6:42 THIS, and the independent co-discovery of it (at least to an extent) afterwards is why not only Open Source Work, but also even documenting unfinished projects is so huge. Even something you think is just a neat trick can be EXTREMELY useful for someone else.
It's truly absurd to think ancient people wouldn't do these calculations. It's not like they had phones to distract them or google... or grocery stores.
Before logarithms there were other tricks to multiply numbers. The babylonians used similar tables to quickly lookup x^2. Using this you can get a*b by calculating ((a+b)^2 - a^2 - b^2)/2.
@@sachs6 right. You could even make a ridiculous looking babylonian slide rule that does this with several steps. Each step is just addition, subtraction, or a function of a single value. The kind of things slide rules do.
I had a circular one (as well as the more traditional sliding one) in the early 70s. It fits in the palm of my hand with my fingers being able to hold it. But, by 1976, I had a TI electronic calculator. I still have the circular slide rule (and the sliding one but some of the useful white part has broken off from the bamboo wood ). Lost the TI calculator ages ago.
@@two_tier_gary_rumain I switched to the Sharp calculators that had a great 49 char LCD AND a "playback" feature so you could check your formula (or reuse it, as it could draw from 6 memory locations). It was my last, and very best, calculator. My circular calculator got cracked because I carried it in my back pocket and sat on it frequently.
Still have my 70 year old Post Versalog slide rule and books of logarithm tables I used in engineering school in the early 50s. My work in radar and military electronics systems used logarithmic measurement techniques extensively also. ( Y db = 10 log X ) And, neither needed batteries or solar power to operate.
I remember my dad giving me a slide rule and book of logarithms in 1966. I lived in Cleveland where was the Chemical Rubber Company (CRC). The CRC put out handbooks on math, physics, and chemistry. They would donate unsold copies of ‘the CRC’ to local school children. I would peruse the pages of the handbooks wanting so much to learn what all the symbols and formulas meant. I too wound up in radar and military electronics engineering.
Was Phasor Analysis important to early radar work? Radar did such a nice job of knocking Nazi bombers and fighters out of the sky. In school it seemed like such a cool way of solving linear time invariant control systems. (Everything is magnitude and angle (frequency dependent)).
Back in the day of my day, Log Tables and/or the Slide Rule was what you had available to use. Calculators had just come on the scene and were not allowed for exams, etc.. :-)
if my dad (middle school math teacher) caught his students using their phone in class for non educational purposes 3 times he'll give them a wheel similar to this one and that student would not be able to use a calculator during exams as a punishment XD
An even easier way to remember how to multiply and divide on the slide rule is to remember that scales establish a ratio which stays constant everywhere on the scale. a/b=a‘/b‘. By choosing one of them to be 1, the calculation becomes mere multiplication or division. By the way, there are very simple circular slide rules with just two scales - inner and outer. Target customers are accountants/merchants. From now on I’ll recommend struggling teens to first watch this video and then 3b1b on logarithms. It‘s often easier to understand things by taking inventors perspective. If I‘m allowed to compare - it‘s like Kathy Loves Physics but for math. Thanks for this very nice video.
They still use them or at least always carry one in their bag and know how to use them, electronics have a bad habit of failing just when you need them which is not good at 30,000 feet@@douglasmagowan2709
@@lucasrinaldi9909 In this case, it just means "the other one". a and a' would be two different values on the same scale, b and b' two different values on the other scale. Sometimes it means other things.
I'd love a video on prosthaphaeresis, the original way to do complex operations using a table 😊 It's so much nicer in my opinion just vecause trig functions are so classical!
@WelchLabsVideo There was an *EXACT* formula for computing an arbitrary digit of the number pi in base 16 (hexadecimal base.) Can one make a geometric explanation of why that formula works (perhaps by using areas within and lengths of a unit circle and the Babylonian quarter of squares tables for computing multiplicative products) ? ab = 0.25 (a+b)^2 - 0.25 (a-b)^2 Maybe the quadrants of the unit circle have some relationship to do with the hexadecimal base ? What is it ?
I used to own a log and trig table book, also referred to as math tables. Used it in high school. Did math like shown in this video but base 10 instead. It was a cheaper book, so it only had a 4 place mantissa. Used a sliderule extensively in a couple navy courses. Wasn't until 1977 that calculators got cheap enough to replace sliderules and math table books.
That's when math education in America collapsed. We left the slide rule and lost a generation of mathematicians. The teachers who came up on slide rules didn't know how to teach "new math" and the students didn't get it either.
@@OKOKOKOKOKOKOK-zn2fy If you really know math and aren't just dependent on tools, it's easy enough to teach. For me, rather than the intuitive crap they sell, I can only learn it by writing it in code. They've lost the logic and the patterns and are stuck with their tools and formulae.
Loved this video EXCEPT all the times the narrator said “math” when they clearly meant “arithmetic.” Using a logarithmic system to simplify computation IS math. Reading a graph IS math. Finding the relationship between two objects IS math. Other than that quibble, solid stuff.
Yup, this is a logarithmic book, but that term was very young, still not related to exponentiation. It was the name given by Napier to the mapping between a geometric and arithmetics series. The arithmetic series giving you an "Arithmetic Number" aka "Logos Arithmos" in greek. So -> "Logarithmos" -> "Logarithm"
I remember my high school math book said the first logarithm table was made with a base of 1.0001, but that mathematicians figured out 10 was a more sensible base (which may be a simplification, but base 10 is what became big). I had basically forgotten that little tidbit until this video reminded me.
From a strictly mathematical point of view (rather than practical usage) e is the most obvious base. If I remember rightly that's what Napier used in his first published set of log tables. e turns up all over mathematics, whereas 10 is not really mathematical at all, it's just an artefact of our chosen notation for interests and decimal fractions.
@@trueriver1950 Sure. But the angle in this video is engineers and actual calculations, not mathematical calculus and analysis. 10 is better in that case, although the difference is marginal, especially to anyone who uses them several times a day.
@@trueriver1950 If you use base 1.0001 (1+10^-4), number e is close to 10000th entry 🙂Because e=(1+1/x)^x for x approaching infinity and 10000th entry is (1+1/10000)^10000=1.0001^10000. Napier used 0.999999 or something like that - same trick which burgi used: you can rewrite number and subtract 1/1000000 (burgi used adding 1/10000). Maybe that's why Burgi is mention in the video - napier work was less intutive and also his table started by 90 degree and each entry was one minute lower with corresponding sine of angle (maybe cosine and tangens too, i don't remeber).
@@tiranito2834 Not at all. Because you can eyeball 10-logarithms with a minimal amount of practice, way easier than you can eyeball e-logs or 1.0001-logs (for instance, even _without_ practice, I can tell at a glance that the 10-log of 3145 is about 3.5). The two other logarithms take a lot more practice, or alternatively, they require a separate step of some mental multiplication with a memorized constant. No, 10 is much more convenient for practical use.
I have a suggestion to most everyone, to relearn logarithms. Everyone uses it all the time (correctly), but because of calculators, they don’t have a good grasp of it.
Maybe if the goal is to build more intuition about logarithms, it's good to mentally calculate floor(log x) i.e. the largest integer we can raise the base of the log to and still be less than x.
Fantastic video! Just pulled out my grandfather's old slide rule (he was an architect). I'd love to get my hands on a copy of the original tables like you show in your video (I assume you printed out a copy for the video).
if you google the following (including the quotes) you should find it: "Bürgi, Jost:Aritmetische vnd Geometrische Progress Tabulen" (note that the spelling "mistakes" are in the page you are trying to find, so don't feel tempted to correct 'vnd' to 'und' etc.)
@@michaeldamolsen pedantic note: at the time of original publication what we now think of as the letter V also served as our letter U. So it's only a spelling mistake from a modern perspective. But you are quite right, you have to keep the archaic spelling or the search engine gets muddled...
@@puppergump4117 must try that... Millionpounds.pdf SaturdaysLottoNumbers.pdf Naaaah, doesn't work ... The last one nearly works, and you don't actually need the .pdf -- trouble is it gives you last Saturday's number which is not quite as useful
Volvelles are awesome! I have a collection of facsimiles that calculate everything from the day of the week and the positions of the stars to the hexachords along the circle of fifths. These little paper computers are phenomenal examples of human ingenuity. I could gush about them for hours.
At school in the 70s we were taught how to use logarithms to make our own slide rule out of cardboard - a very easy process. Is that still done ? If calculators have taken over then kids should have to at least breadboard out their own simple calculator imho ;-)
in the 80s we still had giant slide rules on the walls in our maths class rooms, but we skipped the chapters on their use - I later accidentally "rediscovered the slide rule" playing with a few sheets of logarithmic paper, observing that twice the 1 to 10 bit amounted to the length of the 1 to 100 bit... and actually feeling quite dumb that I had to think about _why_ that was... and I did not immediately connect my "discovery" (half a millennium too late, but still) to those giant wall slide rules, because they had multiple bars / scales and I had always assumed these things were way too complex for me to master
Well they now have smartphones, not just calculators, and with a download they can match the most sophisticated "scientific calculators". I'm glad to see the end of the tyranny of TI-86 in the US, but, we're getting to the point where a kid can just verbally ask an AI what the solution is, without any conception of calculations at all.
I studied elementary school from 1987 in Czechia, graduated in 2004. I know slide rule only because it was magical item that was among some pencils, drafting tools, compasses and rulers in a drawer of my table. We basically learned powers and logarithms as a given tool, we learned how to simplify equations and how to calculate interests. But I wasn't able to apply them correctly on some example with radioactive decay where exponents were too big for straightforward solution on calculator. That was at middle education (is it called high school in US?) Then I used them at university applied as exponential attenution, conversion to decibels and back and that's basically all. Conversion of multiplication to addition and how to get square root was something I learned asking my mother about slide rule. Then recipe at uni was basically to convert stuff to decibels per unit lenght, mutliply by length, convert it back. Or voltage on capacitor vs time. University was actually when I used logs.
Thank you! It would be also interesting to see a more detailed analysis, of how many elementary operations you save using these tables. For instance, it takes time just to find the corresponding logarithm in the book, so it is not so obvious how strongly does it speeds up the process.
Fair point. Let's immagine our self to multiply distinct numbers. Number A by number B both with 10 digit. We choose 10 times A, B. In the usual way we must perform 100 multiplications and 90 addictions. By using this method with look up table we perform only 10 addiction. We may conclude as well then although not perfect is less prone to mistakes.
Oh man i love multiplication. Especially by 10^x where x is a positive number My favorite rule is a*b+a*c=a(b+c) since the sum of b+c being equal to 10^|x| is non zero
If you lookup the methods for getting square roots before calculators in the west, you quickly discover... there's a reason the square-root symbol looks like the symbol used for long division (differentiated in look as though a "v" is fused to its initial forward-slanting line)... It's not that it's a form of division so much as the signs are indicative of relation.
Old guy here. I have a graphical table of logarithms-- read it like a slide rule; quick and easy. Never did care for the log-log slide rules that were the standard in the US; prefer Darmstadt rules. One formula that was always useful: log (a to the x power) = x log a. Also note that electronic calculators do not actually do the operations that are keyed in; they use electronic functions to simulate them. Because of that one can do a string of multiplications and divisions of a number, then reverse the sequence and come up with a number different from the original one. It was always difficult to hammer an understanding of significant figures into students' heads, but with the advent of electronic calculators (which display numbers to as many places as their displays will accommodate) it has become nearly impossible.
Well, lookup tables are used all over the place in high performance computations, so it did change mathematics and engineering forever. It could be argued that it was way ahead of its time I suppose.
My brain hurts at the thought of having to transcribe a copy of a book like that. Imagine the tedium of writing a whole page only to find a compounding error somewhere.
Clockmaking was at the forefront of science and engineering at the time, the skills developed allow the invention of a whole plethora of scientific instruments, laid the groundwork for advances in astronomy and navigation, and gave birth to the machine tools that allowed for the scientific revolution.
Reminds me of how Nintendo64 and Super Mario 64 "calculated" sin(x) (and cos(x) by looking them up and how due to hardware reasons surprisingly efficent it was when compared to other alternative/more modern implementations. There should be a quite interesting video on that here on YT.
No, it’s got a different origin. Logarithm was coined by its inventor John Napier from Ancient Greek λόγος (lógos) meaning word and αριθμός (arithmós) meaning number. Logbook referred to a book in which ships recorded their speed for navigation purposes. They got the name from the device used to record speed, the chip log. It was a piece of wood tied to a roll of string that was dropped into the water. The rate at which the string was unwound was used to measure speed. Logbook now refers to a record of anything stored in written media. The chip log got its name from the fact that the drag (the piece of wood that gets dragged by the water) is a simple piece of wood that is chipped off from a log. Log in this sense comes either from Old Norse lóg/lág meaning fallen tree, which itself comes from the verb liggja, meaning to lie (on the ground/a surface); or it came from Norwegian låg, also meaning fallen tree. “Log in” is a phrasal verb constructed from the fact that when one accesses a computer account, one is adding an entry into a log. Log in the sense of an append-only sequence of records written to file was probably derived from logbook by analogy. Note that logbook as in a record of a vessel’s progress or as a general written record was already being shortened to log long before computers. Logarithm came first (John Napier introduced the term in New Latin as logarithmus) in 1614. Logbook was first attested in nautical records in the 1670s. Log in was first used in 1963 by the MIT Computation Center, in reference to people time-sharing their mainframes.
A ship compass and ship’s log were used to track direction and speed. Then a traverse board was used to keep track of those values for the four hours of a watch. At the end of four hours, the data was entered into the logbook. Then started over for. The next four hour watch. (I read the Aubrey-Maturin series of books.)
A captain's log is commonly found in the chamber pot in a captain's cabin. A cabin boy would check daily if there was a log in it. If there was, he would empty the log, called a log out. He would then inform the ship''s doctor who would record it in his log book. This was the doctor's log. A doctor's log is commonly found in the ...
Me: "Hey, this looks like a fun video... WAIT. WHO made this???" Welcome back! I have been telling all of my precalc students about your Imaginary Numbers Are Real series.
Unlikely. You'd have a terminology crash with accounting, and as the video hints at, the mathematical functions being described have a *much* broader reach than a multiplication aid.
you said the power is calculated by adding row and column. and you also mentioned that burgie calculated total 23k and 27 numbers. but in the video what I can see is the column is 2.3lac. so can you clearify on that?
I'm afraid that, many of such significant work done today will be lost exactly due to the opposite reason. They'll get drowned in millions of publications per year, and get bogged down in impact factor and citation quagmire.
It already is a problem. It's hard to find fundamental in-depth information cause google always returns results which are similar to each other and wiki and basically mainstream. Maybe it's to reduce risk of "misinformation" or low quality information in search results. Imagine that just two weeks ago I had to derive how to extract center of rotation from transformation matrix - i was not able to google it, chatgpt said it's too complex problem. Yes, it needed to multiply three matrices by hand and solve three equations of three unknows having trigonometric functions - but extracting angle was easy so more like two equations without trigs. Another solution was to use some math library and extract eigenvectors. All that google found was the opposite - how to get matrix knowing center of rotation and angle. And that was not only problem I had. Sometimes it disregards some search keywords and I was not able to find exact article on my blog. Maybe it filters spam, I don't know. Also it's very hard to find some negative yet informative articles on someones blog - once i could not find that some problem exists, then one guy randomly linked in-depth research about it in youtube video.
wow, that is beyond genius. the Nobel prize for mathematics does not exist because it was not given to him. actually Nobel prize does not deserve him, in fact this book is MORE valuable THAN Nobel prize.
someone write an AI browser add-on that erases youtube vid titles and replaces them with accurate titles this one: "This book should have changed mathematics forever" -> "Slide Rule"
As the last generation to use slide rules and tables, you don't understand the power of calculators and computers we now possess although I think hand graphing be it on linear, log, semilog, or polar graph paper and working with other graphical methods (nomographs, families of curves, stability plots, mixture phase, Mollier and other chemical and thermal parameter diagrams and other binary, ternary plots, etc.) gave one an intuition on creating functions to simplify calculations with correlations that can be used in computers.
Somewhat sad that these figures have previously been calculated. Would have like to take a stab at computing them by hand. That being said, I appreciate the monumental efforts of past generations to accelerate our calculations. Leading us to future questions and constructs.
I would argue that this is not “mathematics” but arithmetic. Math is closer to theory. This book is closer to practice. If you accept that, then I would argue that the book did indeed change arithmetic forever. This was one (as other comments have pointed out) of several systems leading up the logarithms.
Mathematics is to arithmetic as vehicle is to car. When I was a kid in the 50s in elementary school we had arithmetic class. Nowadays it is called mathematics class. No big deal.
"Should have changed mathematics forever" is a crazy thing to claim. He could have been the inventor of logarithms. A contemporary beat him to it. Meh....
At 1: 51, it looks like the number from the book for 2.475 is 90,630 - but you wrote 9,063. Why is that? (9,063 seems to be the correct one according to my calculator).
Very nice video! A question : does anybody has any idea as to how the book was printed? It seems hand made, and I was wondering how one would obtain such a nice result
Remarkable! It sounds like these mathematicians (Burgi and Napier) were maybe starting to…. Grasp at what Karatsuba discovered after thinking about Kolmogorov’s opinion on the time complexity of the standard multiplication algorithm? Could taking the reasoning behind Napier’s and Burgi’s work make even better multiplication algorithms for computers? Karatsuba essentially improved upon the standard multiplication algorithm by creating the scheme where you do less actual multiplication and more addition (which I guess saves time complexity since addition is foundational to multiplication, and, therefore, a less ‘complex’ operation, computationally?), which seems to be what Burgi has done with his book of Red and Black numbers (Not unlike Red-Black tree data structures in CS - the Red-Black dichotomy pops up in a lot of unexpected places in Math and also Philosophy….) Hmmm…. …
Wondering if the Red-Black paradigm shows up in many places because they were literally the two colors of ink available in the standard stylus/quill pen holder with two ink wells that sat on everybody's desk.
@@amitabho That’s actually how Sedgwick et al did the Red-Black Trees at Xerox. Lol. Simply because - yes, you’re right…. They just had Red and Black pens lying around…. And that was long after the age of quill pens…. (Ball-point pens in the 70s/80s, Lol)…. I guess Charles Sanders Peirce was just trying to be more dramatic about it in his essay on Probability Theory that was titled “The Red and The Black”…. Also, a metal band called one of their songs “The Red and The Black” …. Anyway …
I'm about to fix math and get my Nobel prize. Here's a hint - division by zero is defined as the null set - the set with no numbers, but, you can put any number in a division by zero and it checks out. Division by zero isn't the set with no numbers. It is the set with ALL numbers. That is why the universe has apparent motion - it reached a division by zero error and exploded.
Am I the only one thrown off by the fact tha the red number in the table doesn't line up with the red number in the written card? 160500 is what I see in the table but what is written is 16096. Is there a missing number place somewhere???
The math works out for any basis, but 1.0001 is a good compromise between precision and a set of tables small enough to be publishable. In the real world, we rarely need even three digits of precision. Outside of the real world, I work with people who compute spacecraft trajectories. They need a lot more than three digits of precision. Fortunately, that wasn't much of a consideration 400 years ago.
Perhaps this number give the enough amount of reference vs effort to time ratio , imagine calculating all that by hand , that is really some good work.
He calculated powers rather than logarithms. To get a table you can calculate either. It is a lot easier to calculate powers than logs. To get any precision by repeated multiplication of powers you need a number close to 1. If he used base ten his table would tell us the black number 10 has the red number 1, 100 is 2, and so on. The other thing is that having spotted that he wanted a number close to 1, having one further digit of 1 makes manual multiplication that much easier. Tedious but easy. How many zeros? The more zeros before that final 1 the better the accuracy; but with two costs: the numbers take longer to calculate, because you need to do more multiplications, and the book becomes correspondingly thicker and more expensive to print. I guess he thought that a precision of one in 10^5 was a sensible trade off
Basically a table of logarithms base 1.0001, I suppose he chose that base to simplify the operations to calculate the table, that are just shifts and additions. Very clever.
Ok I'm not a math wiz by any stretch of the imagination, and TBH I never understood log tables but with your explanation here I'm beginning to get a gist of how and why they were used. At about 2:28 you stated the answer with 3 decimal places. Did you limit to 3 decimal places because that was the max number of decimals to the right of the decimal point???? Then you said with an extra interpolation step you could get accuracy out to 7 digits. Can you explain that??? I'm lost as to what you mean, lol I got a flash light ... but still can't find my own arse ..... hmmmm I guess my mom was right.
![The Most Useful Curve in Mathematics [Logarithms]](http://i.ytimg.com/vi/OjIwCOevUew/mqdefault.jpg)
![The Most Useful Curve in Mathematics [Logarithms]](/img/tr.png)
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6:42 THIS, and the independent co-discovery of it (at least to an extent) afterwards is why not only Open Source Work, but also even documenting unfinished projects is so huge.
Even something you think is just a neat trick can be EXTREMELY useful for someone else.
holy sh** it really is the first engineer's tool! I LOVE the idea that slide rules existed around the birth of modern scientific method (Galileo)
What if this is not the FIRST engineer's tool , but just ONE of it.
Hello here comes the ancient Egyptians? Ancient Egyptians didn’t have engineering tools to build those pyramids…..hmmm
@@user-lu6yg3vk9z One thing has absolutely no relation to the other.
@@user-lu6yg3vk9z Mentally insert the "ALIENS!" meme here
It's truly absurd to think ancient people wouldn't do these calculations. It's not like they had phones to distract them or google... or grocery stores.
Before logarithms there were other tricks to multiply numbers. The babylonians used similar tables to quickly lookup x^2. Using this you can get a*b by calculating ((a+b)^2 - a^2 - b^2)/2.
I think they had half square tables directly, so they didn't had to half anything.
@@sachs6 right. You could even make a ridiculous looking babylonian slide rule that does this with several steps. Each step is just addition, subtraction, or a function of a single value. The kind of things slide rules do.
This is a bit slow. It is quicker to calculate ab as ((a+b)/2)²-((a-b)/2)²
@@caspermadlener4191 Not if the calculation is more complicated , it is kind of like using the calculator.
And it's important to realize that the Babylonians did this perhaps 4000 years ago!!
I love the use of dots for repeated digits
You probably will also enjoy IPv6 addressing, then
When I studied engineering in the 80s, I had a circular slide rule. Awesome video.
I have a small salad plate sized circular slide rule that maps out to a linear slide rule nine feet long.
When I studied engineering in the 80s, I had a TI-41CX.
I had a circular one (as well as the more traditional sliding one) in the early 70s. It fits in the palm of my hand with my fingers being able to hold it.
But, by 1976, I had a TI electronic calculator. I still have the circular slide rule (and the sliding one but some of the useful white part has broken off from the bamboo wood ).
Lost the TI calculator ages ago.
@@two_tier_gary_rumain I switched to the Sharp calculators that had a great 49 char LCD AND a "playback" feature so you could check your formula (or reuse it, as it could draw from 6 memory locations). It was my last, and very best, calculator. My circular calculator got cracked because I carried it in my back pocket and sat on it frequently.
@@jeffweber8244 Hewlett-Packard made the 41CX (i.e. HP-41CX), not TI. My HP-41CX still works, although I usually use an HP-42S.
Still have my 70 year old Post Versalog slide rule and books of logarithm tables I used in engineering school in the early 50s. My work in radar and military electronics systems used logarithmic measurement techniques extensively also. ( Y db = 10 log X ) And, neither needed batteries or solar power to operate.
I remember my dad giving me a slide rule and book of logarithms in 1966. I lived in Cleveland where was the Chemical Rubber Company (CRC). The CRC put out handbooks on math, physics, and chemistry. They would donate unsold copies of ‘the CRC’ to local school children. I would peruse the pages of the handbooks wanting so much to learn what all the symbols and formulas meant.
I too wound up in radar and military electronics engineering.
Was Phasor Analysis important to early radar work? Radar did such a nice job of knocking Nazi bombers and fighters out of the sky. In school it seemed like such a cool way of solving linear time invariant control systems. (Everything is magnitude and angle (frequency dependent)).
Of course you used solar power... otherwise you would have been in the dark!😅
holy moly, you're old man. stay with us for about 5 years
Nice video. I've studied the history of logarithms, but was not aware of Burgi's work. Thank you for publishing this.
What brains those guys had! Just unfathomable.
Back in the day of my day, Log Tables and/or the Slide Rule was what you had available to use. Calculators had just come on the scene and were not allowed for exams, etc.. :-)
if my dad (middle school math teacher) caught his students using their phone in class for non educational purposes 3 times he'll give them a wheel similar to this one and that student would not be able to use a calculator during exams as a punishment XD
An even easier way to remember how to multiply and divide on the slide rule is to remember that scales establish a ratio which stays constant everywhere on the scale. a/b=a‘/b‘. By choosing one of them to be 1, the calculation becomes mere multiplication or division.
By the way, there are very simple circular slide rules with just two scales - inner and outer. Target customers are accountants/merchants.
From now on I’ll recommend struggling teens to first watch this video and then 3b1b on logarithms. It‘s often easier to understand things by taking inventors perspective.
If I‘m allowed to compare - it‘s like Kathy Loves Physics but for math. Thanks for this very nice video.
Pilots used to use circular slide rules (the e6b flight computer) right up until the invention of the iPad.
might I also recommend vihart's mini-series on dragons, fractals, and logarithms
Now I feel stupid. I have no idea what that symbol above the letters means in mathematical notation.
They still use them or at least always carry one in their bag and know how to use them, electronics have a bad habit of failing just when you need them which is not good at 30,000 feet@@douglasmagowan2709
@@lucasrinaldi9909 In this case, it just means "the other one". a and a' would be two different values on the same scale, b and b' two different values on the other scale. Sometimes it means other things.
I'd love a video on prosthaphaeresis, the original way to do complex operations using a table 😊
It's so much nicer in my opinion just vecause trig functions are so classical!
Yeah I've bumped into this a few times - will but this on the idea list!
"prosthaphaeresis"???? Do you kiss your mother with that mouth???
@WelchLabsVideo
There was an *EXACT* formula for computing an arbitrary digit of the number pi in base 16 (hexadecimal base.) Can one make a geometric explanation of why that formula works (perhaps by using areas within and lengths of a unit circle and the Babylonian quarter of squares tables for computing multiplicative products) ?
ab = 0.25 (a+b)^2 - 0.25 (a-b)^2
Maybe the quadrants of the unit circle have some relationship to do with the hexadecimal base ? What is it ?
@@solconcordia4315can you share this pi formula?
The formula you wrote is just a multiplication formula
@@ok-hv1or It's the Bailey-Borwein-Plouffe formula.
A really interestic topic, a cool book, and the best explanation on log tables i have seen!
Thank you!
To see where the notion of slide rule ultimately went, check out Thacher's Cylindrical Slide Rule. (I actually used one at work in the early Sixties.)
Thank you for sharing this. I am also commenting to help your brilliant channel with the algorithm so that more people see it.
I used to own a log and trig table book, also referred to as math tables. Used it in high school. Did math like shown in this video but base 10 instead. It was a cheaper book, so it only had a 4 place mantissa. Used a sliderule extensively in a couple navy courses. Wasn't until 1977 that calculators got cheap enough to replace sliderules and math table books.
That's when math education in America collapsed.
We left the slide rule and lost a generation of mathematicians.
The teachers who came up on slide rules didn't know how to teach "new math" and the students didn't get it either.
@@OKOKOKOKOKOKOK-zn2fy If you really know math and aren't just dependent on tools, it's easy enough to teach. For me, rather than the intuitive crap they sell, I can only learn it by writing it in code. They've lost the logic and the patterns and are stuck with their tools and formulae.
Yes. I remember using log and trig tables at school and then a slide rule at university. I still have my slide rule! I ended up in sonar.
In India students still use log and trig tables in their examinations.@@rogerphelps9939
Loved this video EXCEPT all the times the narrator said “math” when they clearly meant “arithmetic.”
Using a logarithmic system to simplify computation IS math. Reading a graph IS math. Finding the relationship between two objects IS math.
Other than that quibble, solid stuff.
isn't it logarithms?
Yup, this is a logarithmic book, but that term was very young, still not related to exponentiation. It was the name given by Napier to the mapping between a geometric and arithmetics series. The arithmetic series giving you an "Arithmetic Number" aka "Logos Arithmos" in greek. So -> "Logarithmos" -> "Logarithm"
I remember my high school math book said the first logarithm table was made with a base of 1.0001, but that mathematicians figured out 10 was a more sensible base (which may be a simplification, but base 10 is what became big). I had basically forgotten that little tidbit until this video reminded me.
From a strictly mathematical point of view (rather than practical usage) e is the most obvious base. If I remember rightly that's what Napier used in his first published set of log tables.
e turns up all over mathematics, whereas 10 is not really mathematical at all, it's just an artefact of our chosen notation for interests and decimal fractions.
@@trueriver1950 Sure. But the angle in this video is engineers and actual calculations, not mathematical calculus and analysis. 10 is better in that case, although the difference is marginal, especially to anyone who uses them several times a day.
@@trueriver1950 If you use base 1.0001 (1+10^-4), number e is close to 10000th entry 🙂Because e=(1+1/x)^x for x approaching infinity and 10000th entry is (1+1/10000)^10000=1.0001^10000.
Napier used 0.999999 or something like that - same trick which burgi used: you can rewrite number and subtract 1/1000000 (burgi used adding 1/10000). Maybe that's why Burgi is mention in the video - napier work was less intutive and also his table started by 90 degree and each entry was one minute lower with corresponding sine of angle (maybe cosine and tangens too, i don't remeber).
@@MasterHigure if the focus is engineering then e and 1.0001 are still more useful than 10 lol.
@@tiranito2834 Not at all. Because you can eyeball 10-logarithms with a minimal amount of practice, way easier than you can eyeball e-logs or 1.0001-logs (for instance, even _without_ practice, I can tell at a glance that the 10-log of 3145 is about 3.5). The two other logarithms take a lot more practice, or alternatively, they require a separate step of some mental multiplication with a memorized constant. No, 10 is much more convenient for practical use.
That circular design was almost certainly inspired by Raymond Llull and his combinatory wheels
I have a suggestion to most everyone, to relearn logarithms. Everyone uses it all the time (correctly), but because of calculators, they don’t have a good grasp of it.
Maybe if the goal is to build more intuition about logarithms, it's good to mentally calculate floor(log x) i.e. the largest integer we can raise the base of the log to and still be less than x.
Can you give an example of that uses of logarithm? I can't see so clearly this commom use... thanks in advance!
@@r2d277 3blue1brown has at least 3 good videos about logs.
@@r2d277 I tried to tell you about some content but it got removed, twice 🤷♂️
@@r2d277 3blue1brown, log
We used 4-figure tables in highschool. Had logs, sin/cos/tan and antilog etc. Good memories.
what's antilog?
"This 400 year old book"
Proceeds to manhandle it for 8 minutes and 46 seconds
bro it's obviously a copy ;-; he wouldnt slice up an actual 400 year old book lmao
@@asdfasdf-dd9lk Off course it's a copy lmao. It was a joke. Myyyy days.
Excellent! We build upon the shoulders of those that went before us! How true in mathematics / philosophy. Well done.
so it's the logarithmic ruler, but in book form
in English it's also called the slide rule
@@norude in czech slide rule is what you call calipers in english :-)
Fantastic video! Just pulled out my grandfather's old slide rule (he was an architect). I'd love to get my hands on a copy of the original tables like you show in your video (I assume you printed out a copy for the video).
Where did you get that paper copy of the original book? Is there a good public domain scan of the original book available?
if you google the following (including the quotes) you should find it: "Bürgi, Jost:Aritmetische vnd Geometrische Progress Tabulen"
(note that the spelling "mistakes" are in the page you are trying to find, so don't feel tempted to correct 'vnd' to 'und' etc.)
Printed it myself from this pdf: bildsuche.digitale-sammlungen.de/index.html?c=viewer&bandnummer=bsb00082065&pimage=8&v=100&nav=&l=de
You can find literally anything by typing the name and .pdf at the end
@@michaeldamolsen pedantic note: at the time of original publication what we now think of as the letter V also served as our letter U. So it's only a spelling mistake from a modern perspective.
But you are quite right, you have to keep the archaic spelling or the search engine gets muddled...
@@puppergump4117 must try that...
Millionpounds.pdf
SaturdaysLottoNumbers.pdf
Naaaah, doesn't work ...
The last one nearly works, and you don't actually need the .pdf -- trouble is it gives you last Saturday's number which is not quite as useful
Volvelles are awesome! I have a collection of facsimiles that calculate everything from the day of the week and the positions of the stars to the hexachords along the circle of fifths. These little paper computers are phenomenal examples of human ingenuity. I could gush about them for hours.
Oh, I would love to acquire a copy of this book. SO BEAUTIFUL
The Space Shuttle was built with the slide rule
At school in the 70s we were taught how to use logarithms to make our own slide rule out of cardboard - a very easy process. Is that still done ?
If calculators have taken over then kids should have to at least breadboard out their own simple calculator imho ;-)
I still have my slide rule from my 70s maths classes, not sure I could use it now without some research.
in the 80s we still had giant slide rules on the walls in our maths class rooms, but we skipped the chapters on their use - I later accidentally "rediscovered the slide rule" playing with a few sheets of logarithmic paper, observing that twice the 1 to 10 bit amounted to the length of the 1 to 100 bit... and actually feeling quite dumb that I had to think about _why_ that was...
and I did not immediately connect my "discovery" (half a millennium too late, but still) to those giant wall slide rules, because they had multiple bars / scales and I had always assumed these things were way too complex for me to master
Well they now have smartphones, not just calculators, and with a download they can match the most sophisticated "scientific calculators". I'm glad to see the end of the tyranny of TI-86 in the US, but, we're getting to the point where a kid can just verbally ask an AI what the solution is, without any conception of calculations at all.
I studied elementary school from 1987 in Czechia, graduated in 2004. I know slide rule only because it was magical item that was among some pencils, drafting tools, compasses and rulers in a drawer of my table. We basically learned powers and logarithms as a given tool, we learned how to simplify equations and how to calculate interests. But I wasn't able to apply them correctly on some example with radioactive decay where exponents were too big for straightforward solution on calculator. That was at middle education (is it called high school in US?) Then I used them at university applied as exponential attenution, conversion to decibels and back and that's basically all.
Conversion of multiplication to addition and how to get square root was something I learned asking my mother about slide rule. Then recipe at uni was basically to convert stuff to decibels per unit lenght, mutliply by length, convert it back. Or voltage on capacitor vs time. University was actually when I used logs.
Briggs logarithms used to be called "Bürgi's logarithms" in Switzerland...
Thank you! It would be also interesting to see a more detailed analysis, of how many elementary operations you save using these tables. For instance, it takes time just to find the corresponding logarithm in the book, so it is not so obvious how strongly does it speeds up the process.
Fair point.
Let's immagine our self to multiply distinct numbers.
Number A by number B both with 10 digit.
We choose 10 times A, B.
In the usual way we must perform 100 multiplications and 90 addictions.
By using this method with
look up table we perform only 10 addiction.
We may conclude as well then although not perfect is less prone to mistakes.
My high school physics class was slide-rules and math tables. Even in college math tables were referenced as-much as the new handheld calculator.
So, logs. Fascinating.
Wow this reminds me the round device Dr. Strangelove used. Is it the same thing? Circular slide ruler?
Yes.
3:31 ... so far, i have seen nothing other than a simpler veraion of {,anti}log table.
Oh man i love multiplication.
Especially by 10^x where x is a positive number
My favorite rule is a*b+a*c=a(b+c) since the sum of b+c being equal to 10^|x| is non zero
Looks like a natural log table book. However, here we see the Importance of publishing critical works as open-source.
this is completely mind-boggling! imagine how much time they took to make that book!
This reminds me of the log tables I was taught to use along with a slide rule in 7th grade. Oh so many years ago.
If you lookup the methods for getting square roots before calculators in the west, you quickly discover... there's a reason the square-root symbol looks like the symbol used for long division (differentiated in look as though a "v" is fused to its initial forward-slanting line)...
It's not that it's a form of division so much as the signs are indicative of relation.
You made it so we don't have to see the Ad. Out of respect for your work, I sat through it. Thanks for sharing.
This disc later evolved into the Nipkow disc, which in turn became a famous TV price in the Netherlands.
Old guy here. I have a graphical table of logarithms-- read it like a slide rule; quick and easy. Never did care for the log-log slide rules that were the standard in the US; prefer Darmstadt rules. One formula that was always useful: log (a to the x power) = x log a. Also note that electronic calculators do not actually do the operations that are keyed in; they use electronic functions to simulate them. Because of that one can do a string of multiplications and divisions of a number, then reverse the sequence and come up with a number different from the original one. It was always difficult to hammer an understanding of significant figures into students' heads, but with the advent of electronic calculators (which display numbers to as many places as their displays will accommodate) it has become nearly impossible.
We learnt this in high school!
And learned to appreciate those before us who worked on this.
I didn’t
Man, Welch Labs is such a great youtuber. Definitely deserves more and more.
Well, lookup tables are used all over the place in high performance computations, so it did change mathematics and engineering forever. It could be argued that it was way ahead of its time I suppose.
just consider how difficult it used to be to calculate, and be happy it is so easy now.
My brain hurts at the thought of having to transcribe a copy of a book like that. Imagine the tedium of writing a whole page only to find a compounding error somewhere.
Man if he wasn't making clocks and focused on science - we'd have flying cars by now
no
We *have* flying cars, now. They're expensive as hell, and still need a pilot's license.
Right, but perhaps because he worked on clocks, I have a pair of watches with slide rule bezels.
Clockmaking was at the forefront of science and engineering at the time, the skills developed allow the invention of a whole plethora of scientific instruments, laid the groundwork for advances in astronomy and navigation, and gave birth to the machine tools that allowed for the scientific revolution.
Did you just trick me into learning logs?
Reminds me of how Nintendo64 and Super Mario 64 "calculated" sin(x) (and cos(x) by looking them up and how due to hardware reasons surprisingly efficent it was when compared to other alternative/more modern implementations. There should be a quite interesting video on that here on YT.
I wondering if the computer store the Burgi's table and do a computation using that rule, is the computer can compute faster than usual?
Just asking, is the "log" in "logarithm tables" the same "log" in "logbook" and "log in"?
No, it’s got a different origin.
Logarithm was coined by its inventor John Napier from Ancient Greek λόγος (lógos) meaning word and αριθμός (arithmós) meaning number.
Logbook referred to a book in which ships recorded their speed for navigation purposes. They got the name from the device used to record speed, the chip log. It was a piece of wood tied to a roll of string that was dropped into the water. The rate at which the string was unwound was used to measure speed.
Logbook now refers to a record of anything stored in written media.
The chip log got its name from the fact that the drag (the piece of wood that gets dragged by the water) is a simple piece of wood that is chipped off from a log.
Log in this sense comes either from Old Norse lóg/lág meaning fallen tree, which itself comes from the verb liggja, meaning to lie (on the ground/a surface); or it came from Norwegian låg, also meaning fallen tree.
“Log in” is a phrasal verb constructed from the fact that when one accesses a computer account, one is adding an entry into a log. Log in the sense of an append-only sequence of records written to file was probably derived from logbook by analogy. Note that logbook as in a record of a vessel’s progress or as a general written record was already being shortened to log long before computers.
Logarithm came first (John Napier introduced the term in New Latin as logarithmus) in 1614. Logbook was first attested in nautical records in the 1670s. Log in was first used in 1963 by the MIT Computation Center, in reference to people time-sharing their mainframes.
@@nathanoher4865 aww, pretty sad they don't share the same root
well at least we know navigators were using both logs quite a lot in the 17th century
A ship compass and ship’s log were used to track direction and speed. Then a traverse board was used to keep track of those values for the four hours of a watch. At the end of four hours, the data was entered into the logbook. Then started over for. The next four hour watch. (I read the Aubrey-Maturin series of books.)
A captain's log is commonly found in the chamber pot in a captain's cabin. A cabin boy would check daily if there was a log in it. If there was, he would empty the log, called a log out. He would then inform the ship''s doctor who would record it in his log book. This was the doctor's log.
A doctor's log is commonly found in the ...
@@nanamacapagal8342 Yep, I can’t imagine how much time it took to plot out routes without computers!
Me: "Hey, this looks like a fun video... WAIT. WHO made this???"
Welcome back! I have been telling all of my precalc students about your Imaginary Numbers Are Real series.
Holy crap! 2 videos within 2 weeks?!
What kind of person calculated all of those numbers. It must have been soul crushing work
What? All not to publish it????
I wonder what terminology we would have got if he had published first. Instead of a log function, would we have a red function?
Fun question! I don't know - but I do like the name red numbers more than logarithms.
Unlikely. You'd have a terminology crash with accounting, and as the video hints at, the mathematical functions being described have a *much* broader reach than a multiplication aid.
you said the power is calculated by adding row and column. and you also mentioned that burgie calculated total 23k and 27 numbers. but in the video what I can see is the column is 2.3lac. so can you clearify on that?
At 1:51, it doesnt map to 9063 but 90630
Good eye! Yeah I put a note about this in the description.
The progress of math and science was really held back by how often people kept things secret.
Welch Labs! Welcome back man!!
I'm afraid that, many of such significant work done today will be lost exactly due to the opposite reason. They'll get drowned in millions of publications per year, and get bogged down in impact factor and citation quagmire.
It already is a problem. It's hard to find fundamental in-depth information cause google always returns results which are similar to each other and wiki and basically mainstream. Maybe it's to reduce risk of "misinformation" or low quality information in search results. Imagine that just two weeks ago I had to derive how to extract center of rotation from transformation matrix - i was not able to google it, chatgpt said it's too complex problem. Yes, it needed to multiply three matrices by hand and solve three equations of three unknows having trigonometric functions - but extracting angle was easy so more like two equations without trigs. Another solution was to use some math library and extract eigenvectors. All that google found was the opposite - how to get matrix knowing center of rotation and angle.
And that was not only problem I had. Sometimes it disregards some search keywords and I was not able to find exact article on my blog. Maybe it filters spam, I don't know. Also it's very hard to find some negative yet informative articles on someones blog - once i could not find that some problem exists, then one guy randomly linked in-depth research about it in youtube video.
They should have gone over this in school or something to make it more interesting. Would be nice to see the progression of math over time.
Your videos on Logarithms have inspired me to create my own Napier tables.
wow, that is beyond genius. the Nobel prize for mathematics does not exist because it was not given to him. actually Nobel prize does not deserve him, in fact this book is MORE valuable THAN Nobel prize.
Excellent!
someone write an AI browser add-on that erases youtube vid titles and replaces them with accurate titles
this one:
"This book should have changed mathematics forever"
->
"Slide Rule"
very clever!😉
So are log rythems more, synthetic or we just didn't have the rapid commuting power to exploit them
The slide rule does the same thing more conveniently, but it didn't change mathematics "forever", either. Looks like an overblown title to me.
Mathematics IS FOREVER .♾️👍
As the last generation to use slide rules and tables, you don't understand the power of calculators and computers we now possess although I think hand graphing be it on linear, log, semilog, or polar graph paper and working with other graphical methods (nomographs, families of curves, stability plots, mixture phase, Mollier and other chemical and thermal parameter diagrams and other binary, ternary plots, etc.) gave one an intuition on creating functions to simplify calculations with correlations that can be used in computers.
Oh wow. So cool!
I never knew this. The is awesome.
Somewhat sad that these figures have previously been calculated. Would have like to take a stab at computing them by hand.
That being said, I appreciate the monumental efforts of past generations to accelerate our calculations. Leading us to future questions and constructs.
It took Napier 20 years to compute his tables. Do you really want to repeat that amount of work?
I feel like this sort of thing will eventually allow us to move around inside pi and find any number in it. Like GPS but for Pi. (PPS?)
I would argue that this is not “mathematics” but arithmetic. Math is closer to theory. This book is closer to practice. If you accept that, then I would argue that the book did indeed change arithmetic forever. This was one (as other comments have pointed out) of several systems leading up the logarithms.
Mathematics is to arithmetic as vehicle is to car. When I was a kid in the 50s in elementary school we had arithmetic class. Nowadays it is called mathematics class. No big deal.
This is kinda analogous to Laplace transform, no? The way LT takes away dealing with differentials by replacing w multiplication in š-space
"Should have changed mathematics forever" is a crazy thing to claim. He could have been the inventor of logarithms. A contemporary beat him to it. Meh....
To be fair, his method would likely have people widely using slide rules a fair bit earlier.
At 1: 51, it looks like the number from the book for 2.475 is 90,630 - but you wrote 9,063. Why is that? (9,063 seems to be the correct one according to my calculator).
Very nice video! A question : does anybody has any idea as to how the book was printed? It seems hand made, and I was wondering how one would obtain such a nice result
yes it was typeset by hand and printed in two colors -- this would have been about 130 years into the era of printing press
After 2 and 1/2 minutes my guess is that this will be about slide rules
at 1:50 its shown that the red number is 9063 , but shouldnt it be 90630? Was just wondering, idk if im wrong xD
Good eye! I put a note about this in the description.
Remarkable!
It sounds like these mathematicians (Burgi and Napier)
were maybe starting to….
Grasp at
what Karatsuba discovered after thinking about Kolmogorov’s opinion on the time complexity of the standard multiplication algorithm?
Could taking the reasoning behind Napier’s and Burgi’s work
make even better multiplication algorithms for computers?
Karatsuba
essentially improved upon the standard multiplication algorithm by creating the scheme where you do less actual multiplication and more addition
(which I guess saves time complexity since addition is foundational to multiplication, and, therefore, a less ‘complex’ operation, computationally?),
which seems to be what Burgi has done with his book of Red and Black numbers
(Not unlike Red-Black tree data structures in CS -
the Red-Black dichotomy pops up in a lot of unexpected places in Math and also Philosophy….)
Hmmm….
…
Wondering if the Red-Black paradigm shows up in many places because they were literally the two colors of ink available in the standard stylus/quill pen holder with two ink wells that sat on everybody's desk.
@@amitabho
That’s actually how Sedgwick et al did the Red-Black Trees at Xerox.
Lol.
Simply because -
yes, you’re right….
They just had Red and Black pens lying around….
And that was long after the age of quill pens….
(Ball-point pens in the 70s/80s, Lol)….
I guess
Charles Sanders Peirce
was just trying to be more dramatic about it in his essay on Probability Theory that was titled
“The Red and The Black”….
Also, a metal band called one of their songs
“The Red and The Black”
….
Anyway
…
Very interesting history.
I'm about to fix math and get my Nobel prize. Here's a hint - division by zero is defined as the null set - the set with no numbers, but, you can put any number in a division by zero and it checks out. Division by zero isn't the set with no numbers. It is the set with ALL numbers. That is why the universe has apparent motion - it reached a division by zero error and exploded.
Too bad there is no Nobel prize in math.
Haven't seen this channel in a while
OK. So you have what is essentially a table of logarithms. Welcome to the slide rule age.
In a couple of decades, perhaps.
Am I the only one thrown off by the fact tha the red number in the table doesn't line up with the red number in the written card? 160500 is what I see in the table but what is written is 16096. Is there a missing number place somewhere???
Great video
The title "this book should have changed mathematics forever" is clickbait. Logarithms did change mathematics forever.
Yes, but someone else's book created the change, not this one.
Interesting a program could be written to turn the wheel and find the answer seamlessly.
Why was 1.0001 chosen as the base?
I assume for high precision with integer powers.
The math works out for any basis, but 1.0001 is a good compromise between precision and a set of tables small enough to be publishable. In the real world, we rarely need even three digits of precision. Outside of the real world, I work with people who compute spacecraft trajectories. They need a lot more than three digits of precision. Fortunately, that wasn't much of a consideration 400 years ago.
Perhaps this number give the enough amount of reference vs effort to time ratio , imagine calculating all that by hand , that is really some good work.
He calculated powers rather than logarithms. To get a table you can calculate either. It is a lot easier to calculate powers than logs.
To get any precision by repeated multiplication of powers you need a number close to 1.
If he used base ten his table would tell us the black number 10 has the red number 1, 100 is 2, and so on.
The other thing is that having spotted that he wanted a number close to 1, having one further digit of 1 makes manual multiplication that much easier. Tedious but easy.
How many zeros? The more zeros before that final 1 the better the accuracy; but with two costs: the numbers take longer to calculate, because you need to do more multiplications, and the book becomes correspondingly thicker and more expensive to print.
I guess he thought that a precision of one in 10^5 was a sensible trade off
607009099 times 121
Basically a table of logarithms base 1.0001, I suppose he chose that base to simplify the operations to calculate the table, that are just shifts and additions. Very clever.
Nope, it is basically approximating e as a base. For example (1.0001)^10000 ~= 2.718... ~= e.
@@meowtheroflearning2320 No, e was found like 120 years later. But it's a nice "coincidence" that it appears as 10000th entry.
Mind.Blown.
I want a copy of this book that will fit into my shirt pocket
Please make a video on your story like what you do and were doing.
Ok I'm not a math wiz by any stretch of the imagination, and TBH I never understood log tables but with your explanation here I'm beginning to get a gist of how and why they were used.
At about 2:28 you stated the answer with 3 decimal places. Did you limit to 3 decimal places because that was the max number of decimals to the right of the decimal point????
Then you said with an extra interpolation step you could get accuracy out to 7 digits. Can you explain that??? I'm lost as to what you mean, lol I got a flash light ... but still can't find my own arse ..... hmmmm I guess my mom was right.
You basically interploate between values on the table to get more accuracy.
The circular tool aside, isn't this just a type of logarithmic table ?
Ah having now got to the end of the video I see this is basically admitted to.