Subdeacon Joe Posted July 8 Share Posted July 8 https://www.sciencedirect.com/science/article/pii/S0315086017300691 A modern trigonometric table is a list of right triangles with hypotenuse 1 and approximations to the side lengths sin𝜃 and cos𝜃, along with the ratio tan𝜃=sin𝜃/cos𝜃. We propose that P322 is a different kind of trigonometric table which lists right triangles with long side 1, exact short side β and exact diagonal δ – in place of the approximations sin𝜃 and cos𝜃. The ratios 𝛽/𝛿 or 𝛿/𝛽 (equivalent to tan𝜃) are not given because they cannot be calculated exactly on account of the divisions involved. Instead P322 separates this information into three exact numbers: a related squared ratio which can be used as an index, and simplified values b and d for β and δ which allow the user to make their own approximation to these ratios. If this interpretation is correct, then P322 replaces Hipparchus' ‘table of chords’ as the world's oldest trigonometric table — but it is additionally unique because of its exact nature, which would make it the world's only completely accurate trigonometric table. These insights expose an entirely new level of sophistication for OB mathematics. https://personal.math.ubc.ca/~cass/courses/m446-03/pl322/pl322.html The ratio c/b is equal to (p2 + q2)/2pq = (1/2)(p/q + q/p). Therefore this ratio, the square of which appears in the first column of the tablet, will have a finite expression in base 60 if 1/p and 1/q do. The Babylonians almost certainly understood the difference between finite sexagesimal expansions and repeating ones, and in particular we have found tables of reciprocals 1/p for many values of p where the expansion is finite. Such numbers p are called regular by Neugebauer. It is not likely to be a coincidence that the values of both p and q associated to the rows of the tablet are regular, and in fact that in all but one case the expansions of 1/p and 1/q appear in the tables of reciprocals that have been found. It seems plausible, therefore, that the Babylonians knew how to generate primitive Pythagorean triples. 3 Quote Link to comment Share on other sites More sharing options...
Linn Keller, SASS 27332, BOLD 103 Posted July 8 Share Posted July 8 Now THAT is INTERESTING!!! 1 Quote Link to comment Share on other sites More sharing options...
Sedalia Dave Posted July 8 Share Posted July 8 Our ancestors were not near as dumb as most people believe. 1 Quote Link to comment Share on other sites More sharing options...
watab kid Posted July 9 Share Posted July 9 there are so many things we think we know that can be expanded on and there are some that really cant , what always amazes me is when the certain types want a consensus to override the facts , i am amazed at those that want to change the constitution ,[because they are so smart] and those that claim so much scientific knowledge in medicine that are wrong [on purpose] that cause so much harm , my wife thinks im always looking to say it cant be done - is wrong - but , really im an optimist , i believe we will succeed , but not by short cuts , quick fixes that make some rich , or false narratives to serve a political end , i believe we succeeded when we see the problem , look for solutions and prove they are workable by letting them evolve in the system , , you cannt force feed an answer , you cannot subsidize a solution , you have to let it be thought out and then implemented , im still good with the three four five giving me a right angle as long as i use adequately long legs to get there , 2 Quote Link to comment Share on other sites More sharing options...
Subdeacon Joe Posted July 9 Author Share Posted July 9 11 hours ago, Sedalia Dave said: Our ancestors were not near as dumb as most people believe. That's why I stopped watching History Channel and Discovery, too much of what I call "speculative archeology." People with a string of degrees, thinking that they can take a 6 hour crash course in something, try recreating what our ancestors did, failing miserably, and declaring that it couldn't have been done by conventional means, so must have been aliens. Time and lots of muscle power can get a lot of stuff done if you have been raised to do it that way, and started your training in stonework, or carpentry, or whatever skills when you were 7 or younger. 1 Quote Link to comment Share on other sites More sharing options...
Subdeacon Joe Posted July 10 Author Share Posted July 10 More on things our ancestors did: In the 1980s, archaeologists uncovered a remarkable Neolithic wooden road near Nieuw-Dordrecht in the Netherlands. This ancient peat road, dating back an impressive 4,573 years, stretches for at least 800 meters. Through the meticulous science of dendrochronology, or tree-ring dating, experts have pinpointed its construction to 2,549 BC. This discovery not only highlights the advanced engineering skills of Neolithic communities but also offers invaluable insights into their transportation and trade practices. The road's preservation in the peat bogs has allowed for a rare glimpse into the sophisticated infrastructure of prehistoric Europe. https://www.dehondsrug.nl/hotspots/veenweg/?lang=en#:~:text=An ancient wooden road through,or causeway of Nieuw-Dordrecht. 1 1 Quote Link to comment Share on other sites More sharing options...
T.K. Posted July 10 Share Posted July 10 My brain aches just tryin to read this stuff! 1 Quote Link to comment Share on other sites More sharing options...
Larsen E. Pettifogger, SASS #32933 Posted July 10 Share Posted July 10 This is a Roman dodecahedron. They are found all over the empire but concentrated in the northwestern part (Gaul, Germania and Britania). All the holes are a different size and no one has any idea what it is for. 1 Quote Link to comment Share on other sites More sharing options...
Subdeacon Joe Posted July 10 Author Share Posted July 10 16 minutes ago, Larsen E. Pettifogger, SASS #32933 said: This is a Roman dodecahedron. They are found all over the empire but concentrated in the northwestern part (Gaul, Germania and Britania). All the holes are a different size and no one has any idea what it is for. 1 Quote Link to comment Share on other sites More sharing options...
Wild Eagle Posted July 10 Share Posted July 10 They wasted all of their time on stuff like this, and we just now figured out that we need tampon machines in the men's restrooms. Thankfully we have pretty much given up on math and engineering and gotten to really important stuff like gender studies. Nothing can stop us now! 2 Quote Link to comment Share on other sites More sharing options...
Larsen E. Pettifogger, SASS #32933 Posted July 10 Share Posted July 10 Yep and the wizards include pregnant man emogis on your computers, tablets and smart phones. And these same wizards say not to worry about AI. Quote Link to comment Share on other sites More sharing options...
Sedalia Dave Posted July 10 Share Posted July 10 17 hours ago, Subdeacon Joe said: That's why I stopped watching History Channel and Discovery, too much of what I call "speculative archeology." People with a string of degrees, thinking that they can take a 6 hour crash course in something, try recreating what our ancestors did, failing miserably, and declaring that it couldn't have been done by conventional means, so must have been aliens. Time and lots of muscle power can get a lot of stuff done if you have been raised to do it that way, and started your training in stonework, or carpentry, or whatever skills when you were 7 or younger. +1000000000000000000000000 1 Quote Link to comment Share on other sites More sharing options...
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