How people came up with the natural logarithm and the exponential function
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- čas přidán 31. 05. 2024
- I discuss the history of the introduction of the natural logarithm and exponential functions, answering the question of how the logarithm was first discovered and how it's properties were derived, and also why it appeared when it did. I consider several other areas of inquiry that could have led to a theory of exponentials and logarithms, including: compound interest, population/economic/technological growth, physics, gambling, navigation, and pure mathematics. Then I explain the work of John Napier in constructing calculation tables for astronomers that introduced the natural log in 1614.
0:00 Intro
1:05 History of compound interest
6:55 Why exponential growth was not a recognized feature of life in the past
8:10 Exponential functions in physics
10:58 Exp and log in gambling
14:34 Navigation, meridional parts, and the integral of the secant
18:43 Calculation tables and Napier's introduction of ln(x)
This is episode 6 in a series called Tricky Parts of Calculus, a series about the subtle and difficult parts of calculus that are usually glossed-over in a calculus class. Tricky Parts of Calculus playlist: • Tricky Parts of Calculus
Also check out my channel for general advice and opinions about math, as well as the Daniel Rubin Show podcast for interesting conversations about math and other topics: / @danielrubin1
For a discussion of how e^x shows up in analysis of the Martingale betting strategy, see this great video from Numberphile: • Gambling with the Mart...
References:
Hald, A History of Probability and Statistics and Their Applications before 1750 amzn.to/2ThUmip
Havil, John Napier: Life, Logarithms, and Legacy amzn.to/2UpMjR7
Newton, Principia amzn.to/3ikO5Ln
Toeplitz, The Calculus: A Genetic Approach amzn.to/3kvNwRE
Williams, From Sails to Satellites: The Origin and Development of Navigational Science amzn.to/3hLHY3w
(I get a small commission from purchases made from these links.)
sumerianshakespeare.com/70701/...
I forgot to mention the Fibonacci sequence! Leonardo of Pisa (Fibonacci) described the sequence F(0)=0, F(1)=1, F(n+1) = F(n) + F(n-1) in his book _Liber Abaci_ from 1202 in the context of a word problem about the breeding of rabbits. The sequence was known in Indian mathematics centuries earlier. The sequence does grow exponentially, but the formula F(n) = 1/Sqrt(5) (((1+Sqrt(5))/2)^n - ((1-Sqrt(5))/2)^n) was only written down in 1843 by Binet.
The so called Golden ratio. Actually it should be called Natural or Circular ratio, because it is related to the real value of π (natural value).
@@PASHKULI ...? no, it is not?
@@glowson3844 Just look for inscribed n-grams in a circle, understand the concept behind finding √x geometrically and you will understand why π and φ are directly related. Today we still use the linear approximation of π, which disregards the very small arc cuts, which contribute to the circum. and area. Hence π (the real, natural value) is just a tad larger.
This is very important for astronomy and later will be for discovering dimensions.
You haven't mentioned the contribution of Ibn Hamza al-Maghribi who used logarithm functions long before John Napier.
@@fslurrehman True, but since the late medieval times (western) Europeans (brits, french, spanish, dutch) were the most "expansive" (means conquering) people, thus they took all the credits for "inventions". Iven the battery was invented in Mesopotamia\Iraq... but that is a different subject.
I loved this! Some more history of math should be taught in high school when introducing concepts, we take for granted a lot of impactful work that really had centuries of refinement and adjustement that now seems given by definition on a blackboard that has no relation to reality.
Absolutely!
Thank you finally someone 🙏
Exactly.. this way of methodology of teaching should be encouraged in physics, Chemistry, Literature, Biology as well as Mathematics. It gives spirit and enthusiasm to learn more about the subject. It will add value to what the teachers trying to convey teaching these formulas or subjects. I have always witnessed in the class asking some of our practical minded classmates “what all these knowledge will do good in my real life?” With the traditional teaching method we think that the knowledge we have been learning is to get high grades to be better than our classmates and to be excepted to better universities racing and succeeding in tests or exams in front of them. This kills the thirst for knowledge and thirst to try to do better than what is thought to us in school..
No way
And some history of history should be reduced
People have always been so smart. These origin stories are fascinating.
AMEN!
When there is a problem everybody needs to solve, people will find amazing solutions that look impossible at a first glance.
I find that education system in most advanced countries are lacking on creativity, which I think it really impacts children to think of new ideas.
Yes people have always been smart enough to screw others with compound interests, and then smart enough to outlaw compound interests because they destroy common folks.
@@gelinrefira all interest is compound interest. It just describes how interest works over time. For example, you have 100$ in saving account with a 3% interest rate yearly. After one year you have 103 (3% of 100$ = 3 more dollars). After year 2 you don't have 106$, you have 106.09$ (3% of balance 103 = 3.09 more dollars). No one "invented compound interest". People came up with interest to get some money back on money they've loaned out. The math just describes how this accumulates over time. You are able to benefit from interest or compound interest just as much as anyone else. If you have a bank account you're benefitting from compound interest. If you have a single stock you're benefitting from compound interest (so long as you chose a good stock whose value goes up over time)
@@gelinrefira non-compound interest isn’t used outside of math.
When I asked my math teacher where did these magical logarithm numbers come from, she said they are from the back of the math book.
I’m sorry you had a shitty math teacher; math is humanity’s most beautiful / important creation, and it’s truly a tragedy that more people don’t appreciate it.
@@justinwatson1510 while I never did very well with other parts of the math curriculum, I did find an almost mystical fascination with geometry, so much so that I finished the whole years worth of geometry in six weeks. Occasionally I wish I had a similar experience with algebra. Maybe I will in my next life.
I had the exact opposite experience; I sucked at geometry in high school (I blame the shitty Georgia public school partly; the teacher was almost worse than useless) but algebra just made intuitive sense to me. I wasn’t able to appreciate geometry until after I got through differential equations or maybe linear algebra.
you act like its the teachers job to teach you that... they are there to see if you pass exams, stop being ungrateful. and people like you say stuff like "college should be free". education is not free stop whining
😂
Since I last studied math for engineering about 45 years ago I had a very hard time keeping up with this video but I still find it fascinating that people were able and willing to produce such complicated work hundreds of years ago. I often wondered how and where all the log tables came from. I think I was one of the last engineering classes that used the slide rule for some of our classes.
History, math, economics, and physics in one video. I love it
+ 1
I believe math should be learned based on problems that need to be solved, rather than blindly memorizing formulas. So, to solve a specific problem, you'd try out many different methods with increasing sophistication, by first using the method used in the stone age, then with the method used in the year 1600, etc, until you arrive at the most efficient/advanced method, which also provides the correct answer. This way you acquire real knowledge and understanding of what you're doing.
But then, you would have to learn trig. before algebra. Good luck with that. I suppose it could be done. It would be interesting to learn how to solve things by numerical methods first now that computers are available to us.
math should not be studied for a particular purpose rather it should be accepted as a form of art
@@ecavero1 That's true. Numerical methods are the way to go for basically all of math taught in schools and colleges. Its easier for us to think about and understand discrete rather than continuous sets. Who the hell knows what infinity means anyway -- e.g. limit goes to infinity? Give me a break. Math needs to be reformulated.
@@ecavero1 you need the math foundations to do it, im helping someone with a project on calculations for a rocket launch to orbit and ISS but everything is a struggle because they lack that foundation.
@@theremin_monkey limits are literally so usefull though. and you cant brute force simulate everything. even with supercomputers you cant (or it would benefit alot from a hybrid approach).
I don't get to learn about this part of math, the historical part, in school but it gives me so much more vigour and interest in mathematics. Thank you
I actually have a slide rule collection. 😅 I find them superior to electronic calculators in many ways, especially when the slide rule is specialized to perform a specific job. There are slide rules out there specifically designed to compute the frequency of resonant circuits, perform dead reckoning navigation, convert between Imperial and Metric...
Daniel this is amazing. I will have to watch it many times to process everything. Thanks for taking the time to create this.
Glad you enjoyed it!
On the question of usury, it's a bit more complicated than that. Merely having Interest on loans was not condemned as usury by basically anybody; Rome had simple interest, and this wasn't completely outlawed even with the conversion of the Empire, or in the Byzantine period.
The classical definition of usury for Catholics is charging interest either with what is known as an intrinisic title to interest, or charging interest in excess of any extrinsic title to interest.
An intrinsic title would be built into the loan: I give you $1000, and insist you now owe me $2000, without any need for me to do any work or sustain any costs to deserve it.
Extrinsic titles would be where you factor in stuff like the work I need to do to loan you the money, paperwork, paying for collection, and later on stuff like lost opportunity costs, inflation, lawyers to write up contracts, etc.
The basic moral premise here is that the quid and the quo in a quid pro quo need to be equal, but in different modes; I pay you, the loaner, in accordance with the value of the service you provide, factoring in the work you did, and the costs you sustained. I don't owe you any excess of that, because you would be charging me nothing for something; it would be a scam.
Islam has a similar position on usury, and is usually a little stricter about it, depending of course on the school. For example, there are a number of Islamic banks today that make interest voluntary; there is just imense social pressure to tip well, and these banks often do quite well actually.
Absolutely LOVE the history weaved throughout. Amazing, amazing video. Please do more - you have a gift for mathematical story telling. There are more and more people coming who speak the language of math - and this quality of video raises the bar. Reach out.
Thank you! Will do!
Your sir, have impeccable hand writing. Haven't even watched the whole video, but just had to say it!
Came across this gem today.
I cannot express enough, how fascinating these contents are .
Thanks a lot
Glad you enjoy it!
Greetings from the U.K....Thank you for this... as a layperson interested in maths, all to often actually uses and examples are not talked about and even rarer the historical background to mathematical concepts... this helps immensely! Though I must say that even now, fifty years after leaving school I’m still waiting for that moment when a knowledge of trigonometry will save my life, as I was constantly told it would!
Glad it was helpful!
When Math was taught to me...Nobody paid much attention to...or even talked about the history. Great Stuff.
History of such kind is so rare...please keep uploading more videos like these...
Love from Bangladesh 🖤🖤🖤🇧🇩
I've wondered about this for so long. I've always guessed that e^x was intentionally created as a means to have a function that equals its own derivative
Hello from Adelaide, South Australia. I came here for the maths, but the history is just as interesting, thanks.
Glad you enjoyed it
Another application of the logarithm not mentioned here is the well-tempered musical scale. Interesting coincidence that the invention of this scale was contemporary with invention of the logarithm.
Please, make a video about that for us if you know the theory, it sounds really interesting.
@ Daniel Rubin: what a fantastic 30 mins. I can appreciate the amt. of research and hard work that has gone into this.
I have History of Science and math integrated into my STEM course for school students and 2 days are devoted to this entire subject (devoted to relationship between e and compound interest; the need for logarithms etc. ). this video has given me a refreshingly new perspective to make the class more enjoyable.
Thanks a ton for this video
Thanks! Glad to be helpful to teachers
This is such an important video. I mean the historical context in which the maths was developed really helps one understand in which other contexts they are applicable.
Thanks, Jaco! That's a big part of why I made this video.
@@DanielRubin1 no, thank you! It's because of this video that a^x=e^xlna isn't just another identity in a long anxiety inducing list of identities that I need to memorise and hope to recognise out of a pile of problems, but now I GET it. This has helped for topics in my applied maths AND pure maths courses
Glad I stumbled across this. Math is great. I took many years of math courses at university.
Math history, however, is _chef's kiss_
Oh it feels like a eternity passed by since the last video I saw. Good to see you.
Humanity had so high level math so many thousands years ago its unbelievable to think of it
Not humanity, but only a few ten thousands mathematicians, philosophers, and scholars. In modern time humanity also does not have so high level math. The majority of people, maybe 97% of people do not know this. Only professors and good students know it.
Fascinating summary. Great idea that the need to record the earliest financial transactions drove the development of writing. Thanks.
Amazingly in depth, thank you for this resource.
Glad you enjoyed it!
8:26 ok, the handwriting is amazing, but that straight line with perfectly delineated tick marks for Galileo's inclined plane was truly next level.
Love the history, knowledge is endlessly vast.
This is phenomenal. Im glad the algorithm graced me with your channel
Glad to hear it!
History and Math together, what fun. Thanks.
I've always loved Dante's visual of the Usurers' circle of hell
Just want to say I understood nearly none of this (made it through Calc B but that’s it) but watched the whole thing because of how engaging of a speaker you are, and what a great history lesson too!
So, *that's* why ln is sometimes refered to (in Spanish, anyway) as Napier's logarithm.
I was privilaged enough to learn how to use tables of logarithms for calculations.
Very comprehensive explanation. Thanks.
Interesting fact is at 11:25 the dice that you drew were illegal dice. A fact of standard dice is that opposite sides must add to 7. If you rotate one of the dice through what would presumably be the z axis, then you would have one side adding to 6 and the other adding to 8. This was not a criticism of your excellent video but an interesting piece of knowledge that I wanted to share.
oooo cool fact
That was fascinating! It is amazing all the variations on techniques that came before whats is taught today.
Thanks for including references.
Glad they're appreciated! Lots more great material in those books.
This is how history should be taught
Thanks for the high praise! I'm not a historian, just someone with some specific questions about how and why certain things happened, especially within math, and I just put together what I found into a kind of story that satisfied me. Real history should probably be more careful and complete.
I wrote that like ten minutes in. By the eighteenth minute I became convinced I had found the best video on CZcams. Job well done.
Thank you for this great insightful lecture
Amazing. Thank you and hope you keep making these types of videos.
fantastic video, thanks for making it!
Awesome video! Thank you!
Absolutely amazing video!
Glad you enjoyed it!
Useful recap. Never to be used again.
Have not watched this video yet , but I certainly will. This is wonderful subject! A few years ago I read the book "e The Story of a Number" I highly recommend this book. Everyone that is interested in math even at the algebra level should know what is natural about the natural logarithm base. Of course the natural log base is critical to the study of calculus. Looking forward to watching this video. Thanks in advance!
You're doing an awesome work, it's really pleasent to watch :).
Thank you very much!
All abstract concepts should be taught with anchors to historic or social context, this was great. Instant subscribe.
Such a good high quality video!
Great analysis
Yes, thought this might be solved very quickly with a novel solution, though on second thought ,worked through the Solution using the quadratic formula and Logs. Thankyou for the video.
Awesome video Mr. Rubin, can you do how people came up with the Fourier series next? I know how to compute the coefficient and use them in differential equations, but I would like to know how we can come up with Fourier series assuming we have no prior knowledge about it.
Thanks! I'll definitely have some things to say about Fourier series eventually. Their invention is a better-known story. Preview: Although you can point to things like the theory of epicycles as a proto-trigonometric series, Fourier deserves credit for the idea that every periodic function can be represented as an infinite trigonometric series. He was trying to solve the heat equation in a finite interval, and managed to find a non-trivial cosine series on (-pi/2,pi/2) to represent a constant. In his original paper, I believe he did not yet have the idea of integrating and exploiting orthogonality to find the coefficients, and he instead set up a not so rigorous infinite linear system to equate coefficients of Taylor series.
Superb !
Thanks a lot
I'm always happy when I hear someone mention usury these days, especially in a practical historical context. There are many good reasons why it was considered a sin for the longest time, which seem to have been forgotten by our modern societies.
Nice presentation, and nice handwriting!
Awesome video! I love this channel! 💘
greatly enjoyed this!
Awesome, thank you!
Excellent work and presentation Daniel. I have subscribed.
Greetings from Melbourne Australia
Harry.
Many thanks!
great video! thanks alot
Thanks, great research and presentation. Although I didn't follow everything thoroughly. Did you only use resources in the references or are there any other source?
Awesome !
This is amazing, Thank you !
Glad you like it!
duude m not into this thing at all, math makes my head spine ! but ooh i reaaally enjoyed this video ! your voice, storytelling, and handwriting are soo calming and satisfying !
Really interesting. I've been interested in how one might 'build up' mathematics 'from scratch' if, say, all the world's knowledge but a few 'foundational' texts were to be wiped out some day. Thus, understanding how one might build various math tables, but especially the principles and techniques behind them (i.e. how to actually compute the logs in an efficient way), is always of interest to me.
This is not a critique, but more of a suggestion: While I appreciate how much friggin' info you've managed to cram into this video, I must admit that I actually had to play a large portion of it at 0.75x speed just so I could follow along! 😅 Also, there were several (and I mean several!) places in the video where you cover a topic so quickly -- I'm sure a fellow mathematician wouldn't have trouble with it, but I'm not such a mathematician! 😅 -- that I could imagine an entire video presentation on just that one topic that by itself would be very worthwhile (IMHO) and could take up minimum 10 minutes on its own, if only just presented at a slower pace and 'spelling out' a few more of the details.
Perhaps it might be worth your consideration to make some videos that go at a more pedagogical pace for a broader audience of math-interested lay people. Two excellent channels that I'm sure you've heard of are Mathologer and 3blue1brown. Mathologer in particular is able to tackle pretty math-heavy topics yet at a pace that a fairly broad audience can still handle. Of course, they use animations, and I'm not asking for that; just for the overall pacing.
For example, I imagine that this one 33 minute video could probably be expanded into at least 5 or 6 'episodes' or 'parts' on this one topic and/or sub-topics, and each such video somewhere in the neighbourhood of 10-20 minutes at a more casual pace.
Again, this is just a suggestion. It may take more time to make such videos (but then again, if it's more leisurely, and there's less pressure to 'fit everything in', then maybe it might actually take less time?), so I'd absolutely understand if that's not a direction you'd like to go in. Just an idea! 😊
Yes I also think it's very important to know how to build up math from scratch, it's the only sensible way math should be explained in my opinion.
Knowing where a certain piece of knowledge comes from is even more important than the piece of knowledge itself. Unfortunately almost noone adopts this approach, people want to skip ahead to the "good" part and often give circular arguments, or derivation using more complicated theory that may even depend on the truthfulness of the fact they're trying to explain itself.
One thing I've learned while studying maths is that understanding the whole tree of axioms and proofs that lead up to a fact, that you use as a given, is one of the most valuable thing you can have, contrary to what 99% of people(mathematicians included) might say.
Yeah, I stopped trying to follow along.
Impressive, very nice.
Glad you like it!
Perfect handwriting there .. impressive
Through the years, I have bought some great books about history of mathematics. Some of them I remember being about great men who advanced math knowledge. Some on the other hand were more tuned to math itself. Too bad, I now could locate only one of the history tuned ones, Davis & Hersh: The Mathematical Experience. All my presently on-hand books are more about newer math tables and functions. However, somewhere I still have a book with a title something like "Personalities in Math", just don't remember where. But as a side note, I used to have two slide rules, one pocket size and another longer one for more challenging calculation that I kept in my briefcase. Then came the pocket calculator years. But a couple of years ago I found on eBay another 12 inch slide rule and bought it. It is now within easy reach on top of my desktop PC. Loop closed...
I could not make it past the part talking about ancient Sumerian banker. That is fine and interesting and all, but probably should have been another video all to itself.
I tuned in hoping to hear about the history and origin of exponent and log function. A small blurb about how they have their origins in ancient banking and compound interest (see video linked in the description!) would have sufficed. Just my two cents.
My understanding is that Napier's book had 90 pages of tables with 30 entrees per page, or 2700 entrees total. That would be 45 degrees starting from 90. From 45 degrees to 0 would be redundant information.
This is knowledge explosion!
fantastic!
Thank you! Cheers!
@@DanielRubin1 i get the amount of research, editing and knowledge that goes into making these videos. excellent contents and presentation. keep up the good work!
Thanks man. I am from India , not good English but still easy to understand your English accent.
6:35 if you want to know more about the ancient economy i highly recommend "Debt: The First 5000 Years" by David Graeber.
How did this video get so little likes? It's amazing! Keep up the awesome work
I was wondering the same thing. Could always use more! Thanks!
Interesting. Animals that store food are banking. I never thought of it that way.
Thanks!
we learnt how to use log tables and slide rules in the 70s, probably amongst the last kids who did once electronic calculators came on the scene
Hello Daniel,
Watching your video made me realize how dumb I am and what extraordinary talent you are. I'm just jealous of your brain. Exceptional work. Thanks
that's a lot of knowledge
I been asking this and here i am.
Please take a look at “The Cultural Foundations of Mathematics” by Professor CK Raju. It is viewable as a Google book. The Fibonacci series was actually found by Hemchandra when he tried to solve a Sanskrit music problem of metering. Sin is actually a howler for the Sanskrit Jya which means a cord in a circle. Jya got mistranslated by the Arabs as Jeb(pocket). Jeb got translated to Sinus and that is how you have Sin theta.
Kepler: equal areas of arc in equal periods of time
Coasting in the Mediterranean Sea had two impetuses: avoiding the Barbary Pirates and Coastal Trade Routes
...converted to "our" units is 128 bln bushels...
yeah, thanks a lot, really helped me out...
Subscribed! 🤓
29:00 when calculating these values, was Napier using scrap paper? a chalkboard? I'm not aware that paper was disposable like it is today.
My father used a slide rule in college and I own it now. I learned to use it as a child.
Cool
Interesting algorithms.
amazing! very impressed. could you make a video about how briggs' construction of common logarithm was made? and i also recommend you to make a videos for layperson like me. you deserve more subscribes.
I already knew all that! 😁
⭐️
the fact the mathematicians in 1500 were gamblers is wild
Have you had a chance to do a video on Lambert's W function?
I had to look it up. I've never used that function. Do you use it for something?
Please do a video on old school celestial navigation
Thanks for this video. I have met people and worked with people in different trades and different professions people who knew their profession or their trade better than possibly anyone else, than what you could expect any expert in their field could ever accomplish but yet they did not have the gift of knowing how to break it down simple and teach and convey it to others. I have known people in the welding field for example, they were very good welders , but they didn't know how to teach it to others, they would have been some of the worst teachers and would have just turned people off. I think when teaching anything to another person you have to go slow, you can't speak the jargon or the lingo relevant to that field very fast, like you're running a race, and you have to explain every little segment as you go along, because if you go too fast, some people are just gonna get overwhelmed with the lingo that they've never heard before and they're just gonna tune in to some other place hoping that somebody else will be gifted to be able to teach it and break it down so they can learn it more easy , And that's why we see in a lot of our schools , or let's just say, across the nation , that other countries, I think we're 29 on the list of being below other countries when it comes to math knowledge, there are teachers that can teach you math and make you love it but then there are other teachers that try and engage themselves to teach you and make you hate the very thing that you wanted to embark upon. China turns out more engineers and scientists than the USA, they prioritize their students to learn math if nothing else. Some teachers can teach calculus so easy, some other teachers will only further confuse students and make them forsake their major in college that required high grades in calculus, it's so sad that it's like this in the USA, we have to import nuclear physicist from other countries, or the Manhattan project never would had been successful. Teachers at the beginning of the year need to explain to their students first WHY THEY NEED THIS MATH KNOWLEDGE AND HOW IT IS RELEVANT TO THEIR FUTURE CAREER, because many students sit in class daily, bored, asking themselves how is this or knowing it going to help me in life, then they sadly divorce their attention and thoughts from the math lectures being given to them by their teacher daily, and flunk it. Math teachers 3 times a year ought to expound to their students their need to know all math through calculus. But nothing is going to change for the better in this country. When you have a Supreme Court that legitimizes men marrying men and women marrying women, then you have set the stage and advanced a nation to be ripe for the judgment of God, every segment that comprises human life from that point on races forward into perversion and degradation, darkness is upon this land and people, the highest court in the land have shaken their fist at God, as God spoke through the prophet NAHUM, (I will pour abominable filth upon thee), so now we see this in a country that God previously blessed, but no longer, this nation is cursed in the food market and gas pump, and it's going to elevate into cataclysmic horror, when God fights against a nation that backsides from His commandments and laws, believe me, there are no brighter days ahead folks. Start storing food, water, fuel, and all necessities, because the locust storm heading our way will burn up our food supplies, martial law will become the rule of the day, and fear of daily existence will rob people of what peace and contentment they once knew. God is no respector of persons, the very elected leaders that we see daily make and act upon insane drafted policies , designed to steal and rob us of peace, safety, and prosperity, are part of the punishment from God, who is now fighting against a nation that embraces perversion and prosecutes those that are against it. IM NOT A LOOSE CANNON, I'm just reverberating what investigating journalists tell us daily in local and nationwide broadcast, granted the freedom to do so by the constitution of the USA, and the freedom of information act. The truth hurts and offends those that are guilty of violating the principles of wisdom, who seek to walk in their own ways that are destructive to theirselves and others. Back to math knowledge, watch what I proclaim, next year USA students will fall even lower in the nation, a nation that through its court systems have shaken their meager fleshly fist at God and rebelled horribly against Him.
Mr.Kingston: Paragraphs and coherence of topics would be greatly appreciated.
When NATURE decides that the time has come to erase a particular group of people from the universe, NATURE makes them go insane gradually, and slowly but surely they will implode.
Why does this happen ?
It happens because of "karma",
an ancient law of the universe.
No living person can hide from karma !
You lost me at “judgement of God.” A shame, you started by making sense then lost it.
I guess. Usually it is up to the student to succeed despite being around a lot of losers. Only people who care enough to seek knowledge will find it. Your other points about the nation in disarray do not seem well connected to the content and spirit of the video so not sure how to respond to such a passionate tangent.
Thanks for recommending this to me, The Algorithm.
Great video. I noticed that your markers are bleeding through the paper to the one beneath it. I have the same problem as I use many colored sharpies for drawing or other things. What I did was take a sheet of cardboard from the back of one of those lined notepads that don't have a cover and the pages flip up instead of to the left, I put that cardboard sheet underneath the sheet of paper I'm working on to avoid ruining the sheet(s) below it. I noticed that you often flip the underneath page over a lot, I'm assuming that's because of the ink bleeding onto the sheet. I could be wrong, just what I'm assuming based on what I see. If I am correct then I hope this tip helps you out a little bit. If I'm wrong, then I apologize for the misunderstanding. Peace out and keep up the excellent work!
You're right, my markers do bleed through. Still looking for better ways to do these videos. Will try the cardboard.
Oh wow, your handwriting looks exactly like mine in this video ✍️😋👍
Very nice video. Thank you. I am sorry to say that I saw an often made mistake. The map of mercator is cilindrical, but not central. That wold lead to a growth of the lattitude proportional to \tan(\phi), actually the lattitudes, in order to strech a loxodrome to a strait line, is to be proportional to \log(\tan(\phi+\pi/4)). So the projection is non-geometric, Gerard Mercator did not used logarithms, he probable used a paper globe.
Off topic comment:
Handwriting is beautiful.
Thanks!