I found Katherine Johnson's actual calculations at NASA

[@S3SSioN_Solaris]

Tibees! Holy cow... It's so easy to forget how far computers have come. Those IBM 704's are huge! Wow, I feel so humbled, from Desktop Calculators to Pocket Smartphones...

[@TomLeg]

They were pathetic compared to your cell phone. :-)

[@Nulley0]

Smartbuildings back then

[@HeritageWealthPlanning]

and yet we're told we no longer have the technology to go back to the moon. We'de destroyed it somehow

[@dumbidiot1119]

@@HeritageWealthPlanningwe do, no one is denying that. We don’t have a reason to spend a billion dollars to show off anymore

[@jaewok5G]

"there's an app for that" "I'd feel safer if Katherine did it."

[@удивительный-б8х]

what are you even talking about lol

[@jdl13b]

Same!😆

[@rodjohnson2632]

I didn't see the movie, but read the book "Hidden Figures". I enjoyed the book, but as an engineer, I wished at times it went into more technical detail. Of course, that would appeal to a very small audience, so I didn't expect a lot in that area. It's great to see one of the papers Katherine Johnson contributed to, and some of the actual calculations she did. Also, I still have my copy of "CRC Standard Mathematical Tables 19th Edition", which I bought in high school in 1972. And I still have my old slide rules!

[@bryanpaulsicard4798]

My grandfather used to work on programming for the big supercomputers of the 1950's, so when we finally convinced him to get a PC & taught him how to use things like Zoom, it was as if the universe had come full circle

[@lawrencedoliveiro9104]

IBM machines were just “mainframes”, not “supercomputers”. The first “supercomputers” came from CDC, in their 6000 series. They were around 50 times faster than anything from IBM -- that was really “super”.

[@cybervigilante]

I too came from the punch card era. You can't unpunch a punch card. Difficult things to use, especially if you dropped the deck of cards and mixed them up.

[@JMiskovsky]

If he is up to task: How about showing him modern programming language like F#? Not quirky think like c++ but F# or Python would be easier to use. How he likes excel? How he likes modern PC?

[@andrewlalis]

​@@JMiskovskyD is better

[@shikhadawer5742]

@@JMiskovsky C++ still give better performance than python.

[@edgarfranco367]

This is so beautiful. It literally made my eyes tear. Her calculations and all that she represents to me is pure inspiration, and example of resilience, and perseverance.

[@oriraykai3610]

against "the patriarchy" and evil men I suppose?

[@TheHardcard]

@@oriraykai3610 How about toward the task that needed to be done?

[@narcissesmith9466]

@@TheHardcard Who thinks like that nowadays ?

[@TheHardcard]

@@narcissesmith9466 I am not sure if you intended to reply to me. My comment was concerned Katherine Johnson. Are you asking me who thinks like her now? Please elaborate on what you mean by “like that.”

[@MC-tm2uy]

@@oriraykai3610 you being triggered by this comment said more about you than the OP. And it's not good btw

[@mayflowerlash11]

As an engineering graduate who learned nothing about celestial mechanics it was interesting to get this insight. How Katherine developed the the initial equations would be fascinating to know. I also did not learn Euler's modified method as an undergraduate but when my son was studying engineering he came to me with a problem that required Euler's modified method. I taught myself and my son how to apply it from a text provided by the professor. And I was a bit surprised at myself. I did not use much of the maths I learned at uni after I graduated but when provided with a text I was able to understand it. I took a bit of mental effort I should add. I think this illustrates one of the advantages of a tertiary education. You learn how to learn how to solve problems. I do not regret all the hard work I put in uni to gain that skill. It has enabled me to solve many other unrelated problems. Great video Tibees, more please.

[@azzajohnson2123]

Yep how do you eat an elephant? One bite at a time 😂

[@Skank_and_Gutterboy]

Katherine didn't develop the initial equations, most of those came from Kepler.

[@duane2344]

Thank you for doing this episode. I never had heard of Katherine Johnson in history books. The amount of hard work, perseverance and struggles she and other women of that time had to endure. Makes me wonder how much knowledge we have lost throughout history not allowing everyone have as much education and opportunity that they desire. I am going to watch the movie again.

[@ThreeEarRabbit]

I have idolized Katherine Johnson since I was a child, so I am glad she's finally getting the recognition she deserves in recent years!

[@rksnj6797]

Absolutely! I believe all those women deserve recognition. Where would our space program be without their contributions in the pre/early computer days?

[@azzajohnson2123]

@@rksnj6797 same with Margaret Hamilton. She literally wrote the book on error correction and accepting human errors.

[@dylanparker130]

I love seeing the STEM archives explored! Especially fun to see something that's been typed on a type-writer. My Mum typed someone's Dissertation for them. She said that the Chemical equations were a nightmare because of all the adjustments she had to make to the paper's position for superscripts and subscripts!

[@SleepyAnt-oy8bl-w5j]

Toby, your videos are awesome! Whether you look at exams, explain math or simply talk about your academic endeavours. Your voice is very authentic and calming, which is an extra bonus as well :).

[@ClaireYunFarronXIII]

Tibees! 💗 It is always a pleasure whenever you upload, because I know I will learn about something I have always been interested in, and I will learn it through a relaxing ASMR-like way. 💗

[@daveterret3958]

The paper is an excellent find and this is another great video! I love the care you show to these delicate thoughts which have advanced all of humanity, and I very much concur in the respect you show to their (often obscure) originators. I appreciate your enthusiasm for the topic. This is very worth doing.

[@rwjoyner]

Great video, Tibees! I love that you showed how intricate the work was that Katherine Johnson performed. I also love that you worked in detail through the example calculations.

[@janhaveejaware9138]

Oh wow! This is amazing! Thank you Tibees for bringing this to us!

[@lawrencedoliveiro9104]

12:06 The last time I played with iterative formulas involving trig functions, I was getting third-order convergence. This was fast, even though each step was quite fiddly to compute. Looks like the convergence is similarly fast here as well.

[@daltontinoco7084]

So cool, thanks for hunting this down and talking about it! Also, I am working on programming the Wolff algorithm using multithreading in rust for the ising model... i could not imagine being able to do this without the work done here and by others to get where we are now. It's simply mind boggling to me that she did most of this by hand. Insanely cool. Also, your pointer finger nail is like a cute laser pointer, love it.

[@exploratoria]

Brilliant as usual, Tibees - thank you for this video! Please keep sharing your knowledge with us!

[@jurjenbos228]

I love it how that you analyze the movie by figuring out what book was used, and checking against what is said in the movie. More of that, please!

[@EldritchGod666]

Made me feel a bit old. I was using slide rules and log tables up through my second year at university and we still used punch cards to program the main-frame computer. 😃

[@vk2ig]

I knew of one guy who was quite lazy with punched cards - we were supposed to number them so the order in which they had to be fed into the card reader was known. This guy was always a bit tardy in numbering his cards; and one day - after he had just finished punching quite a long program - he walked into the computer lab and tripped ... and all the cards went all over the floor. He had no idea which cards were in what order ... LOL. As someone remarked at the time: "Now that's what I call a _floating divide_ error!"

[@Vexins]

If it makes you feel better, I just finished my chemistry and math studies, but built and used a slide rule for some things because it was just cool seeing log laws and other math quirks in action, but it also made me respect significant figures and estimate my last digit a lot better for all measurements.

[@parthasur6018]

Me too - and I used to show off my Aristo Hyperbolog! It even did hyperbolic functions!

[@wcottee]

On my first job at an aerospace company in 1979 (22 years old), a colleague of mine dropped a full box of punched cards on the floor. I remember seeing him sitting on the ground "laughing" . How far we have come.

[@jdos2]

@@vk2ig "Always cross your decks." (and you're right- throwing them in the sorter because they were numbered would have been a LOT easier!)

[@Sunset.Rising]

Women are involved in many different fields, but their names are rarely known. Thank you for discussing Katherine Johnson's accomplishments at NASA. Could you possibly make additional videos about women in science.

[@richardbloemenkamp8532]

Great idea to have dived into this a bit deeper. "Hidden Figures" is indeed a great movie.

[@fieldofscreams]

Oh my heavens. I never saw anything like this on KZbin. I’m wavering between whether you are exceptionally brilliant or was this an in-depth investigation. Regardless, inspiring and well spoken.

[@vk2ig]

Really interesting video, thanks Toby! I haven't seen _Hidden Figures_ all the way through yet, but look forward to doing so. Once in the mid-80s I heard news of an amateur radio satellite which had been launched by the USSR only a day or so before. A friend and I decided to try to hear it - I calculated the likely position and time at which it would rise over the horizon (definitely *not* to the same precision as Katherine Johnson, LOL) and was quite pleased to hear the satellite's radio beacon on the receiver around the time I predicted. What was somewhat more interesting was we thought we could hear the beacon a few minutes earlier before the satellite rose, and for a few minutes after the satellite set. We heard the same effect on the next pass, about 90 minutes later. We reasoned that the beacon signal (on 28 MHz) was penetrating the ionosphere somewhere before the satellite rose above the horizon, and the signal was propagating via normal ionospheric and earth reflections to arrive at our receiver ... likewise after the satellite set. All in all it was a very enjoyable learning experience. :D

[@avii8793]

Wow that is so cool !

[@avii8793]

It's not everyday you read a story so wholesome and relatable in an applying maths to life way

[@mateuszcielas3362]

im at awe that someone is so smart to figure ths stuff out and is sure enough to take the responsibility

[@KitagumaIgen]

The Euler-method was discussed for the reentry-trajectory calculation when drag had to be included in the equation of motion. That was something me and my group of colleagues thought was a bit odd - since Euler's method is not that efficient - which one would assume would be of interest if the calculations were to be done by hand.

[@mina_en_suiza]

I wondered the same. Higher order methods were well known at the time. The Euler method, whilst still working, is more a tool to prove the existence of a solution rather than a real calculation method.

[@tabsc3489]

Great video! I really enjoyed this breakdown on the math and calculations. I especially like how you brought up the parts in the movie to refer to them. Impressive that the old textbooks look so similar to the modern ones I"ve used too! looking forward to more videos from you since it's always a delight to learn more about the world and what we've come up with to describe it mathematically

[@synapsee_]

You really have a pleasing voice.

[@1123aka]

Thanks for another amazing video. Can we have a similar video on differential equations? :)

[@Pedrohnm]

Heróis do mar nobre povo! Nação Valente e IMORTAL! Levantei hoje de novo o esplendor de Portugal!

[@tombufford8659]

Thankyou toby, Critical mathematics work from 1960's. You hear how expensive and difficult it is to correct satellite positional errors.

[@jamesmorgan1063]

Well deserved posthumous acknowledgement of her contribution to NASA and Apollo missions.

[@tombufford136]

Most of us use hand held calculators for these calculations without Log tables. Amazing progress in technology. Real team work going on here with the Astronoughts, Engineers and Helpers working together. I wonder about the first calculations involved. The methods likely to be tried and tested by this time. When on holiday in Pakistan I was reading about NASA's request for detailed atmosphere data in the region and their offer to supply 'Ajax' rockets and other 'Kit' to survey the 'Sky' above the Indian Ocean.

[@lenpalmeri6228]

Fantastic research & explanation, Tibees!

[@davemmar]

The movie excited my interest in these three black women. But your explanation gives me a real sense of what they accomplished and just how your understanding of the calculations was conveyed at an understandable level. Thank you, thank you, thank you. I am over the moon with this video.

[@markwerley6965]

Thank you for filling in some of the technical details of one of my favorite films. Really enjoyed this.

[@islamadam8502]

Welcome back 🌷

[@4thesakeofitname]

This was super utter cool ! The calculations are extremely exciting not only because they are quite complex, but also because what they mean (where they were used)...

[@MrChartab]

I wish I was as interested in life as you are in these old notes.

[@bja009]

Touch grass

[@TheAmos1]

really enjoyed the video, as always interesting and fantastic facts, good job 👍🏻

[@erenoz2910]

what an awesome example of applied math and numerical methods! i wish my numerical methods prof talked about this a little bit

[@MUTHU_KRISHNAN_K]

Thank you very much for this video and recommending the movie. I have watched today this amazing movie from which I learnt many things. It was so inspirational and gave an idea about the hard work behind journey to space and the return

[@lindseydejesus1877]

the log math is so interesting! i never thought about applying them like that, to easily handle large number multiplaction/division

[@guitarttimman]

Euler's method utilizes slopes to get the approximations closer and closer to the solution of the differential equation. You can approximate the solution as close as you want to by approximations. I remember doing that in calculus class.

[@g.v.6450]

Great job with your “screen grab”. I had several failures before I gave up.

[@austinmoehring6216]

So interesting. I really enjoyed your video. Thanks for sharing the processes and pictures

[@kmm2442]

A log is like shaping down a number or putting one number in terms of another, so we can work more easily with it.

[@brianofozark]

Thank goodness you are back. It has been a while.

[@misterpopo3736]

The Euler equation scene in Hidden Figures annoyed me. It is unlikey these engineers, especially at NASA, wouldn't know about it.

[@anthonvanderneut]

7:42 for anyone trying to look that up: Google maps only takes longitudes between -90 and 90, so search for "28.5, -80.55"

[@majidalaaribi1642]

hi Tibees I really enjoy this video and as a high school student I need to repeat the same video about three times to get into the topic

[@lawrencedoliveiro9104]

7:00 Actually it was Babbage’s Difference Engine that was designed to compute mathematical tables. The Analytical Engine came later. Or at least the idea for it did--he was never able to actually build one.

[@Skank_and_Gutterboy]

I still remember seeing pictures of the Isograph, a mechanical device that calculated the roots of polynomials up to degree 15. It was built in the 1920s and had the footprint of a large conference room table. It came to a sad end. When it was deemed obsolete, it was donated to Princeton as a teaching tool, where it was left out on a loading dock all night uncovered. There was a heavy rain that night and the resulting rust ensured that it was a precision instrument no more.

[@Silligk]

I can't be the only person who listens to these to fall asleep to. So relaxing

[@JaimeMontoya01]

Thank you Tibees, very cool. The Case A, Eastward launch position is at pad 37A at Kennedy Space Center FYI

[@darkwand8800]

LOVE this!! :D Keep shedding light!!

[@patryan8743]

New to your videos … proof how math can be so beautiful

Your content is great! Thanks!

[@Anonymous-m9f9j]

What an interesting topic. Thanks for sharing.

[@melissamybubbles6139]

Thank you for finding that. It's so special to look at what she accomplished, and what tools she used.

[@igorkulishov8285]

Thank you Tibees! Tremendous work with all the details, original documents, calculations and even the equipment they used. Always enjoying and waiting for new videos! 🚀🛰🔭

[@lesselp]

Finally, something that is rocket science.

[@yassiranaibre5712]

if you understand this you can say that "it's not rocket science" about rocket science lol

[@Evan490BC]

@@yassiranaibre5712 I was going to write something similar.

[@markholm7050]

The caption of the photo at 6:00 says it is at JPL in 1936. The clothing, hairstyles and desks do not look appropriate for 1936. Also, JPL was just in its infancy in 1936 and not likely at that time to employ a room full of people. It’s more likely this photo is from the 1950’s, probably the late 50’s, or early 1960’s. Those steel desks with hard rubber tops were very common in the United States. I think they appeared during or after World War II. I worked at one in the 1980’s. It was a very serviceable desk.

[@vk2ig]

Definitely not 1936, which is somewhat obvious given the context of this video and the sorts of machines being used.

[@tibees]

True, the source of the image had 1936 in the caption but I should have tried to find some more info to verify the date

[@markholm7050]

@@tibees I also suspect those Friden calculators are a model later than 1936, but I really don’t have enough experience to say that with any certainty. I would easily believe them as 1950’s models. Your point in showing the image was to show a situation similar to the one Katherine Johnson was employed in, and I think you achieved that, except for the minor matter of being on the opposite side of the continent.

[@markholm7050]

A room with a dozen or more of those electromechanical calculators worked by skilled hands could be a fairly noisy place! Not deafening, but hardly peaceful.

[@claudeblaschette9597]

Cool videos. I really like how you explain complex facts !!! Greetings from Northern Germany.

[@AS-xt2rp]

Hey Toby... love this video! I was wondering if you could bring back your series Joy of Mathematics because I really enjoyed it! ❤Much love

[@anupamdebnath1884]

Yes I want that Joy of Mathematics series to continue too. It was so enjoyable. Thank you so much for saying it. ☺️

[@LyndreNel]

Fantastically interesting, thank you Tibees

[@kmm2442]

Love you Tibees.

[@kalkumaravel7552]

I reckon you can work at NASA Tibees

[@learnmoreushe3678]

Yeah Tibees for NASA. NASA for Toby.

[@henlo6813]

She is NOT American

[@Kevin-cy2dr]

She's Australian

[@markoverton5858]

Loved the vlog and the finger nails good touch , I was born in 1951 I remember the recent film, great leveller ,my farther sparked my interest in ballistic science, it helped a friend of mine to build a catapult to launch small sugar beat into the path of a jet that regularly buzzed his dads chicken sheds , in-spite of many complaints it kept happening, so we calculated the force needed to launch carefully weighed beat into the air, at a point in time the flight path was fixed by the jets we paced out the launch spot once the jet past over it the beet was launched the very first attempt was a bulls eye the jet lost one of its engines with a plume of smoke, within a few ours police and RAF personnel were all over his chicken farm, it seamed the beat had been spiked with Norfolk flint very hard indeed 🤣😂🤣 the only unknown calculation was the speed, we used stop watches over a quarter mile and hand signals, the impact height was 250 ft proving the RAF had been lying about its 500ft limit they insisted they flew at, since that day they never fly over the farm again, it is possible to beat the system 👍the beet six in number were of different weights to give a spread, of approx 30 ft in depth it was a lottery that the intakes would pass through it, if they had not insisted lining up over the track ever time with great accuracy it would not have been possible, they were very very angry puffing up there chests about cost but it seamed they were so embarrassed about the whole event it died away, there was some talk that some one shot at the plane with a shot gun 🤣😂🤣. We had intended to cause a hit under the aircraft causing the pilot to pull up and abandoned his run, you can imagine the panic when we hit the engine 🤪

[@digiclectic807]

This is a hilarious story. Like a scene from Dwight Schrute's beet farm in the American version of "The Office".

[@abhaynath5833]

Your voice is enchanting

[@lauriebarton2579]

Nice job on the video! Thank you!

[@roberttarquinio1288]

Interesting For orbital mechanics final we had to calculate by hand launching a space shuttle, orbital burns, launch of a probe to send to mars and landing the space shuttle; it took 24 hrs to do 23 pages of calculations We got to use hand calculators I saw the film Hidden Figures. I enjoyed it very much

[@basithassanqureshi7025]

Yes, I found that four years ago and shared that with my Numerical Computing students.

[@sandbagger1912]

Amazing video. Great photos and getting to see the original paper. I remember having a book of log tables and a slide rule in my trig class. Amazing we got to the moon and back using such materials and computers. Thank you, Katherine Johnson and the other gifted NASA mathematicians.

[@mickodillon1480]

Amazing! Love the daisy nails btw.

[@kenc2257]

As an old guy, I still have my CRC tables (can't recall which edition) in a box in the garage. Of course, last time I used them was over 50 years ago...

[@Khwartz]

So Interesting and Well Documented! Very Well Done, Dear :) 👌

[@hansklimstra5987]

Thank you for this video. I remember asking you some time ago if you would look into the Hidden Figures episode. Hoping that I was the inspiration for this.

[@oscarcharliezulu]

Super interesting thx for researching and presenting this!

[@FritsvanDoorn]

Best topic ever! Thank you!

[@DonLuc23]

It is amazing what all was accomplished using these "crude" methods. The rockets, motors, boosters, etc were all done the same way...

[@reneejones6330]

I actually went through that report and recreated the computations in Apl.

[@WhyX11]

Awesome vid as usual :)

[@marcinspace]

Much love to you.. thank you for the educational content:)

[@csvaughen]

Fantastic work!!! Thank you!!

[@guitarttimman]

Who remembers how to use the product rule to solve differential equations of the form dy/dx + P(x)y = Q(x)? I do. It's amazing what the e function can do.

[@erictellez]

Cool video. Thanx for explaining that in detail, I'm a fan of the movie. And your nails look pretty. :)

[@michaelmello42]

Wonderful video! This is a rterrific educational resource.

[@robertmartin7637]

I really enjoyed our video!

[@sakuma6088]

That lady was literally passionate bout what she was doing ... Respect

[@lawrencedoliveiro9104]

13:01 “Nautical miles” ... akk. Remember they were not even standardized (as 1852 metres) until 1976.

[@oldjoe182]

Very interesting! I wonder how much of this math is in KSP? inclinations, oblateness? are these things even there? or they simplified it all to minimum

[@vk2ig]

I forgot to add to my earlier comment: There's a video somewhere on KZbin of one of those calculating aids encountering a divide-by-zero error.

[@pikminlord343]

This was such a cool video

[@asecretone]

Imagine if we lived in a meritocracy. 🤔 These people would be royalty 👑

[@Snotboxrocked156]

Best book of theories I’ve ever seen.

[@mathandsciencewithasideofi1977]

I have a question: What books (not websites) provide proofs and derivations of volume of 3d shapes

[@vasilespirea8375]

Waaw ! Very intresting ! But you know about a romanien scientist who aid NASA to go in space ? His name is Gheorghe Botezatu or George de Bothezat who made for first time this calculations ! And by the way the Werner von Braun was a student of our scientist Herman Oberth (is originar from Transilvania )

[@narfharder]

This gave me a goofy mental picture of two math tools that worked together to bootstrap modern technology: calculus as two loud rich guys arguing about who's on first, and logarithms as the Computers unobtrusively gettin' stuff done.

[@rewirestrike]

I love this. One formula I have been seeeching for but cannot find is the mathematical proof from the Holmdel Horn proving the age of the universe.

[@datascinerd]

I'm so happy right now

[@user-vq3lk]

By the way, great background!

[@lilpopism]

As an aerospace engineer. This tickles my brain