Old Babylonian mathematics and Plimpton 322: A new understanding of the OB tablet Plimpton 322

[@nyar2352]

I am currently working on my PhD in Assyriology, and just found your channel. I am very bad at maths (we had Babylonian maths classes at some point, which still makes me shiver), but I really enjoy the way you are explaining everything. It is wonderful to see interdisciplinary discoveries like this! Thank you!!

[@WildEggmathematicscourses]

Hi Nyar, Great to hear from you, send me an email some time to let me know what you are working on. I do hope you keep trying to learn more mathematics--that was certainly one of the most remarkable aspects of OB and indeed Sumerian culture, and it should be a priority for you to understand the brilliant things they were able to accomplish.

[@HowardCrowhurst]

I nearly cried watching this. This is the exact way of thinking of the megalithic builders in Carnac, France from 5000BC onwards. I have shown in my books (mostly available only in French) how the 119-120-169 triangle was used at the Manio site in Carnac. Its larger angle is exactly the double of the 5-12-13 triangle. To pass from one to the other, use the following method. If a=5, b=12, c=13 then the side lengths of the triangle with an angle exactly doubled is 2ab (2x5x12=120), b²-a² (144-25 =119) and a²+b² (144+25=169). This method was clearly used by the megalithic builders in Carnac. (I learnt it from there). It works for all values of a and b (they must be different and positive). This tablet must be derived from megalithic science. (If interested, check up on my work on Carnac on Amazon.)

[@njwildberger]

Thanks for the nice fact relating the two triangles, I was not aware of that. Perhaps you can elaborate a little bit about the Manio site in Carnac. Can you give us an overview of what that site contained and your interpretation of it?

[@HowardCrowhurst]

Here is a link to a talk I did in English about the geometry of the Carnac alignments and their relationship to Pythagorean triangles. kzbin.info/www/bejne/sHS8qWx7pcutqc0

[@dkenworthy2036]

@@HowardCrowhurst Hello Howard - The 119-120 and 169 triangle can be linked to the work of Algernon Berriman in his Book ' Historical Metrology' published in 1953, this works using Thom's megalithic yard.. I am currently working on a video presentation to show exactly how this happens and will link it here when it is completed.

[@bw0n6]

Professor Wildberger has already been officially recognized for the outstanding quality of his teaching, so I would like to particularly commend Professor Mansfield for what an excellent job he is doing with these videos and with interviews and press releases. His presentation skills are no doubt a significant contribution to what is clearly turning out to be a very positive reception in both the public and math media.

[@njwildberger]

Thanks for that very nice comment.

[@bw0n6]

I look forward very much to upcoming videos; there is so much in this paper! I am still trying to get my head around some of the sections but see enough to realize you have presented a rock solid case for your interpretation, as usual. Kudos!

[@peternockolds1130]

I have long known that most modern historians of maths know nothing about some aspects of the history of maths because of the difficulty they have in understanding how Archimedes arrived at his values for sq rt 3 in calculaing pi, when in fact it is child's olay to do so. The Ancients thought in whole numbers so the idea that they should approach trigonometry in this way makes perfect sense. I think that recovering this will have psycholgical benefits. You have explained everything very well in this video and opened a door for me.

[@michaeltran6072]

I only have utmost respect for both of you. As lecturers I had once had in previous years, it really shows that you have a true passion in Mathematics, not only in research, but passing on the knowledge to future generations.

[@NormyTres]

The look on your faces as you describe this is a delight! This is absolutely fascinating, thanks for making this accessible to non-mathematicians.

[@alexakalennon]

I heard about this today on the sience news on german radio. I thought about making a Comment on your channel right away. Asking if you heard the news too. And here I am, totally blown away. Fascinating as always. Thank you so much.

[@njwildberger]

Thanks!

[@gadxxxx]

Congratulations. You guys made page 2 of Melbourne's The Age newspaper today. Quite an achievement.

[@njwildberger]

Thanks Geoff. Actually we have gotten a huge amount of international attention, including coverage in many of the world's top newspapers and journals too. It has been really great to see such interest in Old Babylonian mathematics!

[@twwc960]

You even made it to one of my favorite sites: Slashdot!

[@Onoma314]

Here's something to note: Hammurabi's stele itself is in the shape a finger, the finger is the Mesopotamian unit that corresponds to a degree in astronomy, Each first column entry of P332 is the square of the cosecant of an angle of a right triangle, and the associated angles are roughly one degree ( finger ) apart On Hammurabi's stele, one of the duties of the ruler dealt with " the flood ", which if you are familiar with Mesopotamian astronomy texts ( omen texts ) refers to the New Moon ( or an eclipse ) The specific term is " bubbulu " , in Akkadian :)

[@islandbuoy4]

what some call a finger in the middle east ... others studying similar objects in the British isles a menhir. the finger is interesting, the Egyptians have a hieroglyph that looks like a finger ... linked to the meaning ... 'truth'

[@genevievetauro6531]

It was a chance find.I marvelled at your ability to study, interpret and share it. Thank you.

[@sandygrungerson1177]

I was going to request a video on this, and lo and behold you were already on it! Awesome, Dr W!

[@landspide]

Given the international attention, this and prior videos about the subject should be getting a major boost to the view counters.

[@raycorrea5147]

yes, they were smarter than we thought, keep the knowledge coming and hopefully we can go back to many archeological sites and name them from what they really are and not just "temples for worship". Great work!

[@vladtepes9614]

Wow. I read the article yesterday and watched the clip, but I didn't expect this. Thanks!

[@brendawilliams8062]

You are my strong hold. Dr Wildberger. Thankyou for everything.

[@mkbestmaan]

It would be great if we had a curriculum and a book on Babylonian mathematics. It seems that there would be prerequisites in order to decipher all these tables more quickly. It sounds like a book to look forward to, with all of the Babylonian tables to make use of, perhaps with exercises and worksheets to work on for mastery.

[@njwildberger]

That’s a great idea. People should know more about the remarkable mathematical sophistication of the OB culture

[@lsjohn]

Very good job of explaining this to an interested non-mathematician. Thank you.

[@datmeme8967]

Didn't Neugebauer already discover and publish this translation back in 1945? How are you guys saying you just figured this out?

[@twwc960]

This is fascinating. Especially that this tablet contains Pythagorean triples long before Pythagoras. Also interesting that they worked without angles. We tend to forget that angles were introduced mainly for the study of astronomy, but before then, people had need of measuring plots of land and that sort of thing, so some form of trigonometry was needed even in the earliest times. Incidentally, have you read the paper by Eleanor Robson? She discusses some different ways of interpreting the tablet and how the numbers might have been generated and what it might have been used for.

[@douggwyn9656]

I don't think that angles were much motivated by astronomy. Issues about angles arise from simple geometric figures, as soon as one focuses on intersections of lines.

[@twwc960]

I disagree. There are many ways to characterize the relative orientation of two intersecting lines: angle is merely one way. Ratios, like the tangent ratio, or the spread (square of the sine ratio, if you think in those terms) are other ways. Where angles emerge as the most natural is when you study uniform circular motion, such as planets orbiting the sun (actually elliptical but the same basic arguments apply), or the stars apparent "orbit" in the sky which of course is merely due to the earth's rotation. This is due to the fact that in uniform circular motion, angle increases linearly with time. One of my biggest criticisms of Dr. Wildberger's rational trigonometry as the method of teaching trig in high school is that it would make life very difficult for a freshman physics professor who had to teach his students about uniform circular motion, or its one dimensional projection: simple harmonic motion.

[@douggwyn9656]

I didn't say that angle was the only characteristic, just that it is natural to think of it while one is thinking about differences in intersections. I generally agree that angle (or circular motion) is useful in physics, so not teaching it would cause problems.

[@njwildberger]

I think the historical record shows otherwise: angles really became important only with astronomy and particularly from the uniform circular motion of the earth on its axis.

[@UntitledTitle]

Thank you for making this, this is fascinating and engaging and interesting and insightful, all at the same time, and just awesome :-) Brings back the idea of "math" into something that's also very much part of culture, and really is a deep part of our history.

[@njwildberger]

Thanks Sergei!

[@NocturnalJin]

Just saw an article on Science News about your paper. Holy crap! Well done to you both. I needed a break from math so I missed all of your videos on this. Bad timing.

[@njwildberger]

Thanks Jin, plenty of time to catch up. This story has been 3800 years in the making, so we cannot expect it to sink in immediately.

[@NocturnalJin]

Will do! The ancient alien people are going to have a field day with this, hehe.

[@bw0n6]

Not just the ancient alien folks, check out this vid from what seems to be some kind of Christian fundamentalist numerologist : kzbin.info/www/bejne/fqqqh36Nj7qLaNU

[@robharwood3538]

Great video on a fascinating topic! Thank you, Daniel Mansfield and NJ Wildberger. I really hope to learn more about this tablet, especially how they generated the triples, if that's possible to deduce.

[@mathspiderman]

Thank you very much for the very interesting and knowledgeable video. But please let me have a very small comment on it which is also important because it's part of the history of our mathematics. That's, Al-Biruni was an Iranian scholar and mathematician from old Khwarezm, which was located in the modern-day western Uzbekistan, and northern Turkmenistan. (In the mentioned time was part of old Iran). Thanks again!

[@danielalexan80]

I guess because angles are difficult to reproduce with precision, the length ratios are a superior tool. Measuring two lines and connecting them to make the desired angle.

[@simonts595]

Exactly what is different from the understanding of Neugebauer who published "Mathematical Cuneiform Texts" including seemingly the exact same interpretation in even much more detail in 1945..?

[@arinoz2307]

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[@vaibhavkrupakar240]

Neugrbauer defined it as tangent and secant which doesn't make sense without angles

[@jehovajah]

Secant^2 = 1 + Tan^2. Or cosecant^2 = 1 + cotan^2 is probably the relevant use of the tables. . You would of course have to index the triangle ror rectangle required , either by row or an angle index.

[@colinjava8447]

I think the significance to current mathematics has been overhyped. We have calculus, sin and cos and other trig functions, infinite series representations of them, as well as surds for say expressing sin60 as sqrt3/2. I skeptical about how useful this tablet will be, but its very interesting from a historic point, thats for sure.

[@Reliquancy]

I wrote a program that found there are 875 triangles with longer non hypotenuse leg, called l in the video, up to 13500, the one in the video, that also have a prime hypotenuse. Interestingly some legs can reach a prime hypotenuse more than one way, for example: 420^2 + 341^2 = 541^2 and 420^2 + 29^2= 421^2, and 541 and 421 are both prime. I think 420 is the smallest number where that works...

[@kidu9618]

Hello, One of the first practical necessities of division in ancient times was the equal repartition of seeds for example : Nisaba the Sumerian goddess of writing is also the lady of repartition of the seed. First question : from when did the Mesopotamian start multiplying by the reciprocal in order do divide a number where before they divided a given quantity ? Other question : have you heard of "99 models of bricks" found in the East temple of TEPE GAURA "with submultiples (halves and quarters)" "that might have served for contability". (Source : Pierre Amiet, L'art antique du Proche Orient, Citadelle Mazenod, p.501) Could these bricks possibly be models for abstract calculations ? I have worked on the epic of Gilgamesh and my conclusions are that the epic is also a mathematical text that explains the foundations of the sexagesimal base, this seems to be a taboo within the scientific community. In the epic, the number 60 is the cedar forest(perimeter = 60 bêru), somehow the big unit the two heroes Gilgamesh and Enkidoo have to conquer. The narrative says that the forest is rounded by two limits one measures 1 (bêru), the other measures 2/3... This ratio 2/3 is a leit-motive not only in GIlgamesh (who was 1/3 human, 2/3 divine) but also in Sumerian myhology in which Lu-Nanna is abgal (sage) in the same proportion. The great unit 60 is the one that can deal with the great round unit Earth, which they saw as an egg, an ark or a mountain : Earth is exposed to the sun (it is the day) on one of its sides when it is in the shadow (it is the night) on the other side exactly like a mountain has two sides, when one is enlightened, the other one is in the shadow... The cedar forest is a reduced model of the big one, Earth, the KISAR, the Earth-entirety... Thank you for sharing your discoveries.

[@douggwyn9656]

Actually the ratios are just tangent and secant. It does explain why the Babylonians chose base-60 (larger set of exact ratios than with base-10).

[@jehovajah]

Starting with a line segment a extend it by a segment x . Then using a+x/2 ( bisect the extended line segment) draw a circle on the line segment a+x . The rectangle a by x is drawn on the line segment a * x. Extending the side x to meet the circle now gives us a Pythagorean relationship between the lengths, and the rectangle and squares on the sides and a diagonal of a co rectangle in the figure.

[@jehovajah]

The importance to me in your discourse is the dynamic nature of the topology. Whether a rectangle or circle the objects are dynamically changeable through several interesting ratios. Pythagoras theorem is a special case of this more general procedure.

[@tomb613]

When we use base 60 the harmony between everything in our universe becomes glaringly obvious. The same numbers showing up to describe time, music, (when tuned with the Pythagorean tuning system) and the planet's, moons, and sun in the solar system and the relation to each other. When the Romans took over all knowledge during the dark ages this way of looking at our natural world disapeared from classical education. Thankfully we are looking at the maths of our ancestors and seeing how ahead of their time they were, or how behind the times we are! It's almost like when the monks had consolidated all the knowledge left from the ancient world they saw the importance of this harmony and by changing the way we think to a base 10 metric system and changing the tuning slightly to a=440 from a=432 all the beautiful harmony disappears. I recently picked up a copy of "the quadrivium" which teaches all about the harmony between music and the heavenly bodies..

[@NorceCodine]

The interesting side effect of Sumerian (later Babylonian) mathematics is that since they had 6 as a factor in every composite number (and thus 2 and 3), they had no prime numbers. From their point of view, prime numbers in our decimal based number system are the tale-telling signs of its dysfunctionality (!) It is entirely possible that the 2500 year old western mathematics with all its theorems and proofs and hypotheses - is the grandest cul-de sac in science.

[@Nah_Bohdi]

This is incredibly fascinating! Thank you.

[@sallybright7172]

So could this new trigonometry be applied to the Egyptian prymids. Because when i look at your examples the first thing that is see in my mines eye is the pyramids.

[@jehovajah]

Fraction rules are based on this procedure: find the lowest common multiple, these denominate the sub units. Multiply the numerator by the cofactor to obtain the scaled value. . In this case the scribes write down the numerator scaled only, the sub units are specified by some table. . In this way Division is revealed to be multiplication by appropriate cofactors.

[@jehovajah]

The reason why ax is set to a unit of 60 say is because the square of that unit is an actual square area. Then, and the Pythagorean rule for squares fully applies without having to precisely give a side length for the longer side of the rectangle. Choosing co factors that do have regular square factors in their factor lists helps identify exact pythagoreanntriples by the Babylonian triples.

[@buenaventuralosgrandes9266]

subs to your channel. so much knowledge that are still missing, and our history mostly still blank. i hope people would have a passion to learn more, seek more, proof more, and share more. thank you so much

[@monkyyy0]

Seeing as how floating point numbers are very very bad for computers, is it possible this will be used for 3d rendering? How much of this old style of math is rediscovered? Could you use these ratios to estimate distance accurately enough for draw que's without any slow operations?

[@astorlevy7673]

very informative. Thank you

[@elpidiodelacruz]

Very interesting way to thinking.

[@alijoueizadeh8477]

Thank you for sharing. Biruni was a persian astronomer and mathematician.

[@joshuatift4640]

Now could bit there triangles had something to do with astronomy as well?

[@jehovajah]

The diagonal to the long side ratio depicts a increasing slope to this maximal diagonal and then the diagonal to short side ratio takes the slope up to vertical. . The circular arc length is by passed

[@markuzi6755]

Saw Dr Mansfield on ABC midday news today, Friday 25 August 2017, talking about Babylonian mathematics.Well presented too. www.abc.net.au/news/2017-08-25/babylonian-tablet-unlocks-simpler-trigonometry-mathematics/8841368

[@markuzi6755]

A modern reconstructed Plimpton 322 tablet could be made with the previously missing rows and columns added back in to the tables. The new work is presumably protected by copyright law, so this version could be licenced by UNSW for mass production as an education tool or a wall ornament. I would buy one.

[@bw0n6]

A nice version would be the projected full table of 38 rows and six columns in cuneiform as in Table 9 of the paper, with column headers. You could put it together in LaTeX using the Babyloniannum font, but a real clay version would be BEST. A challenging job for the top scribes only! Maybe Irving Finkel?

[@jehovajah]

What seems to be the case iß our "Number" systems have led us down a rigid path and obscured a greater flexibility available to those who use just natural numbers, and common sense units of measure!

[@grandpaobvious]

Yeah but who wants to have to learn multiplication tables in base 60?

[@jehovajah]

Lol!xxx pedagogues seem to make us do stuff we don't want to do! But if you are interested in anything you naturally pick up those details by custom and practice. Some of us humans find counting very easy nd enjoyable , and we make tables for those who do not , so they can look up answers. It's not cheating to rely on a table or a calculator, but these make no sense if you can not count , or recognise numerals, values and value ordered sequences., magnitude and quantity.

[@suzannebiagi4346]

Basis multiplied by long side equals squared diagonal divided by 2, considering sexagesimal system ratios.

[@brendawilliams8062]

Dr. Wildberger. Is it possible that 1001035. 135 aren’t playing together with 3133 and 405.

[@Onoma314]

Btw, the Plimpton tablet likely also points to a proof that Pi is transcendental and not algebraic Here's why: If you hadn't noticed, the column (d/y)^2 where the missing chunk from the tablet is, closes in on the square root of Pi 6th column = 1.7851929 7th column = 1.7199837 Square root of Pi = 1.772453... If you recall Lindermann's proof that Pi is not algebraic, this in turn tells us that the square root of Pi is also not algebraic Thus, the 322 tablet also uses geometry to show that the square root of Pi is not algebraic ( inferring from the fact that Pi is not algebraic, neither is it's square ) Just sayin'

[@jehovajah]

13500 essentially has to be scaled to some multiple of 60, then the other sides can be similarly scaled. . By performing the division( into sub units) we find the normalised scaling factor by finding the cofactors for each ratio. . Thus one looks for the least common multiple of 13500 and a multiple of 60, or the lowest multiple of 60 to which 13500 is the highest common factor.

[@jehovajah]

Example: (30+2)^2= (30-2)^2 + 4*30*215^2< 240 < 16^2Good approximation is 15.5 whence by doubling64, 56,31 is a close Pythagorean triple.It is not hard to apply the procedure to other cofactors of a different base say 360 (60^2) with a subsequent increase in accuracy .

[@carlosantuckwell]

Excellent. Because of this better (and most ancient) system of trigonometry, it makes me think that maybe Zechariah Sitchin was at least partly right about Mesopotamia being founded by an E.T. civilisation. But back to more practical considerations, could you please explain those sexigesimal numbers having more than one 'decimal'-point / full-stop? You seemed to be saying that a series of two-integer numbers separated by full-stops represented one number? How does that work?

[@mohammedkhalili1154]

Brilliant as usual professor.. Really thank you

[@muddy1084]

Didn't the kings set the standard measure for his subjects to include a six spoke carriage wheel (sixty degrees) just a thought. a common tool. I seen a clay tablet in a magazine with your numbers on it, it looked like today's plot book. Thank you for sharing. Good luck with your project.

[@jehovajah]

The circular arc length becomes viable with the invention of the wheel . Then more accurate divisions of the arc become possible to measure more precisely. It may be the Egyptians who preferred this arc measurement scheme based on the right angle in the semi circle

[@jehovajah]

The concept of an angle measure is a convolution . The angle measure of 90° is really an index! So like row 15 90° means the ratio of arc length to chord length found on row 90 ! ! Of course as you go toward finer distinctions the row numbers increase and the degree, minutes and seconds row indices are then introduced. . So the trig table structure is the same, but the standardisation is different as you point out. You are saying they standardised on the long side , not the diagonal. , a

[@jehovajah]

Now consider a and x as factors of 60 or some other sexagesimal unit, and you can then derive exact Babylonian triples by a process of trial and error computation. The rectangles and squares they relate to may have significance to altar design and orientation. . As I said , this is proposition 14 book 1 of the Stoikeia.

[@tinerfenachinech6029]

In the first row, why there are 00 ? Sexagesimal system is from 01 to 60 or from 00 to 59?

[@codex8085]

The pulsar Map on the Sumerian Star Map is more troubling

[@vrendus522]

Wow, interesting. They say that the Annunaki that came there, taught sciences to the Sumerians long, long ago. I hear that the Annunaki were from Philadelphia, "cough". _EDIT_ Thanks guys If you just want to know the legend, these particular Annunaki landed in Africa first, in order to mine a variety of ores. This was said 435,000 years ago and then left. They then returned to ancient Sumeria and must have known that soon they would die out as a race. They gave a majority of information in carefully taught classes to the Sumerians, which then in time afforded this to subsequent generations. Oddly enough, there was a Sumerian King that after his kingdom fought with a neighboring nation, was the first to practice forgiveness, as long as that forgiven neighboring country would not try and re-war against them once more. They were said to be small giants and seven to nine foot in height and came there by ships. They mixed their genes in with the Sumerians who in turn passed that gene complex down to other generations. Because their Anunnak's genes were fading out, only certain aspects of this gene would take of humans within that area, as actually the then Earth humans, would be the decider of the genetic formation within the offspring. They not only had known mathematics, as well as other arts, but pass principles of law and somewhat of a social template downwards in time. If there is any race that Earth own a debt of gratitude to, it would be the Sumerian brand of the Annunaki.__*A fellow by the name of Geroge Noory has a lot of interesting things to say about an associated discovered by him spices, known as the Els. **}Do be grateful for the ways and arts that you have learned today, as in the realm of yesterday, these were gifts afforded to Earth humans, by those who were of a star traveler's cloth.__ The bitterest glass of wine to swallow, is one from the history of one's own ancestors./ "All I can academically say". Thanks

[@hermetikos]

This. Is. Awesome!

[@mwj5368]

Hi! I'm the world's worst at math ha! This was very interesting! Thanks for your interpretation. What happened to Babylonian trigonometry? This understanding is so new I'm thinking that is an irrelevant question as no one could know yet what happened to this ratio type of trig since it was just discovered now. I wonder what disciplines in science would come together to figure out why this form of trig never influenced subsequent civilizations. I wonder if they could fathom if it would've influenced trig history then what would have happened to scientific advancements. Also, is this form of trig "better". I can see it results in definite numbers instead of fractions... that's if I interpreted things correctly, a big "if" ha! If it is better, how is it better? Will it replace our present angular based trig? In what way will it influence us now? Why was this form of trig discovered now when this tablet was I think unearthed over a hundred years ago? A very nice presentation even though it was way over my head ha!

[@jehovajah]

The angle has never been more than an index to an arc length cut off by a chord, and then later a semi chord was used. However the Babylonian system uses a square, not an arc. . One can find medieval pictures in which a square protractor or sextant is clearly depicted .

[@Reliquancy]

there are 12 right triangles with longer non hypotenuse leg, called l in your video, equal to 360, that's the most up to length 360, for example 360^2+38^2=366^2, length 5050 gives around 30

[@Reliquancy]

sorry 360^2 + 38^2=362^2, 360^2+66^2=366^2

[@Reliquancy]

I think the interesting thing about the triangle in the video though is that 18541 is prime, I guess triangles where one leg is prime are what they were looking for so they're not reducible to a smaller one...

[@RichardSouthwell]

More great motivations to study geometry using the rationals

[@jehovajah]

I deliberately left out the angle parameter as it is not important unless you use arcs. Here you could replace the angle index by the row index.

[@DarkMoonDroid]

I keep wanting to click on those underlined words on the whiteboard. LOL

[@njwildberger]

The portable whiteboard is a primitive but effective technology coming to us from the late 20th century.

[@jehovajah]

There is a connection to the rational parametrization of the circle clearly

[@geirha75]

So, I understand Babylonians didn't have any concept of angels? Is that correct?

[@joshuatift4640]

Why were all the numbers at an angle now would that have some kind of tie in with the pyramids as well??? Aren’t they on a perfect angle?? And now a days we have learned that angles are able to avoid radars in flight.

[@francismarney9311]

Is it not backwords? I belibe it should be reading or writing that from right to left? where the key breaker ends code line with (1)? For me this is the first computer in clay... near to an Excell for megalitic contractors.

[@borzumehrtoloui2059]

Nice video. Only one thing, Biruni was not an Arab scholar. He was a Persian scholar (As was Khawrizmi, btw, from whom we get the modern term of Algebra and Algorithm.). Thanks.

[@cnawan]

Damn, that's awesome :D Well done folks!

[@jehovajah]

Surfs and transcendental measures distinguish themselves by having no exact ( artios) unit or sub unit. They are eternally approximate( perisos) no matter how much you scale them. . This can not be determined by division algorithms, as they give extensively repeating numerals as results. .whatever scaling factor compounds the result because the error is in the first measurement. A perisos( approximate) measure always lies between 2 exact measures.

[@ThePaulTM]

Very Interesting about Old Babylonian mathematics similar to what I found in 3D PLATONIC ORDER. The Icosahedron works around 6 vectors and the Dodecahedron around 10 vectors. A perfect order is obtained by rotating copies at 120 degrees of duo PHI rectangles to form these two perfect sized polyhedra, this can give us 3D DNA geometry. There are 60 edges in total coming from 6 and 10 these numbers are all in the Old Babylonian mathematics. My geometry has been put down as trivial by scholars of CHAOS.

[@jehovajah]

The co rectangle has the relationship ( a+ x)^^2/4 = ( a - x)^2/4 + ax.

[@nicholasbouillon5191]

I found that calculation works for rank 11 but it don t works for the others exemple for raw 1 if d² =428 415 (in dec or 1.59.00.15 in hex) it doesn t match with (169² in dec or 2.49 ² in hex) Could you say where is the problem please ?

[@UtrechterK]

Awesome work

[@johnmastroligulano7401]

Try to imagine why it took this long, why at this point in history you are "realizing" what should have been realized long ago(imagine if time wasn't as meaningful as you think).

[@CivilDistribution]

Base 60! Wow, just wow :D

[@giuseppelucianoferrero8916]

Consentimi di suggerirti che il sistema sessagesimale si fonda sul triangolo di Pitagora( il Mago dei Numeri e della sacralità dei medesimi) dunque ; 60 =(3x4x5) ; e qui avete compreso anche che la stessa terna pitagorica doveva avere una propria rappresentazione geometrica; i Magi (Saggi- Filosofi di quel tempo) lo sapevano ma lo tenevano in serbo solo per i loro discepoli). Essi avevano rappresentato nel cerchio sia il triangolo equilatero la cui somma degli angoli interni vale (3x60°), e l'esagono , poi il quadrato e l'ottagono; infine dovevano rappresentare il pentagono ,ma qui scoprirono che l'angolo al centro di 72° ,dove ( 72°x5=360°)richiedeva di trovare il modo di dividere il diametro in due parti,tali che; d= 2r=1= A +B = 0,618..+0,382.. ; dove A/2 = 0,309.. = cos 72°(lato del decagono) e tracciato il poligono di 10 lati , con lato =2r*0,309..; poi, unendo i vertici ogni due angoli alla circonferenza si ottiene il pentagono il. cui lato opposto (all'angolo al centro di 72°) vale il (2r*cos 54°)= 0,587785... M qui voglio richiamare l'attenzione di come essi trovarono 𝛑 che , per quel tempi, era veramente una conquista: ( Consideriamo il. diametro unitario =1= dove 60°= angolo del triangolo equilatero inscritto nel cerchio Essi scrissero che 𝛑 = 1/cos 60°- (2cos 72° sen72° cos72°= =2r/0,5 - [ 0,618*0,951*0,309]= 1/ 0,318364368..= 3,141 055032.. in. buona sostanza sapevano che 𝛗 ed il suo reciproco costruivano i poligoni. Insomma, dobbiamo accettare l'Idea che essi diedero, alle generazioni che seguirono, gli elementi per procedere nella conquista del sapere matematico. Ci sarebbe da dire qualcosa sul numero -1= ( coseno 180°) ovvero il numero( ì )dei moderni che essi intuirono , ma geometricamente ,considerandolo coefficiente (-1) da moltiplicare per la radice negativa di 2; ( -1) * (-√2) per indicare che la pendenza della diagonale del quadrato di lato 1 ,indica in quale quadrante si trova nel sistema di assi ortogonali . Insomma, avevano intuito che il numero naturale non si trovava solo sulla retta dei numeri Naturali ma il loro valore numerico era il modulo ed il segno indicava sia la direzione sia il verso e la loro posizione nello Spazio Cosmico. saluti. li, 11 ottobre 2019

[@lauracortez3904]

😮 this is awesome!

[@Morrey77]

A small correction: Al-Biruni was an Iranian mathematician and not an Arab.

[@Velzen5]

I was able to reproduce the list using the parametrization of Pythagorean triples, and only using the triples where the middle one (not the smallest and not the greatest) could be factored using only 1, 2, 3 and 5. I needed to go up to the parameters 125 and 54 to obtain row4 but no further. I did get about 20 additional rows though (21 if I stop at parameters 125 and 54) alas showing excel in the comments is beyond my skills

[@tinerfenachinech6029]

how you transform from decimal to sexagesimal and viceversa?

[@tinerfenachinech6029]

Another question is, how they use a sexagesimal system meanwhile use a decimal representation.

[@aldobonet9194]

Secondo le mie pluridecennali ricerche, questi tipi di tavolette cuneiformi a contenuto matematico hanno in comune un unico e formidabile strumento algebrico-geometrico scoperto dai sumeri grazie all'invenzione del mattone da costruzione che ho coniato come: il Diagramma di argilla. il Digramma di argilla era fatto di mattoni rettangolari da costruzione movimentabili e sovrapponibili avente alla base un modulo detto: a modulo quadrato. Un gioco algebrico inventato dai sumeri e conosciuto da tutte le Civiltà potamiche e poi importato dai Greci. il Plinton 322, nel caso in specie, era molto probabilmente il loro "teorema di Carnot dell'antichità " e che ritroviamo poi nelle proposizioni 12 e 13 nel Libro II degli Elementi di Euclide. Chi volesse vedere le mie ricerche veda qui : www.atuttascuola.it/collaborazione/bonet/ Scriba con il diagramma di argilla. 3.bp.blogspot.com/-O553c5rl8bI/U4Gmr-afMYI/AAAAAAAAqnU/Z7gKYEKedJE/s1600/copertina_2.jpg

[@PlatinumDragonProductions999]

This is awesome because irrational numbers bother me. I'd be excited to use "exact" trigonometry and a sexigesimal system. :-D

[@zaneleangela4243]

I am having a problem with how to recall the Babylonian number system signs. more help

[@brendawilliams8062]

Thankyou

[@scottmcphee3264]

hi and wow! cnc machines use radon's instead of degrees. Is this in the same family?

[@douggwyn9656]

You mean radians, not radons. Radians are directly proportional to degrees; the constant of proportionality is pi/180. On the unit circle, the central angle in radians is just the length of the included arc. The main advantage of radians is that calculus works more smoothly in radians (fewer constants).

[@scottmcphee3264]

thank you. I have sine bar indicated race engine valve angles. I have never seen a (example) 10x5 valve angle done in a 5 axis cnc dial indicate / sine bar to the same. They are always off by a few minutes. Does this say my measurements are always wrong?

[@douggwyn9656]

There are several possibilities, which would require some on-site detective work.

[@dailynewsgrind6116]

Strange that G+ doesn't let me post 1 or Both links in the description of this videos! i reposted months ago same thing. Weird. RED FLAG?!

[@ChrisDjangoConcerts]

Babylonian Exact Sexagesimal Trig = BEST. Thanks for explaining so clearly! I got it now I think! hahaha What I don't understand is how they were able to come up with the best system though. It seems so hard to arrive a such a system. This civilization had to be more advanced than we thought in many ways. This changed my perspective on ancient civilizations. I'm not a conspiracy theorist, but I think this is a mystery?

[@bw0n6]

Seeing this tablet makes me think that this type of calculation may have been one of the motivating factors in the Babylonians, or rather Sumerians, choosing base 60 in the first place. Exactly how they arrived at their system seems likely to remain a mystery unless we get a very lucky archeological find, but I would like to hear Norman's speculation on how they might have been able do this.

[@ChrisDjangoConcerts]

To me, if you are right about this, them being so flexible about choosing a base depending on this calculation, shows how skilled they where. They were so above bases and trig that they could make such decisions. Somehow they got common people interested to adopt the base 60 system. That is what I find mysterious

[@KSignalEingang]

It's pretty simple to understand how you might arrive at a base-60 system or something like it, if you think about thinking in ratios, or divisions. Base-10 is easy for counting, if you have the usual number of fingers, but it's lousy for dividing things up. It's only evenly divisible by 2 and 5. Base-12, on the other hand, is extremely convenient (which may be why so many things come in dozens), being evenly divisible into halves, thirds, quarters, and sixths - that is, by all the numbers 1-6, except for 5. So, go one step further and multiply 12 by 5, and you get 60, a number which can be evenly divided by all the integers 1-6, plus of course any multiples of them like 10, 12, 15 and 30. To get much better than that you'd need a base-420 system, which seems unwieldy, particularly if you're doing all your calculations with nothing more sophisticated than a stick and some wet clay. (420 is not generally considered conducive to accurate calculation anyway, if movies have taught me anything).

[@ChrisDjangoConcerts]

I agree! It seems like we are deeply asleep or dumbed down compared to these Babylonians

[@KSignalEingang]

It may also help to remember that our numbering system (commonly called "Arabic numerals" though I'm told it originated in India) seems obvious to us, but there is nothing especially self-evident about the concepts of "base" and "order of magnitude" that are embedded in it. Similar systems have been (apparently independently) developed by the Mayans, Chinese and others throughout history, but they're more the exception than the rule. The ancient Greeks and Hebrews (among others) used systems where letters of the alphabet were made to stand for various quantities, and one just added up the quantities to get the final value - the "digits" could technically be in any order and mean the same thing. Or consider the Roman system where position is used to determine whether quantities are added or subtracted (VI=5+1=6, but IV=5-1=4). The concept of "base" isn't even really in there, it's more like an evolved version of tally marks, with shorthand notations for various larger quantities. My point is, when you start with no assumptions at all about what a numbering system "should" look like, particularly if you're in the mindset of someone who is developing such a system to solve a particular problem (such as creating a table of trigonometric ratios), it becomes less surprising that the end result might not resemble our system to any great degree. That is not, of course, to take anything away from the Babylonian mathematician(s) who developed this particularly elegant system. Rather, just to say that they never had to think "beyond" or "above" the systems familiar to us, because they would never have encountered them in the first place.

[@jasoncopin]

Probably used for architecture. A builder's speed square.

[@jehovajah]

The right angle or ortho was thus divided by rectangular blocks in which the diagonal and the short side are allowed to increase to the maximal diagonal of the square, when the short side equals the square measure. by where

[@DrJugne]

The book referenced to at 19:16 is: Mathematics of the Heavens and the Earth: The Early History of Trigonometry Glen van Brummelen University Press Group, 2009 ISBN 978-0691129730 I just ordered my copy... P.S.: I come from the reference in the paper! Great video!

[@herbie_the_hillbillie_goat]

Let's not pretend that the ratio of the short side to the long side isn't the tangent of the angle adjacent to the long side. They also chose to write down only those ratios and squares that are exact in base-60, ignoring those that aren't. When the short side equals the long side, the ratio of the diagonal to the long side is still √2, which cannot be expressed exactly in any base.

[@WildEggmathematicscourses]

We don't need the approximations implicit in "angles" and "tangents". That is part of the point. For example, to understand Row 1 with sides 119,120 and 169 in the classical trig language which you suggest, you need to calculate the "angle" say opposite the long side 120. Can you tell us what that "angle" is? Not approximately , but exactly?

[@herbie_the_hillbillie_goat]

Wild Egg mathematics courses I didn't suggest anything. Nevertheless, practitioners of mathematics work with approximations all the time. Mathematics isn't solely about exact calculations nor should it be. What is the exact length of the hypotenuse of a right isosceles triangle with unit base... EXACTLY?

[@joshuatift4640]

I kind of understand this and don’t I never graduated college but I’m also taking this from a biblical knowledge as well, as universal and ancient Egyptian understanding also