Reinventing the magic log wheel: How was this missed for 400 years?

[@Mathologer]

The entries so far in the coding challenge in this video: Cristian Merighi: js.pacem.it/2d/circular-slide-ruler Mike Wessler: phoenixave.com/crule Liam Applebe: tiusic.com/slide_rule.html Juan Ignacio Almenara Ortiz: www.desmos.com/calculator/tul8psjh32 (demos) Thank you to all of you who contributed a modular times table app in response to the coding challenge announced in the last video (the Tesla video). All the apps that I am aware of are listed in my comment pinned to the top of the comment section of that video. The winner of the draw for that coding challenge is Mathis Aaserud. Congratulation! Today’s coding competition comes at the very end of this video. As usual everybody who contributes and app automatically enters into a draw for one of my and Marty’s books :) Here is my wishlist for this app: It would be great if one of you could make a nice online circular slide rule app. Possible features: 1. input fields for numbers that are to be multiplied or divided and then the automatic execution of the slide rule actions with the scales spinning as in this video; 2. infinite precision by making it possible to zoom in on the scales and have the scales refine dynamically as we zoom in; 3. tick box for squaring, as we rotate the two inputs for multiplying are kept the same; 4. incorporation of other look-up scales or even log-log scales :) Also, as usual, lots of background information and links in the description of this video. Enjoy :)

[@Bizarro_na_chapa]

Laughted a lot with the t shirt

[@VAXHeadroom]

I think instead of being a wheel, it's really the cross section of a log. I'll see myself out...

[@madhurgupta2249]

Thank you sir for this explanation

[@madhurgupta2249]

I am happy that video was not that big (I mean not as long as 40min) because it's not easy for me to find such time in which I can see your videos 😊😊

[@Mathologer]

@@VAXHeadroom :)

[@scorinth]

I have literally used a slide rule for a quiz in engineering school around 2014. The battery in my calculator died and I didn't have a backup. But the rules of the class said I could use any device for calculation with the professor's permission. So I dug the slide rule out of my bag, held it aloft and said "Excuse me, professor! May I use this for the quiz?" He - having grown up in the age when digital calculators were still uncommon, at best - laughed out loud and said "If you know how to use it, sure!" He was certainly proud that I aced the quiz with that slide rule.

[@Mathologer]

That's great, thanks for sharing that story :)

[@ozboomer_au]

Similar thing for me.. I first started using slide rules when I was about 10 years old, right before the time when electronic pocket calculators appeared (early 1970s, when the calculators cost $100s for a simple arithmetic model). I frequently used circular slide rules as they were smaller/easier to use... but 'graduated' to a Japanese Hemi in the mid '70s (when we were still being taught to use log books at school). By the end of the '70s, I had a 'scientific' calculator but kept the slide rule around and managed to find a 5-inch linear model. Whilst I was completing my Civil Engineering in the early '80s, some folks laughed at me when I used a W&G 'wooden' slide rule... but their laughing went quiet when I was doing stadia reductions (measuring distances with a theodolite) faster than they could as the slide rule had special scales for doing the triangulation, whilst they were still struggling to work out the trig calculations even with the (non-programmable) calculators. As always, it's always a good idea to have a number of tools available... and to use the best one for the job at hand. ...Oh.. and I still have all my slide rules (about 4 or 5, including an 'E6B' flight computer [a circular slide rule] that my Dad used when he was a private pilot, flying Cessnas and Pipers around Melbourne, .au).

[@frankharr9466]

He should have been proud. Good man!

[@fk319fk]

I was sad to see when calculators were allowed in school. From what I see the hardest part of math is not knowing that 2 x 3 = 6, but what is 2e0 x 3e0 equal to? I had a Chemistry teacher harp on using units. "If your units do not make sense, your answer is very likely wrong." (I have my dad's slide rule, and it does trig and I have the schools, as they gave them away.)

[@calyodelphi124]

That is epic. I love it. :D

[@unvergebeneid]

At first I was sceptical as there is no "real magic". But then I saw an infinitely stretchable rubber band and now I'm convinced magic is real ;D

[@Mathologer]

:)

[@archivist17]

* screams in Hooke

[@Mathologer]

@@archivist17 Actually, Hooke is irrelevant in the discussion of this idea :)

[@colleenforrest7936]

Or, magic is reel :)

[@Valenorious]

@@Mathologer Not exactly, working out that law is all about multiplication and division.

[@RGregElder]

Like others, I'm an old engineer who used a slide rule during my exams at university: cheap calculators hadn't been invented yet. (I still have my slide rules.) I also noted that log graph paper has the scales exactly the same as that of a slide rule. So when I was travelling in the UK after graduation (and before calculators and phones), I cut out two of the axes on the graph paper and taped them to cardboard at the proper point so that I was permanently calculating the conversion between Canadian dollars and pounds Sterling. While shopping, I could easily convert an item's cost to familiar currrency.

[@trumanburbank6899]

My dad was a property assessor, and had many well made straight and circular slide rules. My grandpa was a brick mason, but never used one. I showed him how they work, and he thought it was interesting. It was in the mid-70's. I then showed my grandpa my six-place logarithm table. It's an Austrian book, from 1924 by Dr. Ludwig Schrön. The logarithms are to six figures, and the column/row lookup goes to 5 (but you can use proportions between answers to get a sixth). I showed him how to turn multiplication into addition, and division into subtraction, and how to back-reference to find the answer. He was quite amazed by the whole process. I gave him my table for a while as he was quite fascinated and spent time using it on many different occasions.

[@Mathologer]

Thank you very much for sharing this story with the rest of us :)

[@docjohnson2874]

Great story....used my dad's slide rule from engineering school (Armor Inst. now U of Chicago), to get thru organic/quant. chemistry....still have it along with its 1960's metal counterpart.....great history as usual.

[@skwest]

I still have all of my various slide rules from the 60s, including a circular slide rule like this one.

[@mymatemartin]

Aussie here, born 1962. I too remember those books of log tables we used in the 70's in maths class. To this day I've never quite understood what we were trying to achieve with them but strangely this video helped make sense of it for me. Now, if I could just find my 1970s tyrannical maths teacher and tell him that I finally understand, I think he'd be proud. 😁

[@trizgo_]

the rubber band explanation for logs clicks so naturally, in a way that few other explanations do!

[@Mathologer]

Yes, was definitely love at first sight as far as I am concerned :)

[@iveharzing]

It also visually shows that log(0) doesn't make any sense, which is cool!

[@Mathologer]

@@iveharzing Yes, very cool :)

[@krisrhodes5180]

Shows how different people are different, I found the whole concept impossibly confusing, I think because I just don't have the intuitions about rubber band stretching, and related visualizations, that yall do.

[@Noughtgate]

Correct me if im wrong but id assume the rubber band must be specifically engineered for this, intuition tells me the scale will be "asymptotic"? If you can call it that, not logarithmic. Rubber just feels wrong somehow, like it should be something with even more elasticity, like gum

[@robspiess]

Super interesting! Slight error: @6:11 you go to 3.18 on the diagram (each sub-sub-tick is .02).

[@bobosims1848]

Yup, I was going to say something about that, but clearly there's no need. :)

[@listoe521]

This is especially confusing because 0.318 is the reciprocal of pi. So after they correct the error and rotate both scales locked together, we again see 1 on one scale against 3.18 on the other scale. Meaning 3.14*3.18 = 10.

[@jonathanrichards593]

If you were brought up using a slide rule this just jumps out at you!

[@NoIce33]

Well, better than 3.2.

[@koud29]

@@NoIce33 guess he's not from Indiana

[@timothykeech7394]

As a slide rule user, engineering in the 1960s, I found this a fascinating explanation but the graphics are absolutely brilliant, Even knowing what was going to be shown, I was mesmerised by the animation. So beautifully done.

[@Mathologer]

Glad you liked the video, and thank you very much for saying so :) Makes my day.

[@standing_around]

+

[@78tag]

Thank you, At 71 years of age I thought I would never see anything about slide rules again. When the video started I thought to myself - this could be about slide rules. It was surprisingly satisfying that my suspicion was correct. When I was a senior in HS calculators were just becoming available. True engineers keeping slide rules on hand is like current day sailors keeping and using sextants at sea. Young people are going to learn the hard way that all of this electronic technology is going to fail them at some point.

[@stuehleruecker]

Slide rules are not so out of fashion like most think. Several weeks before i saw a technician in a test center uses a slide rule on calculate some chemical rations to mix together. Asked her on it and she explained that its safer. The slide rule (made out of steel) you can throw into the autoclav for sterilization but calculator or phone not. And no battery can be empty. Sterilization chemical are very strong and calculator and phone display surfaces are destroyed if use them. Slide rule is the most esiest and safe option. But its expensive,. one cheap calculator maybe is 15 euro and a slide rule of medical steel about 250 euro. And you need many cause they go 8 hours into the autoclav after each use. Many test places buy cheap calculators and burn them. It seem cheaper but its not and waste resources.

[@Mathologer]

Interesting. I know of a Japanese company that still produces all sorts of different specialised circular slide rules :)

[@iwanttwoscoops]

@@vlc-cosplayer but then you're emitting contaminated information and sound waves :p don't wanna do that to the other techs!

[@ragnkja]

If the lab plans to operate for at least 17 days, the slide rules are cheaper.

[@stuehleruecker]

@@vlc-cosplayer No screaming! That only spread the deseases further. And only one person each separate lab room for reduce danger. The manually calculcation is only for safety, the mix machines do it automatically, but it could be broken.

[@EebstertheGreat]

I'm surprised nobody makes digital calculators that can survive autoclaving, but maybe the market is just too small.

[@lftkryo]

This log wheel also provides a nice illustration of Benford's Law, the "observation that in many real-life sets of numerical data, the leading digit is likely to be small" (wikipedia). If you pick a random point around the perimeter of the wheel, it is very likely that the number it represents starts with the digit 1, less likely that it starts with 2, etc. all the way up to 9 which is very unlikely. This is independent of the magnitude (decimal shift).

[@jasonrubik]

Hi Chipophone guy ! I love everything you have done, so keep up the awesome work ! btw, that 256B demo is ridiculous !

[@macronencer]

Yes! This is a really good observation, thank you.

[@dannynewton8752]

Bedford's Law is a clever way of catching people "Pencil whipping" their expense accounts. You have to have a large number of entries, they should be several orders of magnitude in size and they have to be based on natural data. If someone is adding ten percent to a sale, it would stretch the cost unnaturally. People making up data are going to have a lot of problems explaining why their data is not random in the last two digits, which is another way to check data. I have always wondered if the changes in Global Warming data to make up for bad or missing data makes the data be non-conformal with Bedford's Law.

[@BruceFairlie]

I am a retired Engineer old enough to have been trained in the art of using slide rules. I have owned several, including one with a Pi folded scale that avoided "falling off the end". However, my most treasured slide rule is one I inherited from my father who used it when he worked as an accountant. It is a cylindrical slide rule with the scale wrapped around two one inch diameter concentric cylinders. The upper cylinder slides within the bottom one and they are overridden by a "cursor" cylinder. On the upper cylinder there were two cycles of 1 to 10, on the lower cylinder just one. The cycles consist of 22.5 turns around the cylinders giving an equivalent linear scale length of nearly 1.8 metres. Given this length, four significant figures was easy and five could be reliably interpolated. Interesting that an accountant would only need answers to five significant figures!

[@Mathologer]

Thanks for sharing this with the rest of us :) Something like this? en.wikipedia.org/wiki/Fuller_calculator

[@mydogbrian4814]

@@Mathologer *Scientific American* had an article on the barrel slide rule some 30 years ago. I never forgot it. Still have my jumbo yellow sliding one. Which was replaced by a HP red LED display mid term. So I didnt get proficient with it. Bur I can still do multiplications. Lol.

[@Allen2]

@@Mathologer My father, Ph.D. engineering professor, had several slide rules and circular slide rules, plus one cylindrical slide rule with a bottom and no top so that it could sit atop the desk and hold pencils and pens. It was more of a curiosity than a daily use item because it's easier to carry a flat slide rule in a briefcase or notebook pouch. The best slide rule came in a leather case with a clip to attach to a belt or belt loop on pants.

[@AlanTheBeast100]

Pilot's computer (E-6B) - which has many other functions as well - including an analog vector adder. When using a slide rule, always estimate the value in your head first to avoid factor of 10 assumptions.

[@Mathologer]

Yes, well even real physical slide rules have other functions built into them, mostly lookup tables though :)

[@AlanTheBeast100]

@@Mathologer A cool thing one can do on an E-6B, is "enter" winds aloft for several altitudes simultaneously (direction and speed), thus one can optimally choose the altitude of flight for time and/or fuel. In a mainly head/tailwind case, isn't worth the effort (unless the winds are very strong), but when crosswind, then there may be surprising options.... (entry is a pencil mark on the vector plate).

[@FlyNAA]

@@Mathologer the E-6B's he's talking about are physical. And I don't know about today, but it can say that 10-20 years ago when I was active in flight training (at both ends) they were sold and taught widely, but starting to be on their way out.

[@Mathologer]

@@FlyNAA I recently bought a new one. Very ingenious :)

[@FlyNAA]

@@Mathologer ah, I misread your post

[@dbf1dware]

My dad worked with very early computers in the late 60s and into the 70s. He always had a slide-rule with him. He showed me how to use it, but I was still a bit too young to get it. This is the first time I have ever seen an explanation of how/why it works. Fantastic!

[@johne7100]

I used my maternal uncle's slide rule at school: 15" boxwood with engraved ivory scales. I believe it goes back to the early 20th or late 19th century. The case was made in Harland & Wollf's shipyard in Belfast, in "repurposed" steel, probably in the early 1920's. I still have it, but I haven't used it for ~60 years.

[@theevermind]

IIRC, a round slide rule is shown in Hunt for Red October when the navigator of the Red October is calculating the timing for the turns through the undersea canyons.

[@JustPassingThrough404]

This was super enjoyable. Loved the bit of history along with a fantastically clear explanation with great graphics. Just great fun.

[@carpemkarzi]

Very cool, been a while since I’ve broken out let alone used my old slide ruler. Also I want that shirt.

[@jayfelty5228]

I was intrigued with the title of this video, 'Magic Log Wheel'. I have been recently diving into the Antikythera Mechanism, and was seeing if this was another mystery. I instantly recognized your devise; 'log' for logarithm; a circular slide-rule. I used two of these in the building industry and construction from 1968 until around 1987 until I relied entirely on my hp-15c. I still have both rules, a pocket version and a ten-inch diameter. Thank you for making my day.

[@stupiduser6646]

I have a Jeppesen Computer from my grandfather who was a pilot since the 1930s. It was/is used to determine airspeed for a leg of your flight to find your way long a map heading. I just pulled it out to learn more about it after watching this video. It appears to be capable of these calculations too. Thanks.

[@Mathologer]

The computer that Spock manipulates in the screenshot that I show in the video is a Jeppesen Computer :)

[@jeojavi]

15:19 Am I the only one who thinks it's funny how he dodges the triangle? By the way, it was a great video, very interesting!

[@sergniko]

No, you are not.

[@PC_Simo]

Nope, you’re not alone. I agree on that. 😅

[@ronaldremmerswaal]

Nice video, I was waiting for description from the point of view of an ODE. Let x(a) denote the position of a point x after the wheel has been rotated by an angle a. And assume that 0 < x(a) < L, where the rubber band has initial length L. Then we can impose that the position of x, relative to the rubber band, is invariant under stretching: x/L = (x+dx)/(L + da R), where dx, da are infinitesimal changes in the position and angle, and R denotes the wheel radius. Some simplification (omitting higher order terms) yields dx/da = x R / L. Hence x(a) = x(0) exp(a R / L). You can then define the length of the remaining rubber band l(a) (length of the part of the rubber band which is not yet on the wheel, after rotating it by an angle a) as the position x(0) for which x(a) = L. Hence L = l(a) exp(a R / L) which implies that l(a) = L exp(-a R / L). Then requiring that l(2 pi) = L / 10 yields R = (L/2 pi) ln 10, and therefore l(a) = 10^(-a/2 pi) L. This also shows that the wheel has circumference of 2 pi R = L ln 10. Anyways, this approach seems needlessly complicated compared to the nice visual explanations.

[@nemoyatpeace]

My dad gave me his circular slide rule that he got back when he was in university. It was produced in 1957. Really cool for me to have and play with, but I love the rubber band explanation!

[@decvoid261]

I was fortunate enough to hit that sweet spot in educational timing that included manual and mental calculation, log tables, slide rules (the main focus in the 60's), mechanical calculators of various sorts, electronic calculators, and some building sized computers that could not even compete with a smart watch. A major part of side rule use was the quick mental calculation of what you would approximately expect the answer to be, then you would know where the decimal point went.

[@timothykeech7394]

Me too - exactly similar. Estimation and determination of the order of the calculation result was an essential skill which has left me with a facility for numbers which I would probably never have acquired otherwise.

[@cyber746]

Proof of the length of the wheel being ln(10) : When the number x reaches the wheel, the rubber band has stretched by a factor 1/x (we consider that the rubber band is numbered from 0 to 1 like in the video). If you now wind the band just a bit more so that you reach x+dx (with dx infinitesimal), then the length added on the wheel is dl=a(x)dx. So the total length of the wheel is the integral of dl for x=0.1 to 1 that is int(dx/x, x=0.1..1), which is equal to ln(10).

[@Mathologer]

Great! Now I just need somebody else who programs the infinite precision circular slide rule :)

[@jonathanrichards593]

@@Mathologer I haven't the coding skills to attempt the challenge, but, just from first principles, isn't infinite precision impossible with a finite state digital computer? Also, it's going to be really hard to prove: who is going to watch infinitely long as we zoom in on the square root of two?

[@victorclaytonbarnett2959]

What happens when it's numbered from 0 to 10?

[@cyber746]

@@victorclaytonbarnett2959 Then, the wheel will have a circumference of 10*ln(10), as you scale the complete picture by a factor 10. This factor ln(10) is the ratio of the circumference of the wheel to the length of the rubber band.

[@victorclaytonbarnett2959]

@@cyber746 yes, I see. You would need to have the appropriate units in your integral.

[@garynagy6479]

I truly want to thank you... This has brought so much clarity, and it's all very natural... And that's the true "key". God bless you

[@justinjustin7224]

My favorite thing about this rubberband stretching construction is that it doesn't matter how big the wheel is or what numerical base you want to use. Simply take a wheel of any size, stretch the band as done in the video for 1 revolution, mark the point on the band where the revolution is completed, unravel and add evenly spaced markings for whatever desired numerical base, then rewrap the rubberband around the wheel.

[@erawanpencil]

This is confusing. @17:20 the original creator says "For this, the circle's circumference is ln(10) times the initial length of the rubber band" which I thought implied that the circumference could have been something else, or that he had to specifically make it a certain size relative to the original straight rubber band. But what you're saying is that the stretchiness of the rubber band and the initial choice of the number of ticks on it, by themselves, automatically created this logarithmic property? Are radix/number base and logarithm somehow unified in this contraption?

[@justinjustin7224]

​@@erawanpencilhmm... It's been a while since I've thought about this... Let's see, the amount of material left in the band decreases exponentially with the wrapping, which is what gives rise to the logarithmic scale... The units in the construction I outlined are proportional to the exponential stretching and the size of the circle... But that doesn't implicitly tie the amount of stretching of the band to the logarithmic scale of the base... That seems likely to be where I would have made a mistake if my previous conclusion is incorrect, in which case you'd then have to find a way to trim the rubber band down to the correct scale... Decided to sit and do some math, and now after playing around with the equation L*ln(b)=C a bit, I feel safe in saying that I was wrong before. Though, there are some relatively easy slide rule bases to construct without dedicated measuring equipment; base 6 for example requires a band that is ~1.75 the diameter(D), or base 2 which has a band length between 4.5*D

[@lesmaybury793]

Marvelous 😃. My Dad bought me a slide rule when I started my Engineering apprenticeship in the 1960s. A Faber Castell. He was an Aircraft designer so knew his slide rules. I used it continuously until calculators came of age in the 1970s. I still have it. Thanks for this great video.

[@yinq5384]

Great video as always! 13:42 Let's focus on the label "1" on the rubber band. Initially it's at the end of the straight part of the rubber band. The original rubber band is 1 unit long. Assume the radius of the wheel is R. Denote the angle of rotation of the label "1" by x (initially x=0). For any given angle x, there is a label at the end of the straight part of the rubber band. Denote it by f(x). We have f(0) = 1 and f(2π) = 0.1. For any x, consider a tiny rotation so that x becomes x+Δx. The length of the straight part of the rubber band decreases by RΔx. The new label f(x+Δx) = f(x)(1-RΔx). Take the limit as Δx->0 and we have a differential equation df/dx = -Rf, solve and we have ln(f) = -Rx + C. From the values f(0) and f(2π) we know C=0 and 2πR=10.

[@jessechisholm4842]

In the early 1970s, my high school disallowed the use of pocket calculators in class. I got permission from the school, then purchased dozens of circular "slide" rules. Re-sold them to students and taught classes on how to use them.

[@vert4836]

It is a good day when Mathologer uploads

[@REXOB9]

Great video, thanks. The rubber band/wheel analogy makes it clear. My class at my university was the last to be taught how to use a slide rule in introductory physics. Texas Instruments came out with an affordable scientific calculator, and slide rules went into the drawer.

[@AlRoderick]

When I was taught about scientific notation as a schoolkid the slide rule never entered into the picture as to why it was useful, but one of the great things about it is that it removes a lot of the potential for error from the "moving the decimal point" part. By forcing the mantissa (the part that's not ten to the something) to be between 10 and -10 you get a problem that's easy to work out with a slide rule (or if you're fancy a log lookup table) and the significand (the part that's ten to the something) becomes simple addition of the exponents. This is also how floating point works in computer logic, that's just a clever binary form of scientific notation. Multiplying in a digital computer relies on repeated addition plus some bit shifts to double and half the numbers and the size of values has a defined limit in terms of number of bits/significant figures.

[@LeeClemmer]

I remember some of these words.

[@LeeClemmer]

A mantissa sounds like some female insect that will bite off the head of a male...

[@AlRoderick]

@@LeeClemmer And the significand sounds like an ampersand that's like really important. *&*.

[@ElectricShadowsZone]

Thanks for your interesting video. I remember making a cardboard version of Napier's Bones when I was kid. These were used to multiply and divide numbers but were not based on logarithms. Also when I started my engineering degree in 1974 I had to use a slide rule for basic calculations. A few months later the first handheld calculators appeared. My first one (Rockwell Unicom) cost me several weeks of my engineering trainee pay. A year later I bought my first scientific programmable calculator (HP-25). That was truly the end of side rules and log and trig tables for me.

[@peterflom6878]

When the rubber band snaps, do you get complex numbers? Maybe surreal! Great stuff as usual

[@Mathologer]

Believe it or not but there are actually slide rules for complex numbers demonstrations.wolfram.com/ComplexSlideRule/

[@xyzct]

No, you get topology.

[@gcewing]

When the rubber band snaps you get the so-called "broken numbers", which were studied in 1763 by mathematicians working in a causally-disconnected region of spacetime, so unfortunately we will never find out what they discovered.

[@frankharr9466]

@@Mathologer NO! That is so cool!

[@santerisatama5409]

@@gcewing Hmm. Finnish for 'fractions' aka 'factured numbers' translates also 'broken numbers'. But I get your meaning. What actually happened was that the region dropped in a quantum hole, and all that remained was hypercausally everwhen-connected Duration of very stretchy rubber balloon surface, that strubbornly hides away if anyone mentions "topolology".

[@timbushell8640]

UK Senior school, back in the day, for ages 11 to 16 y.o., week one in maths class was to use log graph paper to make a cardboard linear slide rule... 8 years later I then got my first Casio 'green' numbered calculator... 4 years later both a new slide rule and a Casio "BASIC" programmable calculator... the via CPM into DOS into Windoze via main frames, skipping most of the HP reverse Polish, etc., etc. But if you make one, and get the real basics of 'log' tables (6 digit ones, not those with 4 digit training wheels) engineering is just so much easier. : ))))) Many thanks for this content, excellent new 'stuff' and memory revivals.

[@kenhaley4]

My high school math teacher (this was back in the 60's; I'm 75 now) made and marketed a very nice circular slide rule. His name was Stanley Arlton. I wish I still had mine.

[@Mathologer]

Well you can still buy them, so not all is lost :)

[@pjaj43]

As an R&D engineer who retired some 20 years ago, I grew up using slide rules. They are still in the cupboard behind me. The watch on my wrist, a Citizen eco-drive (light powered and resets every day from radio time signals) has a rotary slide rule in its bezel. But your mention of the introduction of hand held calculators. reminds me of the first ones we saw. It must have been in the mid 70s. The chief engineer acquired 2 or 3 and handed them out in the lab for us to try. Needless to say, as soon as his back was turned I wanted to know what was inside, so out came the screwdriver. This was a very early model, membrane keyboards were a thing of the future. As soon as I separated the two halves of the case the keyboard disintegrated into its component parts, propelled by the springs. Much to the considerable mirth of my fellow engineers. It took me the best part of half an hour to get it back together and working again.

[@Mathologer]

Thank you for sharing this story with the rest of us :)

[@mathphysicsnerd]

Once more a greatly enjoyable introduction to a topic that is tangentially relevant to a lot of my mathematical interests but I knew little about. I can't wait to watch what you Mathologerize next (maybe some simplification of Mihăilescu's theorem? A big ask, I know)

[@Mathologer]

So easy to state and yet such an incredible killer to prove ... I think I pass :)

[@mathphysicsnerd]

@@Mathologer We'd definitely need to have some crazy visual representations to give it a good try-even the 2005 proof is hard for me to understand. Still, someday I hope it'll be possible

[@justincase5272]

Fascinating. As a U.S. Air Force navigator for two decades, I was blessed to learn my navigational skills in the analog days, flying all over planet Earth using a "Computer, Air Navigation, Dead Reckoning Type MB-4" circular slide rule aka "whiz wheel," pencil, paper, H.O. 249 Volumes and the Air Almanac for celestial navigation, and a number of onboard systems such as directional gyros, radar, radar altimeter, doppler radar for groundspeed and drift, outside air temperature gauge, indicated air speed gauge, and the knowledge to put all the pieces of the puzzle together to arrive at a reasonably accurate picture of where were were at all times. None of this, however, would have been possible without the Whiz Wheel, that circular slide rule issues to all pilots and navigators back in the day (probably still today). Whatever you do, don't ever call it an "E-6B!" That is only ONE of MANY different designs used by pilots and navigators throughout the history of aviation. Furthermore, contrary to popular misconception, including errant claims on Wikipedia which I have repeatedly attempted to correct, but to no avail, U.S. Navy Ensign Philip Dalton did NOT invent the E-6B! Instead, here's a short list of his many designs: 1931: Prototype Plotting Board, as directed by the USS Northampton's Scouting Squadron Commander 1933: Dalton Aircraft Plotting Board VC-2, as released by the U.S. Navy Hydrographic Office - became ubiquitous throughout the U.S. Navy 1934: Dalton Aerial Dead Reckoning Slide Rule Model B aka Dalton Aircraft Navigational Computer Mark VII , as referenced by Weems in his pivotal volume, Air Navigation, along with detailed instructions for its use. 1936: The U.S. Army purchased contractual rights to the and re-named it the Dalton E-6B Thus, we can see that it was the U.S. Army who named it the E-6B, while the model that Dalton actually invented was called the "Dalton Aerial Dead Reckoning Slide Rule Model B, which also carried the designation “Dalton Aircraft Navigational Computer Mark VII” showing altimeter correction scale on the slide rule calculator, patent number and a copyright for 1934." All claims that Dalton invented the "E-6B" are not merely incorrect, they are patently FALSE.Fascinating. As a U.S. Air Force navigator for two decades, I was blessed to learn my navigational skills in the analog days, flying all over planet Earth using a "Computer, Air Navigation, Dead Reckoning Type MB-4" circular slide rule aka "whiz wheel," pencil, paper, H.O. 249 Volumes and the Air Almanac for celestial navigation, and a number of onboard systems such as directional gyros, radar, radar altimeter, doppler radar for groundspeed and drift, outside air temperature gauge, indicated air speed gauge, and the knowledge to put all the pieces of the puzzle together to arrive at a reasonably accurate picture of where were were at all times. None of this, however, would have been possible without the Whiz Wheel, that circular slide rule issues to all pilots and navigators back in the day (probably still today). Whatever you do, don't ever call it an "E-6B!" That is only ONE of MANY different designs used by pilots and navigators throughout the history of aviation. Furthermore, contrary to popular misconception, including errant claims on Wikipedia which I have repeatedly attempted to correct, but to no avail, U.S. Navy Ensign Philip Dalton did NOT invent the E-6B! Instead, here's a short list of his many designs: 1931: Prototype Plotting Board, as directed by the USS Northampton's Scouting Squadron Commander 1933: Dalton Aircraft Plotting Board VC-2, as released by the U.S. Navy Hydrographic Office - became ubiquitous throughout the U.S. Navy 1934: Dalton Aerial Dead Reckoning Slide Rule Model B aka Dalton Aircraft Navigational Computer Mark VII , as referenced by Weems in his pivotal volume, Air Navigation, along with detailed instructions for its use. 1936: The U.S. Army purchased contractual rights to the and re-named it the Dalton E-6B Thus, we can see that it was the U.S. Army who named it the E-6B, while the model that Dalton actually invented was called the "Dalton Aerial Dead Reckoning Slide Rule Model B, which also carried the designation “Dalton Aircraft Navigational Computer Mark VII” showing altimeter correction scale on the slide rule calculator, patent number and a copyright for 1934." All claims that Dalton invented the "E-6B" are not merely incorrect, they are patently FALSE.

[@givenfirstnamefamilyfirstn3935]

In UK and Ireland a Pooleys pilot supply company’s computer is used. CRP-1 for PPL and their big CRP-5 for ATPL nav exams. (The small one would have been perfectly adequate).

[@justpaulo]

13:42 (here is my attempt, but I'm no math demon😊): 1) If at the beginning the rubber band as length = 1, then the length(L) of ⅟₁₀ will have a length L₀ = 1/10. Now we start rotating the wheel (w/ radius r), the rubber band stretches and L increases... 2) Say at some point in the rotation L = L₁. If we rotate the wheel by a very small amount Δθ, the new length L₂ will be L₁ scaled by the factor ≈ (1+r∙Δθ)/1 = 1+r∙Δθ (the "1" comes from the fact that wall to wheel distance is always 1) 3) Given #2 one can therefore write L₂ ≈ L₁ (1+r∙Δθ) ⇒ L₂ - L₁ ≈ L₁∙r∙Δθ ⇒ ΔL ≈ L₁∙r∙Δθ. In the limit, when Δθ → 0, we have: dL = L∙r∙dθ 4) dL = L∙r∙dθ is a separable differential equation that can be solved by separating the L's and θ's and integrating: dL = L∙r∙dθ ⇒ dL/L = r∙dθ ⇒ ∫ dL/L = ∫ r∙dθ ⇒ ln(L) = r∙θ + C (C is the const. of integration) 5) When θ = 0 then L = L₀ = 1/10. That means that C = ln(1/10) ⇒ C = -ln(10). So the final solution for L (the length of the original ⅟₁₀) as the wheel turns is: ln(L) = r∙θ - ln(10) ⇒ L = ⅟₁₀ ∙exp( r∙θ ) 6) Finally, when θ = 2π we want L = 1. Plugin this in the equation above we get: ln(1) = r∙2π - ln(10) ⇒ 0 = 2π∙r - ln(10) ⇒ 2π∙r = ln(10) Q.E.D.

[@Mathologer]

demon enough :)

[@svensorensen7693]

Phenominal, and you explained it all before mentioning logarithms (until the end)! Funnily enough, I've been thinking of getting a slide rule to have around. Maybe now I'll get two, one straight, one circular!

[@janstunnenberg6316]

I'm proud to possess a magic log wheel since I was about 17yo. Now I'm 70. Great presentation!

[@Mathologer]

Yes, as far as circular slide rules are concerned, I did read somewhere that the first physical slide rule was in fact a circular one. What's new about all this is the way the logarithmic scale can be generated using this neat rubber band idea that I highlight in this video :)

[@alaindupont9633]

I've been using A LOT an Aristo rule first and later a Nestler rule when I was in a technical highschool (chemistry) in the end of the sixties. I used it not only to make multiplications but also to calculate logs and exponents and sines, cosines.... And, on the two main strips were also marked the special numbers like PI, e, .... The second 'magic trick' was the LL strip on the rule - the LogLog strip. It was so easy to calculate X to the power Y, or log or ln of X Of course, the accuracy doen't go further than the 2nd or 3rd digit, but it was so fast compared to the hard way (using paper log tables, sine tables ...) What a wonderfull thema to make a second part to that very interesting video :-) Standing ovation for all your videos !

[@ILikeEpicurus]

Almost number of the beast. In Dutch we called the slide rule a “gokstok” (gamble stick). In English this device could be the “wheel of fortune”.

[@AlanTheBeast100]

Many engineer's fortune's depended on sliderules. A famous book is "Sliderule" by Neville Shute - about the R100/R101 airship program in England.

[@genseek00]

Goeiendag collega

[@joepike1972]

right, it was the first thing I checked; 665.9960876. (rounding to fit on a t-shirt).

[@jeffgray9008]

25.8070 is a better approximation.

[@ILikeEpicurus]

@@jeffgray9008 correct, but doesn’t look so good on the T-shirt

[@joeyhemlock]

This is the best description of a slide rule that I've ever seen, and I appreciate the history lesson, too.

[@BobWestWA8YCD]

I had a Post Versalog (10") linear slide rule in junior high school that my Dad gave me. I didn't see a circular slide rule until my senior year in high school when we bulk ordered some for our Physics class. It was a cheap little plastic thing but I had fun with it. It had a vinyl slipcase with "FZIX IS PHUN" stamped on it.

[@michaelbishton9439]

I'm terrible at math, so I am all the more amazed by this explanation I can understand. almost to the end. This is remarkable instruction.

[@SpiralDiving]

Pilots are still examined on using an e6b circular calculator :) And I also once used a cylindrical slide rule that was more accurate (to 4 sig. digits compared to 3 for an ~11" rule) due to scale expansion. The reference to "Napiers bones" has a deeper connection because slide rules were constructed using bone or ivory (plastics were not invented) and bone does not expand/contract too much and wears well.

[@mattbox87]

I used to work in a factory while I studied mechanical engineering. I think it was about 2008? I spent some of that time in the drafting office and the lead draftsman in there was fond of his old Hewlett-Packard Reverse Polish calculator. He explained it to me and I found it fascinating. Now I use RPN in the calculator app on my phone. On a later occasion I was speaking with the head honcho, brought up the old calculator and he excitedly went on to talk about slide rules. He was delighted that I'd heard of them and (vaguely) understood how they worked so he gifted me a little 6 inch promotional slide rule for "Opperman Gears Ltd, Newbury, Berkshire, England". I learned how to use the trig scales on the back and took it into exams as a backup in case the calculator ran out of batteries :) I still have it and I love it.

[@andriusbalukas207]

I have used such circular calculators 50 years ago for designing books, which required complex scaling computation for croping pictures or sizing type. I still have two that I use sentimentally. One is calibrated using inches, and the other in millimeters. I would often use the millimeter dial like a slide rule.

[@ulliwm]

me too, still have it along with two linear sliderules - one pocket sized 5" long and one 10" long

[@SteveFrenchWoodNStuff]

I'm thrilled that you're still making these incredibly well-produced, expertly-explained videos.

[@Mathologer]

Long time no see !!!

[@michaelnorris2522]

I have a vague memory of seeing a circular slide rule once and thinking that it would make sense to do it that way. Of course, it would be harder and more expensive. Slide rules were almost universal when I was in engineering college (1973 grad). In the last year or so a few people had HP calculators with four functions and very little else that cost about $200.

[@hellenback8043]

Also Civil Engineer 1973. The controlling factor of whether electronic calculators could be used for an exam was the number of electrical outlets in the room!

[@samus88]

Thank god you didn't upload this yesterday, I would have thought it was an April Fools joke. This is truly "magical" xD

[@PR-fk5yb]

I remember being amazed at slide rules. We did only the "numbers" with it. The scale (decimal) was in our head only. I remember in a class we were proof checking the results given by the then "new calculator device" in order to see if it was calculating properly!! At the time everyone was amazed at the degree of precision of the calculator but we all knew the slide rule was faster!!! What really killed the SR was the TRIG tables... no more need to look up the tables with the calculator...

[@37rainman]

All sliderules which ever I used had trig scales on them. Dont really see how a calculator would be slower than a slide rule

[@1SLMusic]

I love this super simple explanations of logarithms. It’s just so beautiful. These videos always bring me beck to freshman year, solving problems that have already been solved and making new discoveries from that. Fun times.

[@1SLMusic]

Thanks for the heart man :)

[@donaldasayers]

It's a pie chart of Benford's law. I have an Otis-King helical slide rule, the scale is about a yard and a half long but wrapped around six inch rods that slide telescopically.

[@Mathologer]

Ah, yes, that too :)

[@BMrider75]

@@Mathologer I'm familiar with the Otis King calculator, and was hoping you'd show it ! Great video though, thanks.

[@PeterWMeek]

In this picture [ bit(dot)ly(slash)slide_rules ] the middle 10" circular has a spiral scale that goes around the disk many times (looks like nine) for an effective length measured in yards (or meters).

[@paulromsky9527]

Great video as always! We sent men to the moon using slide rules. When I was a young engineer (circa 1988), I was chatting with one of my mentors that was retiring that day. As he was packing, he pulled out a slide rule. I marveled at it, but I didn't know how to use one. He gave me a quick lesson and then gave it to me with the promise I would learn how to use it and hand it over and teach a worthy new engineer how to use it when I retire. As I approach full retirement, I need to find a young mind that will honor this tradition. I am thinking about teaching STEM in the summer from home. Maybe I can pass it down to one of my students. From that day on, whenever I leave a company for newer horizons, I always give an engineering a trinket of sort to a young engineer (usually right out of school). Since I work with mostly electrical engineers, it is usually a brand new high quality Digital Multi Meter (DMM) - if I know they don't have one. It is a nice tradition. I also dispense some sage advice - I make a little ceremony of it. One made me teary eye as he thanked me for being his mentor.

[@Mathologer]

Thank you for sharing this story with the rest of us :)

[@ahcuah9526]

Wow. You sure know how to make a guy feel old. :-) Took a look in my slide rule drawer (yes, I have one) and found 6 slide rules there. Two of them are Picketts, which was pretty much the standard 50 years ago. Also, one if them IS a circular slide rule.

[@ragnkja]

And here I just have my dad’s old Faber Castell school slide rule (the slightly more advanced version with exponentiation and logarithms on the back of the “tongue”, but still just one).

[@Mathologer]

What's new about all this is the way the logarithmic scale can be generated using this neat rubber band idea that I highlight in this video :) As far as circular slide rules are concerned I did read somewhere that the first physical slide rule was in fact a circular one.

[@ahcuah9526]

@@Mathologer Oh, yes. Totally agree. I was just noting my "stash" and neglected to say how cool the rubber-band stretch idea was.

[@jumpingjflash]

your animations are so ingenious and wonderfully explanatory. I can often understand your talks even without sound

[@jonstoehr175]

In 1976, I was an sophomore engineering student in college. My calculator broke and I couldn't afford a new one. I got through that year with a slide rule and a book of log tables. I still have the slide rule in it's original leather case.

[@TillerSeeker]

This nifty video streched my brain logarithmically and in the process imparted a deeper appreciation of the inherent relationship between addition and subtraction on the one hand and multiplication and division on the other!

[@MrAaronvee]

Just in case nobody else has pointed it out, may I add that the frictional force between the elastic band and the wheel increases exponentially. This is why cowboys in westerns sometimes do not tie their horses to the rail. A few turns around the rail makes the horse just as secure.

[@MrAaronvee]

Thanks for monitoring the comments so assiduously.

[@dougr.2398]

This applies “loosely” as well to mooring boats. Oops, the ship has slipped due to vibrations called waves

[@Le-_-Doc]

@@dougr.2398 Precisely why we have capstan knots that do not untie when loose, but who tighten when pulled on. :)

[@MrAaronvee]

@@dougr.2398 Some of those cowboys do not even wrap the hitch around the rail; they simply slap it against the rail, and inertia does the rest. BTW, if you know boats, what do you think of the claim that it is possible - using just ones arms - to push surprisingly large craft (even ships) away from the dockside?

[@MrNicoJac]

@@MrAaronvee As long as your muscles' strength and your feet's (horizontal) grip/friction on the ground stays higher than the water resistance, it's just a matter of time. You would be able to push away a ship, even one of those massive cargo ones. (assuming there's no wind or current, either of which might overpower your 'feeble meat-bag' myofibrils, especially if the ship is big)

[@robwin0072]

I recognized the circular slide rule once you added the points between 0 and 1. I used a slide rule in the early 1980s at NASA Ames Research Centre in Mountain View, CA. To this day, I have five slide rules in my home. When I want to multiply or divide a few numbers quickly - I grab the slide rule before my TI-Nspire CX CAS Calculator. By the time my calculator powers up and is ready, I have my answer(s) on the Slide Rule. I am teaching Slide Rule multiplication, Division, trigonometry, and also teaching LaTeX to my granddaughter now. I learned and used TeX in the early 1980s while at NASA during that same time. Yes, I am a pilot. Since I was a teenager. Also, an Air Force Veteran. I loved my flight computer. Although now they have electronic flight computers in general aviation flight training ground schools.

[@hughobyrne2588]

Slide rules are *awesome*. I love them. They're like a physical manifestation of mathematics - the most abstract field of study there is. What would it take to make a slide-rule type instrument where the calculation performed is not one number multiplied by a second number, but instead, one number raised to the power of a second number? Bonus question, but maybe only for the most hardcore of mathematicians: what about a slide rule where the operation is addition?! The mind boggles.

[@ragnkja]

A slide rule for addition is easy: just slide two rulers (with the same scale, obviously) next to each other. Exponentiation is harder to explain, but my dad’s old school slide rule has scales for exponentiation on the back of the tongue, so it’s obviously possible.

[@hughobyrne2588]

@@ragnkja Yes! I had forgotten that. There's a logarithmic scale from 1 to 10 on the sliding part, and a mirror of the same scale on the frame of the ruler - this is for multiplication. And, there's another line of numbers on the frame of the ruler. Where the second scale has '1', the third has e, Euler's constant. Where the second scale has '2', the third has e^2. Where the second scale has '3', the third has e^3, and so on. So, if you slide the '1' of the slider to '2' on the second scale, the same place as e^2 on the third scale, then the slider numbers '2', '3', '4', and '5' match up with the second scale at 4, 6, 8, and 10, and match up with the third scale at e^4 (or, e^2, the starting number, squared), e^6(=(e^2)^3), e^8(=(e^2)^4), and (e^2)^5. Putting the '1' of the sliding scale at any value 'x' on the third scale means '2' will match up with x^2, '1.5' will match up with x^1.5; putting the '10' of the sliding scale at any value 'x' on the third scale means '5' will match up with x^0.5, '3' will match up with x^0.3, etc.. But the third scale only works for numbers between e and e^10. There's a fourth scale for numbers between e^0.1 and e, and depending on the slide rule, a fifth for e^0.01 and e^0.1, a sixth for e^0.001 and e^0.01, a seventh for e^10 and e^100, and an eighth for e^100 and e^1000. For multiplication, the second scale works for all powers of 10, but this method for exponentiation needs another printed scale for each power of 10. I'm curious if there's a way to make a (finite) slide rule with just two scales that will compute one number to the power of another, generally.

[@ragnkja]

@@hughobyrne2588 Dad’s slide rule does come with instructions for how to calculate logs and exponents in general.

[@hughobyrne2588]

@@ragnkja If it's the type I describe, yes. And if the base you start with, and the result you end with (or, if you're doing logarithms, if the value you start with, and the answer you end with), are both between e^0.001 and e^1000, that's great. The multiplication scales don't have such restrictions - you could be multiplying quantities in the googolplexes, or googolplexths, or any number of orders of magnitude bigger or smaller. All with just those two scales.

[@robertgumpi7235]

Hi Mathologer. You have a circular slide rule on many Citizen Eco-Drive watches. It’s very handy - for example - if you are abroad in a country with different currency. You set e.g. the currency ratio EUR/USD once and then you can quickly calculate prices. Also it can serve as an easy „tip calculator“. Set to 1.1 for 10% tip and read off your watch.

[@Mathologer]

Good point :)

[@PeteC62]

I used to have a lovely 5" (linear) slide rule with very finely engraved scales, but misplaced it somewhere along the way. I still have my 10" slide rule that I bought from WH Smith's over 40 years ago. It hasn't seen much action recently, though!

[@ffggddss]

To me, the most amazing thing here is that you demonstrated the multiplicative property of a wheel generated this way, without even mentioning logarithms. I was anticipating some explanation involving the physics of uniform elastic stretching onto "sticky" wheels. NB. I grew up in the era of the slide rule, so I was already familiar with the circular variety. This was still a new & fascinating way to look at them! And BTW, an important disadvantage of the circular ones is the increasing loss of resolution with scales that are ever closer to the center of the device. Fred

[@Mathologer]

"To me, the most amazing thing here is that you demonstrated the multiplicative property of a wheel generated this way, without even mentioning logarithms." Shows that you are a real mathematical gourmet :)

[@godfreypigott]

Pretty sure that wheels made of logs is a very old invention ... this one just added special Powers.

[@dirkbruere]

Google will turn up lots of them, some in museums

[@Mathologer]

Just in case this is not just meant as a joke about wheels made from wooden logs, the "missed" in the title of this video refers to the way the logarithmic scale can be generated using this neat rubber band idea that I highlight in this video :) As far as I know this was really only discovered recently. Also as far as circular slide rules are concerned I did read somewhere that the first physical slide rule was in fact a circular one.

[@mlmimichaellucasmontereyin6765]

Thanks GP, but, uh... Do you mean like the supernatural powers of the hyper-tiny magic beebees in quasi-Higgsian QM-materialism? ;-)

[@MusicalRaichu]

log tables were pretty popular too in the rustic old days.

[@LeeClemmer]

You win.

[@johnvossler250]

For years I had a slide rule hung above our driveway entrance. It was a very fine Deci-log-log slide rule that was just over 6 foot long. It was originally manufactured to be used in a classroom to teach/demonstrate the proper use of the slide rule. I really loved that. But the harsh conditions at 8,500 ft elevation in the Colorado mountains and the hard UV radiation took their toll and it finally fell apart. I still have my pocket slide rule (6 inches) I used until I purchased my first calculator (HP-67). I can't seem to bear parting with the slide rule. Even after I got rid of the HP-67 and bought an HP-41CV with all the extras.

[@Gunbudder]

i learned to use a slide rule and vernier calipers despite going to school in the 2000's lol. my grandpa was just adamant that i learn how to use it, but more as a way to impress older engineers i think. which has come in handy! as for backup though, i have a solar scientific calculator that would survive any kind of disaster i could think of that a slide rule could also survive, short of going to the moon. radiation is one element that will always beat an electronic calculator and not a slide rule

[@ragnkja]

With a solar-powered calculator and a slide rule, you are prepared for calculating no matter what life throws at you.

[@jonathanrichards593]

@@ragnkja If life throws you a really dark place, then you're back to counting on your fingers :))

[@ragnkja]

@@jonathanrichards593 As long as I don’t need to count past a gross, I’m good.

[@hxhdfjifzirstc894]

@@ragnkja Do you, by chance... have six fingers on each hand?

[@ragnkja]

@@hxhdfjifzirstc894 No, I count the knuckles on the fingers (but not the thumb; I use the thumb to mark which knuckle I’m on), using one hand for units and the other for dozens. I personally use the left hand for units as I’m right-handed so the left hand is more likely to be free, and pressing my thumbnail into the correct knuckle occasionally is enough to help me remember how many dozens I’ve counted so far.

[@barongerhardt]

One of the cooler features of those flight computers is on the back side is a device that you can mark with a pencil and figure out wind speed and direction without doing any trig. Then you can reverse it to get the heading to hold to fly a strait path to your destination and effective ground speed. Flip it back over, then use that speed and distance on a map to figure out the time until destination.

[@clarencegreen3071]

I still have the E6B that I used in 1966 when I got my pilot's license.

[@MsAlisonWunderland]

I remember those flight computers, they were called Daltons computers. We all had them at FTS in the early 60s.

[@harroman42]

When I was at school in the 1960's we all used slide rules and I actually had a circular one. We got very quick at using them.

[@olerocker3470]

I used a slide rule in college (after I got out of the Navy) in all of my engineering classes from 1978 - 1981. Calculators were much too expensive then. My wife spent a whole month of her paycheck in 1982 and lovingly bought me an RPN HP-11C. I still have it and it works!

[@tracycarver6239]

My dad showed me the straight and circular rules when I was in junior high and for a while I knew how to use them, but didn't at the time have any clear idea why they worked. I wish they'd bring more of this kind of thing back into education.

[@МихаилТерентьев-ц6в]

Fun fact: if you have a circular slide rule as an avatar, there is a lot of questions like "What a strange scale on this stopwatch?"

[@PeterWMeek]

What a fun idea; I'll have to try it.

[@KaliFissure]

The tension distribution analogy was great. I can't believe slide rule and circle rule were dropped. They give a intuitive volumetric connection to things. Math is geometric first and only numerical when you try and pin it down

[@johnchessant3012]

12:19 I was sure that this had to use integrals. e.g. the stretch factor of the rubber band when the number x is at the top of the rubber band is 1/x, so the distance from 1 to x around the circle must be the integral of 1/s ds from 1 to x, which is log(x). I guess I'm too grounded in my non-Mathologer ways to come up with this much cleaner proof!

[@Mathologer]

That's the bit of calculus that I mentioned in the last section :)

[@chrissonnenschein6634]

So what is being said: That it took until after the Spanish Conquest of the Americas, & discovery of rubber, to invent the rubber band to make use of the logarithmic scale on a slide rule! Very astute. ::: Mathologer got me with the “Root of Evil” shirt, subscribed❣️. :: We were still using and taught how to use slide rules thru late 80s in college/university, back in Seattle. calculators still weren’t very advanced so logarithm were the better/fastest way to get physical results... was never sure what the other lines on a slide rule were for though⁉️

[@mz1rek]

The reading at 6:14 is 3.18 not 3.14, but the magical thing really understood you were multiplying by pi :)

[@Mathologer]

As I said, real magic at work here :)

[@user-qf6yt3id3w]

"On two occasions I have been asked, - "Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?" In one case a member of the Upper, and in the other a member of the Lower, House put this question. I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question." Charles Babbage.

[@raymondfarinas3169]

@@user-qf6yt3id3w politicians are usually very learned in rhetoric, and sometimes very little else. Such people would like to take credit for right answers without taking responsibility for bad ones.

[@PrinceAlhorian]

This was fascinating, reminds me a little of the Curta Mechanical Calculator where a device looking like a pepper grinder could do all but the most complex calculations.

[@kenhaley4]

Does the distribution of numbers around this wheel correspond to Benford's law about the distribution of the first digit of random values?

[@tomkerruish2982]

Yes, it does. This follows from the observation that multiplication by a fixed amount corresponds to a rigid rotation.

[@iwanttwoscoops]

I had the same thought, and yes; you can very, VERY easily prove this! Specifically, the length of the circumferential arc between two numbers (e.g. length of circle from 1->2) is equal to the AUC of an exponential distribution (for both being base 10). Easy proofs, just relies on showing both as logarithmic distributions, basically. If you want a tougher related concept to explore, investigate the inverse of the gamma/factorial function/distribution, and try to prove it also satisfies benford's law!

[@Tytoalba777]

Looking it up, Benford's law is related to logarithmic scales, so that's why you see that pop up here.

[@tiger12506]

I got some really weird looks using a slide rule in freshman college courses (2013). They are particularly handy for the types of calculations you tend to find in physics and chemistry... The ancient physics professor I became good friends after that. Quickest way to make them useful is solve for the variable you want without calculating anything, just write the numbers out as a large fraction, then you do all the calculations at once, sliding through incrementally each number. You keep track of the exponents in your head as you do so (remembering the +1/-1 for falling off the end of the scale). At the end, you get your answer from the scale x 10 ^ (exponent)... Scientific notation was MADE for these devices and it rocks.

[@knutboersma4386]

I'm an electrician by trade. Slide rules have fascinated me since I started looking into them, and I've accumulated a few. One of the nice things about a straight rule is that the scales are all the same length, whereas a circular rule sees scales get shorter as they move closer to the center. Straight rules also have room for folded scales, which most circular rules lack. I think the disadvantage of off-scale readings and needing to reset the rule is usually offset by the upsides. That said, I think a better circular rule than ones which saw production could be developed. A lens over the disc with hairline marks at points like pi, pi/4, 360/2pi, 746 (or it's inverse) could simulate folded scales and reduce the need for gauge set marks that rules commonly had for radian conversions or finding the area of a circle.

[@Mathologer]

Definitely a good idea. I did read on a wiki page that there used to be a 2 meter slide rule somewhere with a magnifying device attached to it. Also there are helical slice rules like the Fuller cylindrical slide rule that achieve greater precision by being insanely long. In terms of regular slide rules I do have a slide rule that has a bulging slider which magnifies a little. Probably intentional. Also, I believe there were also slide rules that used the Vernier trick to achieve greater precision :)

[@billmccaffrey1977]

I have several slide rules that I actually used back in the day. My high school participated in state wide competitions. I think I was 12 or 13 when the first 4 function calculators were being developed. There were many variations of circular rules developed for engineers. OHMITE was an OHM's law slide rule; SYLVANIA produced a circular rule for calculating doppler shifts; the GREENLEE tool company produced a circular rule for drill sizes for screws and nails that included characteristics of woods - a must have for carpenters. I have many of these pieces of history and still use them from time to time.

[@markuskrypton604]

14:25 I think it's no coincidence that the opposite side of 1 (half a circle) is about 3.14, which is pretty much pi.

[@jimcameron6803]

It's actually the square root of 10, or about 3.162. It's the number whose logarithm to base 10 is exactly one half.

[@Noam_.Menashe]

It's no coincidence that it is the sqrt(10). The distance is log(10)/2, (let's say natural log), and so to get the units we do exp(log(10)/2)) which is sqrt(10).

[@bscutajar]

it is a coincidence

[@nestoreleuteriopaivabendo5415]

Root of 10 is also a good approximation for the acceleration of the gravity in m/s^2.

[@markuskrypton604]

Thanks for the explanations! 🙂

[@KeepEvery1Guessing]

My circular slide rule in collage had all the usual scales, including the square scales, the trig scales, and the log-log scale. There were separate inner wheels on the front and back, and the index wrapped around allowing you to transfer values between the front and rear scales.

[@UnordEntertainment]

base 10 haters will notice this works in any base :)

[@wyattstevens8574]

@Misali!

[@gregwyatt4555]

What a fascinating explanation of logarithms! I am old enough to have used a slide rule at school. My Faber Castel 52-82 was a prized possession. It was lent to a neighbour's child who was a couple of years younger and I never saw it again - still annoyed at my mother for that! I occasionally search for one but never find one (not in Australia, anyway)!

[@Mathologer]

I recommend ebay :)

[@gregwyatt4555]

@@Mathologer I found a few but the freight to Australia is horrendous

[@NeinStein]

Hello early gang!

[@seedmole]

As someone who dabbles in the hobby of making music, I owe a lot to the invention of logarithms. So much nuance in both the controls and the audio of a synthesizer can be brought out by making use of logarithmic and exponential functions rather than merely using boring linear ones.

[@Mathologer]

Actually, the rubber band idea occurred to Dmitry while playing with the circle of fifths :)

[@seedmole]

@@Mathologer I grew up loving math and music, eventually found out that it's just one thing and loved it (them both?) even more from then on

[@Xubono]

I was disappointed that the original reddit post was a computer generated animation. Ok, the premise is a virtual rubber band, which sticks to the wheel at the exact point it touches initially. Further, the rubber band is infinitely stretchable. Is there any reason to think a real rubber band could perform a similar (obviously limited) function and accurately stretch out in a logarithmic scale as shown? I kind of doubt it … and, having grown up using slide rules in high school, this topic seems to require a much lower level of intellect than your usual gems of genius. Its 1:30am here in Adelaide, insomnia and hunger make this old man very grumpy. I should point out that I think Burkard is brilliantly clever, entertaining and a thoroughly nice person in real life.

[@Mathologer]

Actually as long as you ensure that the (real) rubber band does not slip, it should pretty much EXACTLY stretch out in a logarithmic scale as shown. At least for one revolution. Obviously it will break eventually, etc. but the rest should work as advertised. Still, definitely worth performing a little experiment :) It is possible to prove using a little bit of calculus that an ideal virtual rubber band will generate this logarithmic scale.

[@hxhdfjifzirstc894]

I don't see the point of trying to consider how a real rubber band would work.

[@Xubono]

@@hxhdfjifzirstc894 Well, hxhdfj, the premise starts with a rubber band being stretched around a circle. Mathematically simulating real aspects of the physical world is large part of disciplines such as Applied Mathematics, Engineering and Physics. Elastic and Plastic deformations are real, measurable phenomena. Animations based on real, observable data might be called a “simulation”. Otherwise you could classify this as a “thought experiment”, or even just a “computer game”.

[@BikeArea]

Everybody is talking about the great explanations and animations. Come on, this is Mathologer! I'm cheering on that, too, of course, but this time the uplifting soundtrack steals the show - by a magnitude of at least log(100) if not ld(8)! 😃👍

[@vsikifi]

There is a slight problem in the coding challenge: It can be implemented only on a device which has infinite storage and can do infinite precision arithmetic in finite time.

[@Mathologer]

A minor point ... :)

[@tomharris8712]

At first glance, it seems logical. But...the screen has a final resolution, which means, that you can choose a finite number of settings of the wheels, which in turn means that all possible values of the operands are rational, which can be represented by 2 integers with finite length. This gives, that the product and quotient are both rational, which can always be written as a finite sequence of decimals, possibly followed by an infinite repetition of a finite sequence of decimals. So everything can be represented in a finite RAM, and calculated in finite time. And you can never zoom infinitely in finite time.

[@vsikifi]

@@tomharris8712 You can't zoom infinitely in finite time but you can zoom far enough for the numbers to overwhelm the RAM of any device in finite time. That time may be very long because the application will slow down a lot when it has to do arithmetic with ever higher precision but it is still finite.

[@tomharris8712]

@@vsikifiTrue - If you really want to kill it and spend the rest of your life and civilisation and....😏

[@SaturnCanuck]

That was great. One of the few videos of yours where I was ahead of the curve. I KNEW it was a slide rule. My Dad showed me his as a kid. One of my favourite scenes it the movie "Apollo 13" is when Lovelle asks them to check his math, and three guys in turn use slide rules to figure it out. Great scene! So, slide rules rule!

[@hxhdfjifzirstc894]

Same, I also thought of Apollo 13, and knew the video was leading to slide rules. This is one of the better space related movies of all time.

[@Mathologer]

Here is the scene vimeo.com/34664087 :)

[@frankharr9466]

That looks like my ratio calculator. I bought it in an art supply store. It also reminds me of a video on Binford's law. My dad had a slide rule. He was a veternary toxicologist.

[@philliploe6298]

When we were learning about logs in higher school, our teacher divided us into groups with a slide rule each and our assignment for the day was to figure out A) how to use it and B) why it works. A really fun lesson that sticks with me to this day!

[@Mathologer]

Still collecting slide rules so that I can also run a lesson like this :)

[@justjacqueline2004]

Anyone else used these slide rules for "O" level maths ? Shows my age,eek!

[@rogerhudson9732]

When I did 'O' level Additional maths in the 5th form (year 11 in newspeak) an Aristo linear slide rule was my birthday present, 1966. I got a circular slide rule for 'A' level.

[@jonathanrichards593]

For O-levels, A-levels and my degree course! I still have the rule, although the sliding cursor (which Mathologer omitted, chiz) is a little damaged. It has got scales on it that I've forgotten how to use, however!

[@jonathanrichards593]

@hognoxious Our physical chemistry tutor made us use 9-figure log tables to calculate answers to his seminar exams, specifically so that we could not use 8-digit calculators (which were very new at the time). I think I still have the book of tables somewhere...

[@alanwilson175]

Nice video presentation. This is just an "old fashioned" slide rule. I have 3 of the linear slide rules, and I once had a circular one, but I forgot where it was long ago. Of course, the whole idea is to implement a logarithmic scale for convenient multiplication. The innovations on some of my slide rules was to add additional scales for squares, cubes, logs, and trig functions. This whole technology of slide rules was made obsolete by hand calculators that became common place in 1976 or so.

[@peterflom6878]

I used a slide rule as a kid. Much cheaper and more portable than calculators of the time (I was born in 1959)

[@kimobrien.]

I had both kinds at one time. You can also do square and cube roots by dividing the linear distance between one and your number in half or by a third. The advantage to the circular was the printing was on the wheel and it had two clear pointers you could set the distance between and unlike the stick kind you never had to worry about it falling apart on your way to and from class. Since accuracy is based on distance to print on the circular at four inches around was about as accurate as a 12 in rule. I think they also had trig functions on the wheel or with slide rules along with a linear scale and maybe one marked off from 0 to 90 or 360. Our high school math books came with trig and ln functions printed as tables in the books.

[@erawanpencil]

and on top of rooting, you can also do squaring or cubing by MULTIPLYING linear (circumferential) distances... correct?