I can't breathe'. Court hears alleged murder victim's house was unusually clean and smelt like bleach. Prime Minister says he does not believe he has told a lie in public life.
Massive funnel-web nicknamed Megaspider donated to venom-milking facility. Child rescued from alleged porn operation, as man faces 43 charges. American journalist jailed for 11 years in Myanmar. China Evergrande avoids default, but where is the money coming from?
Chinese consumers welcome Australian products as China sets new record in Singles' Day sales. With Thailand's sex industry shuttered, Dao's savings are almost gone and she's struggling to provide for her family. Popular Now 1. Second COP26 draft agreement softens language on coal and fossil fuel reduction.
Second COP26 draft agreement softens language on coal and fossil fuel reduction Posted 4h ago 4 hours ago Fri 12 Nov at am. SA's Deputy Premier makes defamation threat in attempt to halt conflict inquiry Posted 5h ago 5 hours ago Fri 12 Nov at am. Perth Airport 'missed the mark' in move to recognise traditional owners on boarding gates Posted 5h ago 5 hours ago Fri 12 Nov at am.
Aussies highlight positives from poor T20 World Cup build-up after reaching final Posted 6h ago 6 hours ago Fri 12 Nov at am. Massive funnel-web nicknamed Megaspider donated to venom-milking facility Posted 6h ago 6 hours ago Fri 12 Nov at am. The researchers speculate the tablet could have been used to survey fields or construct buildings.
For example, knowing the height and width of a building, ancient builders would have been able to calculate the exact measurements need to build pyramid slopes. The Greek astronomer Hipparchus has widely been considered the father of trigonometry. During his life, roughly dating to B. Despite being in top condition for a tablet likely created around B.
Glue residue found on the side suggest the break was recent. The team used previous research on Plimpton to speculate that it was originally built with six columns and 38 rows. Duncan Melville is a professor of mathematics at St. Lawrence University who specializes in Mesopotamian mathematics. Melville stated that to accept the study's results would in a sense redefine trigonometry, but Wildberger, who has previously argued for new theories of trigonometry , argued adopting a new mindset to understand how ancient Babylonians may have worked is essential.
The Olmec civilization, the first in Mesoamerica, offers valuable clues into the development of the rest of the region. Allen noted the most important finding from the tablet is the evidence of Pythagorean triples, indicating that Babylonians were seemingly aware of the Pythagorean theorem—years before Pythagorus.
If the UNSW study does show how the tablet was used to find approximate solutions to equations involving triangles, only speculative historical context can determine exactly how the tablet was applied in day-to-day life said Wildberger. If the Babylonians were the originators of trigonometry, say Allen and Melville, it was drastically improved in efficiency and accuracy by the Greeks nearly a thousand years later.
Some of their computations were very accurate. Babylonian arithmetic was rather clumsy, but then so were Egyptian and Greek variations. He noted that mathematicians in the ancient world heavily borrowed from one another, making it difficult to track their origins. All rights reserved. Solving an Ancient Tablet's Mathematical Mystery. A Disputed History The Greek astronomer Hipparchus has widely been considered the father of trigonometry. Does this study dethrone him? Footnote 1.
A number is a finite sequence of digits. Because scribes wrote the null digit as blank space, intermediate null digits appear as gaps within the number. More importantly, any leading or trailing null digits are lost in the vacant space surrounding the number. We separate the digits of a number using a colon. The digit to the left of a colon represents units that are sixty times greater than the digit to the right.
Footnote 2. The loss of surrounding null digits means that SPVN numbers are only determined up to a multiple of For example, the number 16 : 00 : 05 itself is ambiguous and can mean any rational number of the form.
Decimal numbers are fundamentally different from SPVN numbers since the former can have leading zeros, trailing zeros, and a radix point while the latter can not.
Several modern mathematical concepts must be adjusted to accommodate the different and more ancient SPVN number system. Footnote 3 Footnote 4. Before we can address the concept of multiple we must first recognize that many SPVN numbers are actually units, called regular numbers.
Footnote 5. The standard table of reciprocals Table 1 is a canonical piece of scribal equipment found throughout the OB period. It lists every one and two-digit regular number from 2 to 1 : 21 together with their corresponding reciprocals, and was memorized by scribes during their training.
This meant that scribes could instantly recognize all regular and irregular numbers within this range. These numbers play a fundamental role in Mesopotamian mathematics and, following Proust, , p. An elementary irregular number is any one or two-digit number between 2 and 1 : 21 that is not regular. In other words, we regard the elementary regular numbers as obviously regular and the elementary irregular numbers as obviously irregular.
Scribes knew that certain numbers became smaller when multiplied by certain reciprocals. In fact, they knew precisely which numbers could be reduced and how to reduce them. For example, they knew 5 : 55 : 57 : 25 : 18 : 45 could be reduced with multiplication by the reciprocal of 3 : 45, and that numbers like 7 or 5 : 09 : 01 could not be reduced at all. We return to decimal for a moment. For decimal numbers, it is well known that any integer with final digit 2, 4, 6, 8, or 0 is a multiple of two, and any integer with final digit 5 or 0 is a multiple of five.
Other multiples are possible if we look at more digits. For instance, any decimal integer whose final three digits are. Our definition of multiple for SPVN numbers is similar, except that we limit the number of final digits to at most two, omit trailing null digits, and have many proper factors to consider because 60 is a superior highly composite number. This definition is one of the novel aspects of the paper. On the other hand, the possible endings for a multiple of 10 are 10, 20, 30, 40, and Footnote 6.
Scribes would only consider factors that were obvious from the final one or two digits of a number. Because they were not willing to look beyond these digits, our definition explicitly excludes the elementary regular numbers 27, 32, 54, 1 : 04, 1 : 21 and their reciprocals from consideration. To be precise, scribes would memorize lists with the following standard format.
Such lists are called a - multiplication tables and were memorized to facilitate multiplication. Although it seems that some tables served an additional purpose related to the determination of factors. Memorization of this table enabled scribes to easily multiply by 3 : 45, but it also allowed them to immediately recognize any multiple of 3 : 45 by matching the final two digits with this table. How does this definition of factor fit with the Mesopotamian list-based approach to mathematics?
The combined table is another standard piece of scribal equipment. It consists of the standard table of reciprocals followed by a homogeneous selection of about forty a -multiplication tables. Instances of the combined table are found throughout OB times across multiple archaeological sites and, aside from isolated variations, are basically uniform in structure Sachs, , p. While the combined table was almost certainly used for multiplication, the inclusion of the standard table of reciprocals and the curious selection of multiplication tables suggests more is true.
Friberg argued that the combined table could be used to facilitate division Friberg, , p. Perhaps it is more accurate to say that the combined table served as a reference table for all things related to multiplication, including the identification and removal of factors.
The standard table of reciprocals was also included because it showed how to remove these factors. In other words, the combined table relates to the identification and removal of factors in the following sense: the multiplication tables showed which numbers have factors, and the standard table of reciprocals showed how to remove those factors.
Footnote 7 , Footnote 8 , Footnote 9. Scribes used their understanding of factors to reduce large numerical problems to smaller problems that could be solved by reference to standard tables. Here we define factorization as the process of simplifying a large problem through repeated identification and removal of factors, which halts once the problem is simple enough to be solved directly. But this is just one application, it has since become apparent that scribes also used factorization to solve linear equations and find square roots.
Examples are discussed below. When does factorization stop? Here we emphasize that factorization continues at the pleasure of the scribe. It is a tool that helps them solve large problems, and ceases once the problem is small enough to be solved directly. This usually means that all factors are removed, but not always. This subtle distinction is important for our later analysis. This discussion focuses on factorization in general. It is different to the specific analysis of factorization for regular numbers, such as Sachs ; Proust It is much easier to determine the presence of factors for regular numbers because regular numbers have fewer possible one and two-digit endings.
For example, any regular number ending in 2 : 40 must be a multiple of 2 : 40 Proust , p. This level of simplicity does not extend to numbers in general.
Indeed, the irregular number 12 : 40 ends in 2 : 40 but it is not a multiple of 2 : However, we regard it more generally as a specific application of factorization. Here and throughout we used bold-font to emphasize those digits used to determine factors during factorization. Table 3 shows that the scribe computed the reciprocal of 5 : 55 : 57 : 25 : 18 : 45 by iteratively breaking off elementary regular factors.
The reciprocal of each factor was recorded on the right, and then the reciprocal of the whole computed from these parts. To be precise, the task is to compute the reciprocal of 5 : 55 : 57 : 25 : 18 : The answer is not obvious and so the scribe seeks to reduce the problem by removing an elementary regular factor.
This reciprocal is recorded on the right and the problem is reduced to finding find the reciprocal of. Line ten begins with 1 : 34 : 55 : 18 : 45, from which another factor of 3 : 45 is found. The reciprocal 16 is recorded on the right, and the problem is reduced further. Line eleven begins with 25 : 18 : 45, from which still another factor of 3 : 45 is found.
The reciprocal 16 recorded, and the problem is reduced again. Line twelve begins with 6 : The product of these reciprocals is calculated in lines fourteen to seventeen:. In this example, the scribe repeated factorization until no factors remained. Lines 4 to 6 are given in Table 4. Line four follows directly from factorization: the scribe does not know the reciprocal of 17 : 46 : 40 and seeks to simplify the problem by breaking off an elementary regular factor.
The scribe can immediately see that 6 : 40 is a factor. Line five begins with 2 : This is important because factorization has stopped before all the elementary regular factors were removed. This example is important because it emphasizes the idiosyncratic nature of Mesopotamian factorization. Scribes do not necessarily seek to remove each and every factor as with modern factorization.
Instead, an individual scribe will cease removing factors once they can solve the problem directly. Footnote Our next example comes from VAT Friberg, , p. Factorization can reduce this problem to. This is typical, almost all known exercises can be solved with factorization and the combined table alone Sachs, , p. This is followed by a variation on factorization where only square elementary regular factors are removed.
As usual the reciprocal of each factor is recorded on the right, but additionally the square root of each factor is recorded on the left. The product of these individual square roots yields the expected result 1 : 03 : 45 in lines six and seven. In modern language, the scribe made two symmetrical calculations. The scribe computes the square root of this many-place number by successively removing square factors until the problem is small enough to be solved directly. In line three, a square factor of 3 : 45 i.
Then the factor is removed. In line four the procedure repeats. Another square factor of 3 : 45 is identified, its reciprocal and square root recorded, and then the factor is removed.
The problem has now been reduced to finding the square root of 4 : Factorization halts because the answer, 17, can be found directly from a table of square roots Friberg, , p.
In summary, scribal mathematics is fundamentally about lists and procedures. Large problems are reduced to small problems by factorization, and small problems are solved directly from standard tables. A Pythagorean triple is a right triangle whose three sides are all integers where the square of the hypotenuse equals the sum of the squares of the other two sides. The equivalent Mesopotamian understanding of this fundamental object is slightly different.
A diagonal triple is a rectangle whose sides and diagonal are SPVN numbers, as distinct from a Pythagorean triple which is a right triangle over the positive integers. The fourth in this series is given in Table 6. In any case, this example demonstrates a key characteristic of Mesopotamian mathematics: the questions were designed so they could be solved using only standard procedures and tables.
It is because of this tradition, not lucky coincidence, that this square root can be found using only the techniques we have discussed thus far Footnote It is instructive to prefix our main discussion of Plimpton with a discussion of MS , a sequence of OB mathematical exercises published by Friberg in This is followed by the generation of five diagonal triples using the standard method shown above in Table 6.
What can be said about these diagonals? A small amount of simplification is required to deduce that 1 : 00 : 07 : 30 is irregular. This is because once a single factor is removed, what remains is a number without any elementary regular factors.
These calculations present no difficulty for a student familiar with the combined table and versed in factorization. This might seem trivial, but it is not. The question requires the student to use a rectangle with a regular diagonal i. It is not surprising that the student elects to answer this question with the numerically simple rectangle. Naturally, 3, 4, 5 is the preferred triple on account of its three regular sides. The object known as Plimpton Fig.
It was sold by the archaeologist, adventurer, academic, and antiquities dealer Edgar Banks to the famous publisher and collector George Plimpton in about , who then bequeathed it along with the rest of his collection to Columbia University in where it resides today.
Plimpton The tablet is broken on the left, and the position of the break suggests this is only the latter part of a larger original. We summarise some of the significant work regarding Plimpton and present a new theory based on the above understanding of SPVN numbers and factorization.
This is followed by a discussion of some of the many theories regarding its purpose. The extant fragment of Plimpton contains a table with four columns and fifteen rows. The meaning of the fourth column is quite clear, as this is simply a row number Table 8. For simplicity of exposition, but without loss of generality, we assume p and q are relatively prime. A full list of the parameter values, and a discussion of how they might have been chosen, is given in Proust, , p.
The remaining 23 fill the blank space on the tablet which had been ruled as if the author expected to place additional entries there. Alternatively, the starting point may not have been two parameters p and q chosen from the standard table of reciprocals but a single parameter x chosen from a hypothetical larger table of reciprocals Bruins, , p.
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.
0コメント