UPDATE: David Gerber wrote me back! He had conversations with his father about the equameter, and says that it was indeed pronounced with a long "a", so the first two syllables sound the same as in "equation". So I got it slightly wrong. He also sent me photos of an original unit! Follow the link in the description. Thanks David!
Wow, using a precalculated slide rule for coefficients, and simply offsetting for the decimal place. Literally the obvious solution but done masterfully. Gerber was amazing.
Someday I hope to understand the insight that you just posted. Thank you, very much. At least I think I see how the dotted box lines up with your comment… so again, thank you (also that calculator 5:33 rocks).
Just read the Wikipedia article about the Gerber standard for plotters etc., I had no idea since I've never worked on this field. But a valuable hint regarding the phenomenon of Joseph Gerber! What a remarkable person, indeed!
Something that's kind of interesting to me is the potential for using programs designed to mimic the kind of approximations many of these graphical computing devices often use. Developing "good enough" code that's easy to run quickly is hard, pulling out a mathematical approximation that gets you within +- 2% in a fraction of the time is why videogames could use so many vectors without the machine catching fire.
@@nottelling6598 I hate the name of that algorithm, although the algorithm itself tickles the "obfuscated C code" parts of my brain. "Inverse square root" is just squaring. It should be called "fast reciprocal square root".
Still very relevant for looking at old plotted data. I have huge graphical datasets for several aircraft with no good way to deduce the equation until now! Thanks! Also your website appears to be offline with an Error 522
Sorry I'm having networking issues which I hope will be fixed within a week. I'd be happy to email you the files if you really need them now- my email address is easy to find online.
The algorithm is relevant, but the physical device is not. It's more efficient, precise, reliable, and repeatable to scan them into computer and process the images digitally.
Remarkable device and the thinking that went behind it. The physical beauty of the devices is lost in the digital age. I find these machines endlessly fascinating. Thanks!
Amazing work.... Amazing man...... I still don't understand how finding an equation from only 4 data points is obsolete with the advent of computers? I'm an electronics engineer and a PCB designer, never knew Mr. Gerber and the remarkable work that he has done..... wow !
If you can feed in the points to a computer (x and y coordinates), the entire calculation can be done instantly. Try something like this: www.wolframalpha.com/input?i=cubic+polynomial+through+%281%2C0%29+%284%2C2%29+%283%2C1%29+%280%2C3%29
6:55 It's mentioned that the equameter is working out the Taylor coefficients, but it doesn't look like it's considering any derivative data. I think it might be streamlining polynomial interpolation
Yes it is a polynomial interpolation. But the coefficients in the resulting interpolation turn out to equal the Taylor coefficients, even though it's not using derivatives. So it IS the Taylor polynomial, even though it's not computed in the standard way. (There are theorems that show the n-th order Taylor polynomial is the unique polynomial which matches the first n derivatives of f(x) at a given point. This is the usual approach. Another theorem is that there is a unique n-th order "interpolating polynomial" which matches the values of f(x) at n+1 different points. This is what the equameter computes. And ANOTHER theorem shows that the Taylor polynomial equals the interpolating polynomial.)
I have never heard of this, never heard of your channel, but SOO glad TH-cam recommended me this since this sounds exactly what I've been looking for, for some of my programs I write. thanks for the upload! I'm genuinely surprised you're not a bigger channel.
How annoying. My grandfather had one of those Gerber Variable Scales, but my family had tossed it after he passed away. I was happy I was able to snag his Slide Rule though.
the only way id be able to do that in 2014 was to take coordinates off the graph by eye, probably from a photocopy and then shove them into a Ti-83's regression system. this would then give me a polynomial and then i could shove that living and impure entity even into a differential equation. i stress living and impure, its done by eye the interpolation. often times i would draw vertical lines and then fit my ruler's tic marks into it and try to guestimate. you can get down to +/-0.005'' or maybe, just maybe +/-0.0025'' with this method which... may put you in serious contention with the thickness and haziness of the origin lines and your pencil tip.
There might be a coming revolution in computers with analog computing reemerging. Specifically with the way flash memory cells can store data as amounts of charge and how circuits can modify those amounts. I can see a lot of the old ideas coming back with processes like this turned into hybrid digital analog circuits.
Seems like there are several important computing and calculating devices that would be excellent to have in the event of the dreaded EMP....🤔 Perhaps a brief rundown of some of these would make a good future video..?
I would encourage you to try to get in touch with Cliff Stoll. He's also an incredible enthusiast for these old math tools. Numberphile has a few videos of his mechanical calculators.
I think there's an interesting observation with four-syllable words. Instead of pronouncing all syllables with equal emphasis, people feel a compulsion to emphasise one of those syllables, even when it isn't necessary.
A little gripe, you mention Gerber's version is more accurate. I think you mean more precise. Wonderful video, my mind is always blown by how physical objects can simplify what seems to be complex math.
I believe it is both more accurate and more precise- wrapping the scale around effectively makes the scale 2 or 3 times longer. This lets you put more scale markings (more precise), and also lets you read the answers more accurately, I think. Putting finer scale markings on a slide rule only improves precision, not accuracy- but if you also make the rule twice as long then that improves accuracy too, doesn't it? Maybe these terms are not so clear-cut for analog instruments...
Have this vid saved and am constantly going back and watching it. How would the equameter work on multivariable equations? It would seem a computer program version would be more practical for these higher dimensions. P.S. I've checked out your link to wolfram alpha already.
It would probably have overcomplicated things, but I think Vernier scales on the A, B, C,...H lines alternating sides could have lent an extra digit of precision.
A standard Vernier doesn't work on a logarithmic scale due to the subdivisions changing size. This is why you never see a Vernier on a slide rule, even though the extra digit of accuracy would've been much appreciated. There actually was a patent (US 2424713) for a Vernier-type thing for logarithmic scales, but as far as I know it was never used on a real commercial product. It's on my list of possible video ideas for the future, but it might just be too obscure for me. (And that's really saying something!)
@@ChrisStaecker it just uses 1:9 similar triangles and relies on the fact that you can shift them perpendicular to the scale to magnify your interval between markings. Of course straight line is off by some amount because of log scale but on small part of it error is small enough for this weird Vernier to work.
For me "Equameter" sounds greek, like "hexameter" and should rhyme with "thermometer", with the accent on the second syllable. Has nothing to do with the metric "meter", but with the greek for "measure"-
Really interesting, well researched and explained. I never knew WHY Gerber-Files were called Gerber, but now I should know, lets hope I remember for the future! 😁
I found your mistake! On the first measurement, you lined up the zero axis on the graph not with the zero of the curve sheet, but with the frame of the curve graph. That could account for the difference in results.
Very observant! But actually I noticed this while I was filming- I went back and redid everything, but I forgot to reshoot that very first shot. So it is wrong on-screen, but my answers are based on the correct re-do. I think the accuracy that I got is just due to imprecision in estimating values on the scale. When I look at the two curves plotted together, to me it's about as accurate as can be hoped with the scale that I'm using. Thanks for watching carefully!
"Speedometer" is typically pronounced with the "me" as less stressed, I think "equameter" could pronounce with the long 'a' from "equal" if you similarly put less emphasis on the "me" syllable. Whereas you're pronouncing it *EE*-kwah-Mee-tur, I propose you pronounce it ee-*KWA*-mə-Tur.
Yes! if you can feed in the points to a computer (x and y coordinates), the entire calculation can be done instantly. There has been much continuing work in data "interpolation", which is what this thing does, and computers are very good at doing it. You can try something like this: www.wolframalpha.com/input?i=cubic+polynomial+through+%281%2C0%29+%284%2C2%29+%283%2C1%29+%280%2C3%29
Love the videos! I went to go print one of these devices, but your packet of worksheets doesn't seem to be linked. Could you check the hyperlink? Thanks!
Unless he is talking about Valves game platform STEAM this just raises further questions, because I'm pretty sure it's STEM: Science, Technology, Engineering, Mathematics (eg the usefully college courses). Specifically separated from the humanities: Art, Philosophy, gender studies, communication, business, and English majors (the why did I waste four years of my life courses).
@@ChrisStaecker Fair enough... (As long as I can still tease my sister for originally choosing an English major, requiring grad school to become an accountant.) I love your videos show casing these devices!
Yes this is a mistake! Somehow I transposed the coefficients in the "solution" at 7:53. This is obtained by inverting the matrix [[1,0,0,0],[1,1,1,1],[1,2,4,8],[1,3,9,27]]. When I typed up the result as a new system I messed it up. If you look carefully at the worksheet at 9:36 you'll see it's the correct way around, with a, b, c, d in columns. Thanks for the catch!
@@ChrisStaecker thanks Chris I new something was not right, keep up the good work, Gerber was a genius of the first order. I have a electronic engineering background and we used numerous graphical techniques. Perhaps you could let me know what you've got in the pipeline.
So I tried to use your curve sheet. How ever I don't know what to do when my curve dips beneath the X axis. I tested your sheet with f(x) = 2x^2 - 4x + 1 function
You need to add a constant and scale the curve so that it all fits in the overlay area. That means it can’t go below the x-axis. The original equameter was very big so I imagine this wasn’t a serious limitation.
You can create a new x axis that is below your actual offset. Lets say its offset down 5 units. You then get the equation, 2x^2 - 5x + 6. You just take the 5 off the equation, giving the equation of your original graph of 2x^2 - 5x + 1
The good part about calling it an equameter instead of "the equation finder" is that there wouldn't be as many search results for things not relevant. Though in this case it isn't as bad, both searches on Google list less than 1 hundred. With movie and book names, 1 or 2 words keep getting the movie and not something else. All these strange things you show are odd. Not many people know of them. I find out, this was invented! I wonder how many of the millions of patents are for stuff like this.
You speak relatively slow, and I've found your voice to sound more natural at 1.25 playback speed. I mean there are a few audio glitches due to making audio faster without raising the pitch but other than that it makes the video more enjoyable. This is just my observation, don't take it as something mean.
![POLYCAT - ข้อความรูปยิ้ม :) [Music Video]](http://i.ytimg.com/vi/lBhIgPGCgbI/mqdefault.jpg)
UPDATE: David Gerber wrote me back! He had conversations with his father about the equameter, and says that it was indeed pronounced with a long "a", so the first two syllables sound the same as in "equation". So I got it slightly wrong.
He also sent me photos of an original unit! Follow the link in the description. Thanks David!
aqwameter lol
I get why he pronounced it that way which is really cool but sorry no that’s just not how Americans roll, it’s an ekua-meter now.
Wow, using a precalculated slide rule for coefficients, and simply offsetting for the decimal place. Literally the obvious solution but done masterfully. Gerber was amazing.
Someday I hope to understand the insight that you just posted. Thank you, very much. At least I think I see how the dotted box lines up with your comment… so again, thank you (also that calculator 5:33 rocks).
You are a gem of TH-cam. I love these videos. I am thinking “e-kwahm-met-ter.”
Gerber files are still in use today. Much like a DWG file in drafting, they describe semiconductor and PCB layouts.
Wow ive been using gerber files this whole time and i didnt know about the person at all
Brilliant catch
Just read the Wikipedia article about the Gerber standard for plotters etc., I had no idea since I've never worked on this field. But a valuable hint regarding the phenomenon of Joseph Gerber! What a remarkable person, indeed!
Something that's kind of interesting to me is the potential for using programs designed to mimic the kind of approximations many of these graphical computing devices often use. Developing "good enough" code that's easy to run quickly is hard, pulling out a mathematical approximation that gets you within +- 2% in a fraction of the time is why videogames could use so many vectors without the machine catching fire.
Wasn't it Quake that had an absurdly fast inverse square root function that used a quirk in how IEEE floating point numbers were represented?
@@der.Schtefan Quake III Arena used the Fast Inverse Square Root Algorithm to get "good enough" answers that were really close.
@@der.Schtefan yes. it's so fucking ingenious
@@nottelling6598 I hate the name of that algorithm, although the algorithm itself tickles the "obfuscated C code" parts of my brain. "Inverse square root" is just squaring. It should be called "fast reciprocal square root".
Still very relevant for looking at old plotted data. I have huge graphical datasets for several aircraft with no good way to deduce the equation until now! Thanks! Also your website appears to be offline with an Error 522
Sorry I'm having networking issues which I hope will be fixed within a week. I'd be happy to email you the files if you really need them now- my email address is easy to find online.
Thank you! Email sent. Sorry to hear about the network troubles.
The algorithm is relevant, but the physical device is not. It's more efficient, precise, reliable, and repeatable to scan them into computer and process the images digitally.
I find these graphical methods fascinating. They’re very much a lost art.
2:51 IS GOLDEN! You’re making your rounds count my man. Well done, Twitter Sniper.
Remarkable device and the thinking that went behind it. The physical beauty of the devices is lost in the digital age. I find these machines endlessly fascinating. Thanks!
Amazing work.... Amazing man......
I still don't understand how finding an equation from only 4 data points is obsolete with the advent of computers?
I'm an electronics engineer and a PCB designer, never knew Mr. Gerber and the remarkable work that he has done..... wow !
If you can feed in the points to a computer (x and y coordinates), the entire calculation can be done instantly. Try something like this: www.wolframalpha.com/input?i=cubic+polynomial+through+%281%2C0%29+%284%2C2%29+%283%2C1%29+%280%2C3%29
Oh wow. This _is_ the same Gerber as the Gerber (.gbr) file format! I looked up the wiki on this guy and he was prolific!
6:55 It's mentioned that the equameter is working out the Taylor coefficients, but it doesn't look like it's considering any derivative data. I think it might be streamlining polynomial interpolation
Yes it is a polynomial interpolation. But the coefficients in the resulting interpolation turn out to equal the Taylor coefficients, even though it's not using derivatives. So it IS the Taylor polynomial, even though it's not computed in the standard way.
(There are theorems that show the n-th order Taylor polynomial is the unique polynomial which matches the first n derivatives of f(x) at a given point. This is the usual approach.
Another theorem is that there is a unique n-th order "interpolating polynomial" which matches the values of f(x) at n+1 different points. This is what the equameter computes.
And ANOTHER theorem shows that the Taylor polynomial equals the interpolating polynomial.)
H. Joseph Gerber was an incredible person! Totally worth reading his history. His name is still ubiquitous among many who work with electronics.
I have never heard of this, never heard of your channel, but SOO glad TH-cam recommended me this since this sounds exactly what I've been looking for, for some of my programs I write. thanks for the upload! I'm genuinely surprised you're not a bigger channel.
How annoying. My grandfather had one of those Gerber Variable Scales, but my family had tossed it after he passed away. I was happy I was able to snag his Slide Rule though.
I'll bet that's where we get the name for Gerber files. The files that show drill plans and trace layouts on printed circuit boards.
Yes it's the same Gerber- see other comments on this video.
the only way id be able to do that in 2014 was to take coordinates off the graph by eye, probably from a photocopy and then shove them into a Ti-83's regression system. this would then give me a polynomial and then i could shove that living and impure entity even into a differential equation. i stress living and impure, its done by eye the interpolation. often times i would draw vertical lines and then fit my ruler's tic marks into it and try to guestimate. you can get down to +/-0.005'' or maybe, just maybe +/-0.0025'' with this method which... may put you in serious contention with the thickness and haziness of the origin lines and your pencil tip.
There might be a coming revolution in computers with analog computing reemerging. Specifically with the way flash memory cells can store data as amounts of charge and how circuits can modify those amounts. I can see a lot of the old ideas coming back with processes like this turned into hybrid digital analog circuits.
Add in ML unit with literally animal brain inside and you get the best neural network.
Wow, don't know what got you on my feed but totally worth it. Thanks for this insightful video.
Where TF is this when i was in high school, this is so usefull for the homework back then
This series is fun to watch
Quality video. Love learning about math gizmos like this
Seems like there are several important computing and calculating devices that would be excellent to have in the event of the dreaded EMP....🤔
Perhaps a brief rundown of some of these would make a good future video..?
why did you use an electronic calculator for adding when you could've used an addiator for realism
Really like your videos and the history they teach! So cool!!!
I must have this book...!
I would encourage you to try to get in touch with Cliff Stoll. He's also an incredible enthusiast for these old math tools. Numberphile has a few videos of his mechanical calculators.
Amazing !
It's like a nomograph but on a whole nother level. mans probably a gangster, explains why his name is on half my stabbing knives.
and on my baby food
I think there's an interesting observation with four-syllable words. Instead of pronouncing all syllables with equal emphasis, people feel a compulsion to emphasise one of those syllables, even when it isn't necessary.
A little gripe, you mention Gerber's version is more accurate. I think you mean more precise.
Wonderful video, my mind is always blown by how physical objects can simplify what seems to be complex math.
I believe it is both more accurate and more precise- wrapping the scale around effectively makes the scale 2 or 3 times longer. This lets you put more scale markings (more precise), and also lets you read the answers more accurately, I think.
Putting finer scale markings on a slide rule only improves precision, not accuracy- but if you also make the rule twice as long then that improves accuracy too, doesn't it? Maybe these terms are not so clear-cut for analog instruments...
This is super cool! Great audio too!
Have this vid saved and am constantly going back and watching it. How would the equameter work on multivariable equations? It would seem a computer program version would be more practical for these higher dimensions. P.S. I've checked out your link to wolfram alpha already.
This is amazing
It would probably have overcomplicated things, but I think Vernier scales on the A, B, C,...H lines alternating sides could have lent an extra digit of precision.
A standard Vernier doesn't work on a logarithmic scale due to the subdivisions changing size. This is why you never see a Vernier on a slide rule, even though the extra digit of accuracy would've been much appreciated.
There actually was a patent (US 2424713) for a Vernier-type thing for logarithmic scales, but as far as I know it was never used on a real commercial product. It's on my list of possible video ideas for the future, but it might just be too obscure for me. (And that's really saying something!)
@@ChrisStaecker it just uses 1:9 similar triangles and relies on the fact that you can shift them perpendicular to the scale to magnify your interval between markings. Of course straight line is off by some amount because of log scale but on small part of it error is small enough for this weird Vernier to work.
For me "Equameter" sounds greek, like "hexameter" and should rhyme with "thermometer", with the accent on the second syllable. Has nothing to do with the metric "meter", but with the greek for "measure"-
Really interesting, well researched and explained. I never knew WHY Gerber-Files were called Gerber, but now I should know, lets hope I remember for the future! 😁
I found your mistake! On the first measurement, you lined up the zero axis on the graph not with the zero of the curve sheet, but with the frame of the curve graph. That could account for the difference in results.
Very observant! But actually I noticed this while I was filming- I went back and redid everything, but I forgot to reshoot that very first shot. So it is wrong on-screen, but my answers are based on the correct re-do. I think the accuracy that I got is just due to imprecision in estimating values on the scale. When I look at the two curves plotted together, to me it's about as accurate as can be hoped with the scale that I'm using.
Thanks for watching carefully!
Equameter was named by the same team who came up with Blastoise and Charizard
"Speedometer" is typically pronounced with the "me" as less stressed, I think "equameter" could pronounce with the long 'a' from "equal" if you similarly put less emphasis on the "me" syllable.
Whereas you're pronouncing it *EE*-kwah-Mee-tur, I propose you pronounce it ee-*KWA*-mə-Tur.
2:49 "manufacturing" you were well within your rights to correct him. How could anyone not know what stem and steam stand for
For mathematicians: this is just a procedure for Lagrange interpolation
Meter means MEASURE in greek so it means Equation Measurer.
AWESOME vid and much appreciated. Question: is there a modern computer program that would accomplish the same thing?
Yes! if you can feed in the points to a computer (x and y coordinates), the entire calculation can be done instantly. There has been much continuing work in data "interpolation", which is what this thing does, and computers are very good at doing it.
You can try something like this: www.wolframalpha.com/input?i=cubic+polynomial+through+%281%2C0%29+%284%2C2%29+%283%2C1%29+%280%2C3%29
Love the videos! I went to go print one of these devices, but your packet of worksheets doesn't seem to be linked. Could you check the hyperlink? Thanks!
Yes! Thanks for the heads-up. It's fixed now!
Great Stuff! The download seems to have a broken link. Mirrors?
I should write a smartphone camera filter that does this.
Also how long would you have taken if you just used Lagrange interpolation?
0, 1 & 2 are extremely close to your graph. Since the Equameter uses only x = 0..3, it is only the 3 that seems problematic.
graph at 6:12 is so close at 0,1,2 but missed 3
maybe you just misaligned something in the last point?
amazing!
is there a chance that you know of any alternative for the gerber variable scale today on the market?
I don't know of a commercial one, but it's not too hard to make a simple one out of an elastic band, if you just want the basic scale.
@@ChrisStaecker I did that to space electric wire clips at an even distance along a run of wire along my basement ceiling!
Actually you can make a transparent overlay with a range of scales drawn as converging lines.
@@aeromodeller1 I like this a lot! Is that a standard technique? I've never heard of doing this, but it's a great elegant solution.
I vote for eee kwam uh turrr
Unless he is talking about Valves game platform STEAM this just raises further questions, because I'm pretty sure it's STEM: Science, Technology, Engineering, Mathematics (eg the usefully college courses). Specifically separated from the humanities: Art, Philosophy, gender studies, communication, business, and English majors (the why did I waste four years of my life courses).
Google it, stick- STEAM is the new STEM.
@@ChrisStaecker Fair enough... (As long as I can still tease my sister for originally choosing an English major, requiring grad school to become an accountant.) I love your videos show casing these devices!
Not sure how you got equations for a b c and d.
Surely y0= a?
Yes this is a mistake! Somehow I transposed the coefficients in the "solution" at 7:53. This is obtained by inverting the matrix [[1,0,0,0],[1,1,1,1],[1,2,4,8],[1,3,9,27]]. When I typed up the result as a new system I messed it up. If you look carefully at the worksheet at 9:36 you'll see it's the correct way around, with a, b, c, d in columns. Thanks for the catch!
@@ChrisStaecker thanks Chris I new something was not right, keep up the good work, Gerber was a genius of the first order.
I have a electronic engineering background and we used numerous graphical techniques. Perhaps you could let me know what you've got in the pipeline.
I’d like to make one of these, but the link doesn’t work.
Sorry- fixed it: faculty.fairfield.edu/cstaecker/machines/equameter.html
So I tried to use your curve sheet. How ever I don't know what to do when my curve dips beneath the X axis. I tested your sheet with f(x) = 2x^2 - 4x + 1 function
You need to add a constant and scale the curve so that it all fits in the overlay area. That means it can’t go below the x-axis.
The original equameter was very big so I imagine this wasn’t a serious limitation.
Flip it over. But you will need to change some signs in the little boxes.
You can create a new x axis that is below your actual offset. Lets say its offset down 5 units. You then get the equation, 2x^2 - 5x + 6.
You just take the 5 off the equation, giving the equation of your original graph of 2x^2 - 5x + 1
So back in the day those clear overlays would be called ‘foils’. Fact.
The good part about calling it an equameter instead of "the equation finder" is that there wouldn't be as many search results for things not relevant.
Though in this case it isn't as bad, both searches on Google list less than 1 hundred.
With movie and book names, 1 or 2 words keep getting the movie and not something else.
All these strange things you show are odd. Not many people know of them.
I find out, this was invented!
I wonder how many of the millions of patents are for stuff like this.
Yes the weird name makes it very easy to find! (Or not find, as the case may be.)
You speak relatively slow, and I've found your voice to sound more natural at 1.25 playback speed. I mean there are a few audio glitches due to making audio faster without raising the pitch but other than that it makes the video more enjoyable. This is just my observation, don't take it as something mean.
Hmm- I think this is pretty close to my natural speaking voice. Maybe I am constantly boring my loved ones?
Gerber file??
I'm gonna pronounce is "e-QUAY-meh-ter". Similar to thermometer
That is the correct pronunciation- see my pinned comment.
Goddamn manufacturing lol. Guy doesn't know what he's supporting lol
I thought they also made baby food and multi tools.
i gave you the 420th upvote on this video, yay
I say "ekwaameter"
15 minutes of calculating and you didnt even tell us the results? thats messed up
How do you only have 8k subscribers?
3 weeks ago it was 2k!
not understandable due to annoying background music. A shame