penultimate step of factorization, i.
The nature of this error adds weight to our hypothesis that the operation in these columns was indeed factorization. This reinforces the idea that factorization was independent. Whatever they were looking for, it was already apparent from Again, whatever the scribe was looking for, it must be apparent from these numbers without any reduction. Clearly, 1 : 05 has a regular factor but the scribe does not remove it.
The scribe seems to stop removing factors once none are left, or once the numbers are sufficiently small. The error in row 13 can be explained as a copy error as follows. This was suggested by Britton et. Britton et al. For an alternative hypothesis, see Bruins, , p.
See Table 15 for a restoration of the missing parts of Plimpton based on this understanding of its contents. This analysis answers some questions about Plimpton , but raises others. Why is the long side absent? Why would a scribe factorize a number but not record any of the factors? No reason is specified in Plimpton , but one answer resolves all these questions: the sides were factorized to determine if they are regular or not.
In summary, MS and Plimpton are very similar indeed. One tablet invites us to investigate the regularity of the diagonal for five rectangles, and the other invites us to investigate the regularity of the short side and diagonal for fifteen or more likely 38 rectangles.
We have presented a new interpretation of Plimpton as a table of rectangles that shows which sides are regular and which are not. What was the purpose of the text? Answers to this question are both speculative and necessary. Speculative because the text itself does not provide an answer, and necessary because any interpretation must fit within the wider context of Mesopotamian mathematics.
The remainder of this section summarizes the views of some major studies and is followed by a final section where we view Plimpton in relation to recent discoveries concerning contemporary land measurement. Neugebauer believed that Plimpton was a theoretical document motivated by a need to understand diagonal triples. In other words it was known during the whole duration of Babylonian mathematics that the sum of the squares of the lengths of the sides of a right triangle equals the square of the length of the hypotenuse.
Neugebauer, , p. Friberg observed there is a problem with this line of thinking. If we assume that Plimpton was a theoretical investigation into diagonal triples with common factors removed, as both Neugebauer and Britton et.
This view was recently advanced again in Friberg, , p. Friberg is right to object: Plimpton cannot be a theoretical study of diagonal triples with common factors removed. However, this objection may be safely set aside because it now seems clear that Plimpton has nothing to do with the concept of common factors.
Plimpton may well have a theoretical character and we shall return to this idea later. This produces very simple answers to the questions posed above: 10, 40, 50 and 1 respectively. They were carefully chosen to ensure the problems were tractable. In other words, the answers could be found using standard tables.
Friberg, , p. It would have enabled a teacher to set his students repeated exercises on the same mathematical problem, and to check their intermediate and final answers without repeating the calculations himself. Robson, , p. This is a very bold conclusion. A modern analogy would be to say that it contains a mix of elementary school problems alongside the unsolved conjectures of mathematics and hence does not serve as a list of problems for any given audience.
On this point we agree with Britton et al. Any interpretation of Plimpton as a table of trigonometric functions is rightly dismissed by Robson as anachronistic, who is far from alone on this point. There is vast consensus that Plimpton is not about trigonometry as we know it Van Brummelen, , p. However, it is misleading to say that.
Without a well-defined centre or radius there could be no mechanism for conceptualising or measuring angles, and therefore the popular interpretation of Plimpton as some sort of trigonometric table becomes meaningless. Scribes measured and understood just one angle: the right angle. In particular, diagonal triples were used by surveyors to construct shapes with perpendicular sides by least the OB period see below, also Mansfield So while it is meaningless to consider Plimpton as a table of trigonometric functions, it is not meaningless to emphasize the practical importance of right angles in Mesopotamian surveying and consider Plimpton as a table of rectangles.
We are now in a position to refine these conjectures into a new hypothesis: that Plimpton was a theoretical investigation into a certain problem in contemporary land measurement.
We have presented a new analysis of Plimpton as a table of rectangles with information about which sides are regular and which sides are not.
Unfortunately, the answer to this question was lost thousands of years ago and we do not hope for definitive answers. However, it is important to establish that our interpretation fits within the wider context of Mesopotamian mathematics and that there was at least some contemporary interest in rectangles with regular sides.
Tables of rectangles are arguably the earliest mathematical texts Proust, , p. These very early tables of rectangles show the simple relationship between perimeter and surface area. These tables were used by surveyors , mathematically trained scribes specialized in land measurement. Surveying was a highly respected profession in ancient Mesopotamia. These tools and their function in surveying are mentioned in Enki and the World Order where the Sumerian god Enki determines the destiny of Nisaba, the patron goddess of scribes Robson, , p.
A field plan is a sketch of a field and its measurements made by a surveyor. Field plans are found from the very earliest times Lecompte, , p. Most known examples date from the Ur III period — BCE and concern large agricultural fields belonging to institutions such as palaces or temples Nemet-Nejat, , p. Surveyors would measure these fields to estimate the expected size of the harvest, so essentially these early field plans are just agricultural estimates Liverani, , p.
To facilitate its measurement, a field was subdivided into shapes that are approximately rectangles, right trapezoids, and right triangles. The margins of acceptable discrepancy narrowed during the OB period as land ownership began to shift away from institutions and towards private individuals. Cadastral accuracy became increasingly important to avoid private disputes over boundaries Nemet-Nejat, , p.
The senior scribe says. This new phase in surveying is best illustrated by the remarkably accurate field plan Si. Vincent Scheil discovered and cataloged Si. A partial edition was published in Scheil , p. Like earlier field plans, Si. But unlike earlier field plans it concerns the sale of private land and the measurements have been made with unusually high precision.
The rectangles themselves are most remarkable because they actually have opposite sides of equal length, which is unique and suggests that OB surveyors had devised a way to create perpendicular lines more accurately than before. Establishing perpendicular lines is a delicate task that usually requires specialized equipment. How could a surveyor create accurate perpendicular lines with just a measuring rod, rope, and pegs?
The answer lies at the boundaries where we find three shapes two rectangles and a right triangle with the dimensions of diagonal triples. The perpendicular sides of these shapes were likely extended by sight to form the lines found in the subdivision. Diagonal triples were used to create rectangular altars in ancient India Datta, , p. The surveyor who wrote Si. Why were 5, 12, 13 and 8, 15, 17 chosen instead of the simple 3, 4, 5 triple? What determined the scaling factors 1 : 30 and 30? We conjecture that the actual shape of the region would influence which diagonal triple was used—with narrow triples chosen to match narrow regions.
Indeed, in Si. The Roman surveyor Balbus used the 3, 4, 5 triple as an auxiliary rectangle that can be discarded after its sides were used to create perpendicular lines Bohlin , p.
This method is still popular today, which might lead us to expect that the Mesopotamian use of diagonal triples should be similar and there is no need for scaling or other triples. However, the Mesopotamian use of diagonal triples appears to be genuinely different. Instead of creating a small auxiliary shape, OB surveyors would create a whole region with the dimensions of a diagonal triple.
It seems that one side of the region was fixed and used to determine the scaling factor. The field plan Si. The diagonal triples 5, 12, 13 and 8, 15, 17 from Si. In conclusion, OB surveyors created accurate perpendicular lines from a variety of diagonal triples and those with regular sides were particularly useful. Was Plimpton inspired by this cadastral interest in rectangles with regular sides? Perhaps, but we cannot hope for definitive answers to such questions.
Instead, through MS and Si. As Britton et. They drop this assumption and conclude it is mathematics of some kind, as originally suggested by Neugebauer Britton et al. Our analysis agrees with that of Britton et. Here we have dropped the assumption that Plimpton was about diagonal triples with common factors removed, and this has allowed us to reach a new conclusion. Our conclusion is that Plimpton is a study of rectangles.
Four key observations support this interpretation. Through this interpretation, we have reduced the number of errors in Plimpton down to the five listed in Table 9.
This makes our new interpretation more consistent with the text than any previous hypothesis. What is the purpose of the square numbers of column I?
We conjecture, with Britton et al. This is one of the few ways that a scribe could specify that the table concerns genuine diagonal triples and not simply rectangles. The understanding of Plimpton was a long and arduous road, each author having made their humble little contributions. This article is no different. Here we have attempted to make sense of columns II and III in light of an improved understanding of factorization and without the assumption that these columns were reduced synchronously.
Neugebauer originally suggested that Plimpton was theoretical in nature and Price later suggested it was related to practical mensuration. Paradoxically, it now seems both were correct and that Plimpton was a theoretical investigation into rectangles with regular sides that was motivated, directly or indirectly, by the use of these objects in contemporary surveying.
We conclude that Plimpton is an investigation into rectangles with regular sides. It could have been motivated by a particular practical need, or by a purely theoretical interest in geometry. Although it is more likely that the answer lies somewhere between these two extremes. In any case, Plimpton has nothing to do with the modern study of trigonometry developed by Greek astronomers measuring the sky.
The use of an explicit symbol to represent null became more common over time. The use of a colon to separate sexagesimal digits is familiar from modern time-keeping, and following Proust we use the same symbol for SPVN numbers. We also omit any initial zero from the first digit of a number, so we write 6 : 40 instead of 06 : See Proust, , p.
It is worth mentioning that there is no SPVN expression for the number zero. This is hardly a flaw since the number zero is a relatively modern invention that the SPVN system was never intended to deal with. Irrational numbers are different again to irregular numbers. Unfortunately the same can be said for 28 : 45 and 38 : 45, which are not multiples of 3 : The combined table also contained multiplication tables related to practical quantities such as brick weights, see Friberg, , p.
See Friberg, , p. The purpose of the 22 : multiplication table appears to be related to the factorization of regular numbers. The purpose of the 44 : 26 : multiplication table remains unclear. For an alternative geometric interpretation of this text, see Robson, , p. A scribe would have understood a diagonal triple as a rectangle, with the equivalent statement about right triangles a secondary interpretation as per Britton et al.
Bruins Bruins, , p. It is more correct to say that the values of p and q were almost all chosen from the standard table of reciprocals because one additional reciprocal is used Neugebauer and Sachs , p.
Adams, R. Old babylonian networks of urban notables. How were the columns computed? Creighton Buck implied that through mathematics and cunning observation, one could sleuth out the meaning of the tablet and offered an explanation he thought fit the data.
There are a few theories about how Plimpton was created and used by the person or people who made it. On the other hand, some believe it links the Pythagorean theorem known by these ancient Mesopotamians and many other civilizations long before Pythagoras with the method of completing the square to solve a quadratic equation, a common problem in mathematical texts from that time and place.
Some believe the numbers came from so-called reciprocal pairs that were used for multiplication. Some think the tablet was a pedagogical tool, perhaps a source of exercises for students. Some believe it was used in something more like original mathematical research. Academic but readable information about these interpretations can be found in articles by Buck in , Robson in and , and John P.
Britton, Christine Proust, and Steve Shnider from For one, the tablet contains some well-known errors, so claims that it is the most accurate or exact trig table ever are just not true. But even a corrected version of Plimpton would not be a revolutionary replacement for modern trig tables. A trig table would include columns with the sine, cosine, tangent, and possibly other trigonometric functions of angles.
These formulas are based on calculus and can be as precise as necessary. Need the correct answer to 50 digits? Your computer can do it, probably pretty quickly. The sine of an angle is the opposite side divided by the hypotenuse, the cosine is the adjacent divided by the hypotenuse, and the tangent is the opposite divided by the adjacent.
The values of trig functions of most angles are not rational numbers. Mansfield and Wildberger seem to have homed in on the observation that when the side lengths of a right triangle are all integers, these ratios are all rational.
And in fact, the creator of the table set it up so the denominators of all the fractions are easy to represent in base Modern trig tables are based on angles that increase at a steady rate.
Because like other ancient Mesopotamians, the people who produced Plimpton thought of triangles in terms of side lengths rather than angles, the angles do not change steadily. Neither way is inherently superior. Mansfield read about Plimpton by chance when preparing material for first year mathematics students at UNSW. He and Dr. Mansfield added. The research is published in the journal Historia Mathematica. Daniel F.
Plimpton is Babylonian exact sexagesimal trigonometry. Historia Mathematica , published online August 24, ; doi: Archaeology Featured Mathematics. Babylon Clay tablet Plimpton
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