text: We trace the origins of trigonometry to the Old Babylonian era, between the 19th and 16th centuries B.C.E. This is well over a millennium before Hipparchus is said to have fathered the subject with his ‘table of chords’. The main piece of evidence comes from the most famous of Old Babylonian tablets: Plimpton 322, which we interpret in the context of the Old Babylonian approach to triangles and their preference for numerical accuracy. By examining the evidence with this mindset, and comparing Plimpton 322 with Madhava's table of sines, we demonstrate that Plimpton 322 is a powerful, exact ratio-based trigonometric table. ||||| This ancient Babylonian tablet may contain the first evidence of trigonometry 
   
   Trigonometry, the study of the lengths and angles of triangles, sends most modern high schoolers scurrying to their cellphones to look up angles, sines, and cosines. Now, a fresh look at a 3700-year-old clay tablet suggests that Babylonian mathematicians not only developed the first trig table, beating the Greeks to the punch by more than 1000 years, but that they also figured out an entirely new way to look at the subject. However, other experts on the clay tablet, known as Plimpton 322 (P322), say the new work is speculative at best. 
   
   Consisting of four columns and 15 rows of numbers inscribed in cuneiform, the famous P322 tablet was discovered in the early 1900s in what is now southern Iraq by archaeologist, antiquities dealer, and diplomat Edgar Banks, the inspiration for the fictional character Indiana Jones. 
   
   Now stored at Columbia University, the tablet first garnered attention in the 1940s, when historians recognized that its cuneiform inscriptions contain a series of numbers echoing the Pythagorean theorem, which explains the relationship of the lengths of the sides of a right triangle. (The theorem: The square of the hypotenuse equals the sum of the square of the other two sides.) But why ancient scribes generated and sorted these numbers in the first place has been debated for decades. 
   
   Mathematician Daniel Mansfield of the University of New South Wales (UNSW) in Sydney was developing a course for high school math teachers in Australia when he came across an image of P322. Intrigued, he teamed up with UNSW mathematician Norman Wildberger to study it. “It took me 2 years of looking at this [tablet] and saying ‘I’m sure it’s trig, I’m sure it’s trig, but how?’” Mansfield says. The familiar sines, cosines, and angles used by Greek astronomers and modern-day high schoolers were completely missing. Instead, each entry includes information on two sides of a right triangle: the ratio of the short side to the long side and the ratio of the short side to the diagonal, or hypotenuse. 
   
   Mansfield realized that the information he needed was in missing pieces of P322 that had been reconstructed by other researchers. “Those two ratios from the reconstruction really made P322 into a clean and easy-to-use trigonometric table,” he says. He and Wildberger concluded that the Babylonians expressed trigonometry in terms of exact ratios of the lengths of the sides of right triangles, rather than by angles, using their base 60 form of mathematics, they report today in Historia Mathematica . “This is a whole different way of looking at trigonometry,” Mansfield says. “We prefer sines and cosines … but we have to really get outside our own culture to see from their perspective to be able to understand it.” 
   
   If the new interpretation is right, P322 would not only contain the earliest evidence of trigonometry, but it would also represent an exact form of the mathematical discipline, rather than the approximations that estimated numerical values for sines and cosines provide, notes Mathieu Ossendrijver, a historian of ancient science at Humboldt University in Berlin. The table, he says, contains exact values of the sides for a range of right triangles. That means that—as for modern trigonometric tables—someone using the known ratio of two sides can use information in the tablet to find the ratios of the two other sides. 
   
   What’s still lacking is proof that the Babylonians did in fact use this table, or others like it, for solving problems in the manner suggested in the new paper, Ossendrijver says. And science historian Jöran Friberg, retired from the Chalmers University of Technology in Sweden, blasts the idea. The Babylonians “knew NOTHING about ratios of sides!” he wrote in an email to Science . He maintains that P322 is “a table of parameters needed for the composition of school texts and, [only] incidentally, a table of right triangles with whole numbers as sides.” But Mansfield and Wildberger contend that the Babylonians, expert surveyors, could have used their tables to construct palaces, temples, and canals. 
   
   Mathematical historian Christine Proust of the French National Center for Scientific Research in Paris, an expert on the tablet, calls the team’s hypothesis “a very seductive idea.” But she points out that no known Babylonian texts suggest that the tablet was used to solve or understand right triangles. The hypothesis is “mathematically robust, but for the time being, it is highly speculative,” she says. A thorough search of other Babylonian mathematical tablets may yet prove their hypothesis, Ossendrijver says. “But that is really an open question at the moment.” ||||| Babylonian tablet Plimpton 322 will make studying maths easier, mathematician says 
   
   Updated 
   
   A small clay tablet that is 3,700 years old may be about to make studying trigonometry much easier, thanks to the work of an Australian mathematician. 
   
   Key points: Babylonians approach to calculating triangles using simple ratios much more straightforward than our current methods 
   
   Discovery has the potential to make maths education easier for teachers and students 
   
   "It's rare the ancient world teaches us something new", mathematician says 
   
   Daniel Mansfield from University of New South Wales said the palm-sized tablet, covered in rows of tiny numbers in angular cuneiform script, simplifies the study of triangles from using angles and irrational numbers to simple ratios. 
   
   "This gives us a different way of looking at trigonometry. One that's really just passed on ratios. And the beautiful thing about it is that it's much simpler," Dr Mansfield told AM. 
   
   "It only involves ratios, you don't need to study trigonometry through angles: sin, cosx, tan and irrational numbers. You can do it with just ratios." 
   
   The tablet, known as Plimpton 322, was discovered early last century. 
   
   In 1945, mathematicians noticed it contained Pythagoras' theorem — named after the ancient Greek mathematician who actually lived more than 1,000 years after the tablet was made — which shows the square of the side opposite the right angle of a triangle is equal to the sum of the squares of the other two sides. 
   
   Mathematician Patrick Owen, who has spent decades teaching trigonometry to often reluctant kids, said the discovery had the potential to open children's eyes to mathematics in a more engaging way. 
   
   "If that could actually be someone else's perspective, a different perspective on the same knowledge base… so thinking about right-angle triangles, non-right-angle triangles, angles and sides and things like that," Mr Owen says. 
   
   "And someone else sees it differently, that could be that they have this different insight that then others can look at and say, 'oh, that's so much clearer'." 
   
   Why are triangles so important? 
   
   This may all sound esoteric, but trigonometry is fundamental to engineering, architecture, oceanography, astronomy and surveying. 
   
   In fact, our current understanding of trigonometry comes from the ancient Greek astronomers. 
   
   Dr Mansfield said this approach overcomplicated calculations relating to Earth-bound things. 
   
   "We're looking at them just as triangles, we're not importing any of these ideas from astronomy," he said. 
   
   "The angle is perfect for astronomy, it's great, and it's really helped us out with understanding our triangles. 
   
   "But it's not the only way to understand triangles. And yet in our culture we treat it as the only way." 
   
   Today, high school students work out the angles of triangles and lengths of sides using trigonometric functions such as sine, cosine and tangent, which are ratios of sides. 
   
   Modern trigonometry relies only on approximation. 
   
   The Babylonians counted in the same way we count time 
   
   We count in base 10, but the Babylonians counted in base 60. We use base 60 too, when we're talking about time. 
   
   "Ten … only has two exact fractions: half, which is 0.5, and one fifth, which is 0.2. That's problematic if you want to divide," Dr Mansfield explained. 
   
   "For example, one dollar divided by three is 33 cents, with one cent left over." 
   
   The Babylonian system has many more exact fractions, meaning they could calculate for engineering projects, like their famed hanging gardens and stepped pyramids, with much greater accuracy. 
   
   Dr Mansfield said he was not expecting to see school curriculums changing anytime soon, but maybe his discovery would improve the dim view many students have of maths. 
   
   "I hope that teachers might take an interest in it and then pass that interest onto their students, and therefore help everyone like mathematics just that little bit more," he said. 
   
   Based on a comparison of writing styles with other Babylonian tablets, Plimpton 322 has been dated to between 1822 to 1762 BC, which is around the time of the Babylonian King Hammurabi. 
   
   According to UNSW, Plimpton 322 is a trigonometric table that is the world's only completely accurate one, because it only includes Pythagorean triangles with side lengths that are whole numbers. 
   
   "This ensures that the ratios of sides and squared ratios can be represented in the table without approximation," Dr Mansfield said. 
   
   "It's rare that the ancient world teaches us something new. 
   
   "After 3,000 years, Babylonian mathematics might just be coming back into fashion." 
   
   The new study has been published in Historia Mathematica, the official journal of the International Commission on the History of Mathematics. 
   
   Topics: mathematics, science-and-technology, engineering, human-interest, university-of-new-south-wales-2052 
   
   First posted ||||| Dating from 1,000 years before Pythagoras’s theorem, the Babylonian clay tablet is a trigonometric table more accurate than any today, say researchers 
   
   At least 1,000 years before the Greek mathematician Pythagoras looked at a right angled triangle and worked out that the square of the longest side is always equal to the sum of the squares of the other two, an unknown Babylonian genius took a clay tablet and a reed pen and marked out not just the same theorem, but a series of trigonometry tables which scientists claim are more accurate than any available today. 
   
   The 3,700-year-old broken clay tablet survives in the collections of Columbia University, and scientists now believe they have cracked its secrets. 
   
   The team from the University of New South Wales in Sydney believe that the four columns and 15 rows of cuneiform – wedge shaped indentations made in the wet clay – represent the world’s oldest and most accurate working trigonometric table, a working tool which could have been used in surveying, and in calculating how to construct temples, palaces and pyramids. 
   
   The fabled sophistication of Babylonian architecture and engineering is borne out by excavation. The Hanging Gardens of Babylon, believed by some archaeologists to have been a planted step pyramid with a complex artificial watering system, was written of by Greek historians as one of the seven wonders of the ancient world. 
   
   Daniel Mansfield, of the university’s school of mathematics and statistics, described the tablet which may unlock some of their methods as “a fascinating mathematical work that demonstrates undoubted genius” – with potential modern application because the base 60 used in calculations by the Babylonians permitted many more accurate fractions than the contemporary base 10. 
   
   Facebook Twitter Pinterest The tablet could have been used in surveying, and in calculating how to construct temples, palaces and pyramids. Photograph: UNSW/Andrew Kelly 
   
   Mathematicians have been arguing for most of a century about the interpretation of the tablet known as Plimpton 322, ever since the New York publisher George Plimpton bequeathed it to Columbia University in the 1930s as part of a major collection. He bought it from Edgar Banks, a diplomat, antiquities dealer and flamboyant amateur archaeologist said to have inspired the character of Indiana Jones – his feats included climbing Mount Ararat in an unsuccessful attempt to find Noah’s Ark – who had excavated it in southern Iraq in the early 20th century. 
   
   Mansfield, who has published his research with his colleague Norman Wildberger in the journal Historia Mathematica, says that while mathematicians understood for decades that the tablet demonstrates that the theorem long predated Pythagoras, there had been no agreement about the intended use of the tablet. 
   
   “The huge mystery, until now, was its purpose – why the ancient scribes carried out the complex task of generating and sorting the numbers on the tablet. Our research reveals that Plimpton 322 describes the shapes of right-angle triangles using a novel kind of trigonometry based on ratios, not angles and circles. It is a fascinating mathematical work that demonstrates undoubted genius. 
   
   “The tablet not only contains the world’s oldest trigonometric table; it is also the only completely accurate trigonometric table, because of the very different Babylonian approach to arithmetic and geometry. This means it has great relevance for our modern world. Babylonian mathematics may have been out of fashion for more than 3,000 years, but it has possible practical applications in surveying, computer graphics and education. This is a rare example of the ancient world teaching us something new.” 
   
   The tablet also long predates the Greek astronomer Hipparchus, traditionally regarded as the father of trigonometry. 
   
   Wildberger said: “Plimpton 322 predates Hipparchus by more than 1,000 years. It opens up new possibilities not just for modern mathematics research, but also for mathematics education. With Plimpton 322 we see a simpler, more accurate trigonometry that has clear advantages over our own.” 
   
   He and Mansfield believe there is more to learn of Babylonian maths, still buried in untranslated or unstudied tablets. 
   
   “A treasure trove of Babylonian tablets exists, but only a fraction of them have been studied yet. The mathematical world is only waking up to the fact that this ancient but very sophisticated mathematical culture has much to teach us.” 
   
   They suggest that the mathematics of Plimpton 322 indicate that it originally had six columns and 38 rows. They believe it was a working tool, not – as some have suggested – simply a teaching aid for checking calculations. “Plimpton 322 was a powerful tool that could have been used for surveying fields or making architectural calculations to build palaces, temples or step pyramids,” Mansfield said. 
   
   As far back as 1945 the Austrian mathematician Otto Neugebauer and his associate Abraham Sachs were the first to note that Plimpton 322 has 15 pairs of numbers forming parts of Pythagorean triples: three whole numbers a, b and c such that a squared plus b squared equal c squared. The integers 3, 4 and 5 are a well-known example of a Pythagorean triple, but the values on Plimpton 322 are often considerably larger with, for example, the first row referencing the triple 119, 120 and 169.
target: – Greek astronomer Hipparchus is known as the father of trigonometry. But the Guardian reports Babylonian mathematicians may have gotten there 1,000 years earlier. Ever since the real-life inspiration for Indiana Jones discovered a 3,700-year-old clay tablet called P322 in what is now Iraq, historians and mathematicians have debated the meaning of the rows and columns of numbers on it, according to Science. In 1945, mathematicians realized those numbers contained the Pythagorean theorem for deducing the sides of a right triangle a millennium before Pythagoras had coined it, ABC reports. And in a study published Thursday in Historia Mathematica, Australian researchers believe they finally know what those mysterious numbers are: a trigonometric table more accurate than any we have now. “It took me two years of looking at this and saying, ‘I’m sure it’s trig, I’m sure it’s trig, but how?’” mathematician Daniel Mansfield tells Science. One breakthrough came when Mansfield realized the Babylonians had a completely different way of looking at trigonometry than we do, one "based on ratios, not angles and circles." He calls it "fascinating" and evidence of "undoubted genius." And fellow researchers Norman Wildberger says P322 represents a "simpler, more accurate" version of trigonometry. While Mansfield and Wildberger say the Babylonians could have used the tablet to build pyramids and palaces, critics say there's no evidence for that and call the study's conclusions "highly speculative." (Those fortune cookie "lucky numbers" are definitely not unlucky.)