This book should have changed mathematics forever 

What if this is not the FIRST engineer's tool , but just ONE of it.

Hello here comes the ancient Egyptians? Ancient Egyptians didn’t have engineering tools to build those pyramids…..hmmm

@@user-lu6yg3vk9z One thing has absolutely no relation to the other.

​​@@user-lu6yg3vk9z Mentally insert the "ALIENS!" meme here

It's truly absurd to think ancient people wouldn't do these calculations. It's not like they had phones to distract them or google... or grocery stores.

I think they had half square tables directly, so they didn't had to half anything.

@@sachs6 right. You could even make a ridiculous looking babylonian slide rule that does this with several steps. Each step is just addition, subtraction, or a function of a single value. The kind of things slide rules do.

This is a bit slow. It is quicker to calculate ab as ((a+b)/2)²-((a-b)/2)²

@@caspermadlener4191 Not if the calculation is more complicated , it is kind of like using the calculator.

And it's important to realize that the Babylonians did this perhaps 4000 years ago!!

I remember my dad giving me a slide rule and book of logarithms in 1966. I lived in Cleveland where was the Chemical Rubber Company (CRC). The CRC put out handbooks on math, physics, and chemistry. They would donate unsold copies of ‘the CRC’ to local school children. I would peruse the pages of the handbooks wanting so much to learn what all the symbols and formulas meant. I too wound up in radar and military electronics engineering.

Was Phasor Analysis important to early radar work? Radar did such a nice job of knocking Nazi bombers and fighters out of the sky. In school it seemed like such a cool way of solving linear time invariant control systems. (Everything is magnitude and angle (frequency dependent)).

Of course you used solar power... otherwise you would have been in the dark!😅

holy moly, you're old man. stay with us for about 5 years

I have a small salad plate sized circular slide rule that maps out to a linear slide rule nine feet long.

When I studied engineering in the 80s, I had a TI-41CX.

I had a circular one (as well as the more traditional sliding one) in the early 70s. It fits in the palm of my hand with my fingers being able to hold it. But, by 1976, I had a TI electronic calculator. I still have the circular slide rule (and the sliding one but some of the useful white part has broken off from the bamboo wood ). Lost the TI calculator ages ago.

@@two_tier_gary_rumain I switched to the Sharp calculators that had a great 49 char LCD AND a "playback" feature so you could check your formula (or reuse it, as it could draw from 6 memory locations). It was my last, and very best, calculator. My circular calculator got cracked because I carried it in my back pocket and sat on it frequently.

@@jeffweber8244 Hewlett-Packard made the 41CX (i.e. HP-41CX), not TI. My HP-41CX still works, although I usually use an HP-42S.

Yeah I've bumped into this a few times - will but this on the idea list!

"prosthaphaeresis"???? Do you kiss your mother with that mouth???

@WelchLabsVideo There was an *EXACT* formula for computing an arbitrary digit of the number pi in base 16 (hexadecimal base.) Can one make a geometric explanation of why that formula works (perhaps by using areas within and lengths of a unit circle and the Babylonian quarter of squares tables for computing multiplicative products) ? ab = 0.25 (a+b)^2 - 0.25 (a-b)^2 Maybe the quadrants of the unit circle have some relationship to do with the hexadecimal base ? What is it ?

​@@solconcordia4315can you share this pi formula? The formula you wrote is just a multiplication formula

@@ok-hv1or It's the Bailey-Borwein-Plouffe formula.

Pilots used to use circular slide rules (the e6b flight computer) right up until the invention of the iPad.

might I also recommend vihart's mini-series on dragons, fractals, and logarithms

Now I feel stupid. I have no idea what that symbol above the letters means in mathematical notation.

They still use them or at least always carry one in their bag and know how to use them, electronics have a bad habit of failing just when you need them which is not good at 30,000 feet@@douglasmagowan2709

@@lucasrinaldi9909 In this case, it just means "the other one". a and a' would be two different values on the same scale, b and b' two different values on the other scale. Sometimes it means other things.

That's when math education in America collapsed. We left the slide rule and lost a generation of mathematicians. The teachers who came up on slide rules didn't know how to teach "new math" and the students didn't get it either.

@@OKOKOKOKOKOKOK-zn2fy If you really know math and aren't just dependent on tools, it's easy enough to teach. For me, rather than the intuitive crap they sell, I can only learn it by writing it in code. They've lost the logic and the patterns and are stuck with their tools and formulae.

Yes. I remember using log and trig tables at school and then a slide rule at university. I still have my slide rule! I ended up in sonar.

In India students still use log and trig tables in their examinations.@@rogerphelps9939

if my dad (middle school math teacher) caught his students using their phone in class for non educational purposes 3 times he'll give them a wheel similar to this one and that student would not be able to use a calculator during exams as a punishment XD

Thank you!

From a strictly mathematical point of view (rather than practical usage) e is the most obvious base. If I remember rightly that's what Napier used in his first published set of log tables. e turns up all over mathematics, whereas 10 is not really mathematical at all, it's just an artefact of our chosen notation for interests and decimal fractions.

@@trueriver1950 Sure. But the angle in this video is engineers and actual calculations, not mathematical calculus and analysis. 10 is better in that case, although the difference is marginal, especially to anyone who uses them several times a day.

@@trueriver1950 If you use base 1.0001 (1+10^-4), number e is close to 10000th entry 🙂Because e=(1+1/x)^x for x approaching infinity and 10000th entry is (1+1/10000)^10000=1.0001^10000. Napier used 0.999999 or something like that - same trick which burgi used: you can rewrite number and subtract 1/1000000 (burgi used adding 1/10000). Maybe that's why Burgi is mention in the video - napier work was less intutive and also his table started by 90 degree and each entry was one minute lower with corresponding sine of angle (maybe cosine and tangens too, i don't remeber).

@@MasterHigure if the focus is engineering then e and 1.0001 are still more useful than 10 lol.

@@tiranito2834 Not at all. Because you can eyeball 10-logarithms with a minimal amount of practice, way easier than you can eyeball e-logs or 1.0001-logs (for instance, even _without_ practice, I can tell at a glance that the 10-log of 3145 is about 3.5). The two other logarithms take a lot more practice, or alternatively, they require a separate step of some mental multiplication with a memorized constant. No, 10 is much more convenient for practical use.

what's antilog?

Yup, this is a logarithmic book, but that term was very young, still not related to exponentiation. It was the name given by Napier to the mapping between a geometric and arithmetics series. The arithmetic series giving you an "Arithmetic Number" aka "Logos Arithmos" in greek. So -> "Logarithmos" -> "Logarithm"

Maybe if the goal is to build more intuition about logarithms, it's good to mentally calculate floor(log x) i.e. the largest integer we can raise the base of the log to and still be less than x.

Can you give an example of that uses of logarithm? I can't see so clearly this commom use... thanks in advance!

@@r2d277 3blue1brown has at least 3 good videos about logs.

@@r2d277 I tried to tell you about some content but it got removed, twice 🤷‍♂️

@@r2d277 3blue1brown, log

Fair point. Let's immagine our self to multiply distinct numbers. Number A by number B both with 10 digit. We choose 10 times A, B. In the usual way we must perform 100 multiplications and 90 addictions. By using this method with look up table we perform only 10 addiction. We may conclude as well then although not perfect is less prone to mistakes.

in English it's also called the slide rule

@@norude in czech slide rule is what you call calipers in english :-)

if you google the following (including the quotes) you should find it: "Bürgi, Jost:Aritmetische vnd Geometrische Progress Tabulen" (note that the spelling "mistakes" are in the page you are trying to find, so don't feel tempted to correct 'vnd' to 'und' etc.)

Printed it myself from this pdf: bildsuche.digitale-sammlungen.de/index.html?c=viewer&bandnummer=bsb00082065&pimage=8&v=100&nav=&l=de

You can find literally anything by typing the name and .pdf at the end

​@@michaeldamolsen pedantic note: at the time of original publication what we now think of as the letter V also served as our letter U. So it's only a spelling mistake from a modern perspective. But you are quite right, you have to keep the archaic spelling or the search engine gets muddled...

​@@puppergump4117 must try that... Millionpounds.pdf SaturdaysLottoNumbers.pdf Naaaah, doesn't work ... The last one nearly works, and you don't actually need the .pdf -- trouble is it gives you last Saturday's number which is not quite as useful

I still have my slide rule from my 70s maths classes, not sure I could use it now without some research.

in the 80s we still had giant slide rules on the walls in our maths class rooms, but we skipped the chapters on their use - I later accidentally "rediscovered the slide rule" playing with a few sheets of logarithmic paper, observing that twice the 1 to 10 bit amounted to the length of the 1 to 100 bit... and actually feeling quite dumb that I had to think about _why_ that was... and I did not immediately connect my "discovery" (half a millennium too late, but still) to those giant wall slide rules, because they had multiple bars / scales and I had always assumed these things were way too complex for me to master

Well they now have smartphones, not just calculators, and with a download they can match the most sophisticated "scientific calculators". I'm glad to see the end of the tyranny of TI-86 in the US, but, we're getting to the point where a kid can just verbally ask an AI what the solution is, without any conception of calculations at all.

I studied elementary school from 1987 in Czechia, graduated in 2004. I know slide rule only because it was magical item that was among some pencils, drafting tools, compasses and rulers in a drawer of my table. We basically learned powers and logarithms as a given tool, we learned how to simplify equations and how to calculate interests. But I wasn't able to apply them correctly on some example with radioactive decay where exponents were too big for straightforward solution on calculator. That was at middle education (is it called high school in US?) Then I used them at university applied as exponential attenution, conversion to decibels and back and that's basically all. Conversion of multiplication to addition and how to get square root was something I learned asking my mother about slide rule. Then recipe at uni was basically to convert stuff to decibels per unit lenght, mutliply by length, convert it back. Or voltage on capacitor vs time. University was actually when I used logs.

bro it's obviously a copy ;-; he wouldnt slice up an actual 400 year old book lmao

@@asdfasdf-dd9lk Off course it's a copy lmao. It was a joke. Myyyy days.

Nope, it is basically approximating e as a base. For example (1.0001)^10000 ~= 2.718... ~= e.

@@meowtheroflearning2320 No, e was found like 120 years later. But it's a nice "coincidence" that it appears as 10000th entry.

Yes.

I didn’t

Fun question! I don't know - but I do like the name red numbers more than logarithms.

Unlikely. You'd have a terminology crash with accounting, and as the video hints at, the mathematical functions being described have a *much* broader reach than a multiplication aid.

It took Napier 20 years to compute his tables. Do you really want to repeat that amount of work?

What? All not to publish it????

no

We *have* flying cars, now. They're expensive as hell, and still need a pilot's license.

Right, but perhaps because he worked on clocks, I have a pair of watches with slide rule bezels.

Clockmaking was at the forefront of science and engineering at the time, the skills developed allow the invention of a whole plethora of scientific instruments, laid the groundwork for advances in astronomy and navigation, and gave birth to the machine tools that allowed for the scientific revolution.

It already is a problem. It's hard to find fundamental in-depth information cause google always returns results which are similar to each other and wiki and basically mainstream. Maybe it's to reduce risk of "misinformation" or low quality information in search results. Imagine that just two weeks ago I had to derive how to extract center of rotation from transformation matrix - i was not able to google it, chatgpt said it's too complex problem. Yes, it needed to multiply three matrices by hand and solve three equations of three unknows having trigonometric functions - but extracting angle was easy so more like two equations without trigs. Another solution was to use some math library and extract eigenvectors. All that google found was the opposite - how to get matrix knowing center of rotation and angle. And that was not only problem I had. Sometimes it disregards some search keywords and I was not able to find exact article on my blog. Maybe it filters spam, I don't know. Also it's very hard to find some negative yet informative articles on someones blog - once i could not find that some problem exists, then one guy randomly linked in-depth research about it in youtube video.

Wondering if the Red-Black paradigm shows up in many places because they were literally the two colors of ink available in the standard stylus/quill pen holder with two ink wells that sat on everybody's desk.

@@amitabho That’s actually how Sedgwick et al did the Red-Black Trees at Xerox. Lol. Simply because - yes, you’re right…. They just had Red and Black pens lying around…. And that was long after the age of quill pens…. (Ball-point pens in the 70s/80s, Lol)…. I guess Charles Sanders Peirce was just trying to be more dramatic about it in his essay on Probability Theory that was titled “The Red and The Black”…. Also, a metal band called one of their songs “The Red and The Black” …. Anyway …

No, it’s got a different origin. Logarithm was coined by its inventor John Napier from Ancient Greek λόγος (lógos) meaning word and αριθμός (arithmós) meaning number. Logbook referred to a book in which ships recorded their speed for navigation purposes. They got the name from the device used to record speed, the chip log. It was a piece of wood tied to a roll of string that was dropped into the water. The rate at which the string was unwound was used to measure speed. Logbook now refers to a record of anything stored in written media. The chip log got its name from the fact that the drag (the piece of wood that gets dragged by the water) is a simple piece of wood that is chipped off from a log. Log in this sense comes either from Old Norse lóg/lág meaning fallen tree, which itself comes from the verb liggja, meaning to lie (on the ground/a surface); or it came from Norwegian låg, also meaning fallen tree. “Log in” is a phrasal verb constructed from the fact that when one accesses a computer account, one is adding an entry into a log. Log in the sense of an append-only sequence of records written to file was probably derived from logbook by analogy. Note that logbook as in a record of a vessel’s progress or as a general written record was already being shortened to log long before computers. Logarithm came first (John Napier introduced the term in New Latin as logarithmus) in 1614. Logbook was first attested in nautical records in the 1670s. Log in was first used in 1963 by the MIT Computation Center, in reference to people time-sharing their mainframes.

@@nathanoher4865 aww, pretty sad they don't share the same root well at least we know navigators were using both logs quite a lot in the 17th century

A ship compass and ship’s log were used to track direction and speed. Then a traverse board was used to keep track of those values for the four hours of a watch. At the end of four hours, the data was entered into the logbook. Then started over for. The next four hour watch. (I read the Aubrey-Maturin series of books.)

A captain's log is commonly found in the chamber pot in a captain's cabin. A cabin boy would check daily if there was a log in it. If there was, he would empty the log, called a log out. He would then inform the ship''s doctor who would record it in his log book. This was the doctor's log. A doctor's log is commonly found in the ...

@@nanamacapagal8342 Yep, I can’t imagine how much time it took to plot out routes without computers!

To be fair, his method would likely have people widely using slide rules a fair bit earlier.

Mathematics is to arithmetic as vehicle is to car. When I was a kid in the 50s in elementary school we had arithmetic class. Nowadays it is called mathematics class. No big deal.

No

​@@НиколайКошмар-ь7б haha, why? Learning how to use a slide rule would be awesome.

@@NotSure416 It's also completely useless as we have calculators and i also dont see any transferable skill worth of noting. Its also not fun, but tedious.

@@IsomerSoma Learning sometimes requires a bit of discomfort until one masters a skill. Exponentials and logarithms are extremely useful. Once one learns that basics, then it would be acceptable to use a more powerful tool such as a calculator. We still teach how to add, subtract multiply and divide by hand. Should we not teach long division because it's tedious?

@@NotSure416 Mental arithmetic is useful. I use it daily while doing mathematics (mostly proofs for university; small computations). It would not only be inconvient for me to not know mental arithmetic but also seriously diminish my comprehsion of steps in a proof or to come up with some number trick (like +1 -1 is like adding 0) to transfer some statement into something that has better properties. A worse version of a calculator would be completely useless. It gives me no new insight and its just if you are good at it a repetitive, brainless mechanical skill. Afterall that was the entire point of it before it got replaced. It isnt tedious because its hard but because its very boring. How the disk caculator is constructed is however quite interesting and ingenious. I still dont think it serves any meaningful purpose if we would still teach this. We would better be served by giving maths a more problem solving direction in school. For computation we not just have calculators ... we can CODE our mathematical algorithms or just download math packages for python. We can do very complex computations this way. Being good at numerical analysis as well as being able to put the math into code is WAY more valuable than any calculator or slider skills (there are actually competitions and schools for this especially in asia which is stunningly useless imo). It is also cognitively way more challenging thus generating more transferable learning to other parts in life. Tl;dr: It makes no fun, it barely teaches you anything transferable and it is useless in every situation of modern life. The same cant be said about mental arithmetic at all. It is way quicker and more convient to caculate something small in your head than to pull out your smart phone every time. Also being able to do mental arithmetic gives you a basic understanding how operations work which is important to much more abstract and high level mathematics as well as basic every day applications of math in every day life.

Too bad there is no Nobel prize in math.

In a couple of decades, perhaps.

I believe Burgi developed the tables well before Napier, but he did not publish them until after Napier and Briggs made their work known to the public.

6:38 Always obvious when someone doesn't watch the whole video.

He has another video that is just about the logarithms book and the slide rule.

Yes. The numbers between 1 and 10 engraved on a slide rule are scaled along the rule in proportion to their base 10 logarithms. When you slide one scale along another, the combined length is the sum of the logarithms of the two numbers. Because log(ab) = log(a) + log(b), the number under the index on the sliding scale is ab.

Good eye! Yeah I put a note about this in the description.

Fancy finding you here!

@@calebhubbell2290 : I get around. 🙂

The video is describing the HISTORY of log tables. Of course it's "just" logs. The point is to show the thinking that led to the first tools that took advantage of logarithms -- lookup tables, then eventually slide rules.