If we were to graph the position of the tip of the light wave (the arrow head) along the x-axis and the time that has passed along the y-axis, we would get a graph something like this. (This is the kind of graph we are going to turn in to our Feynman Diagrams in Quantum Mechanics.)

So, now we have 4 observers.

• 2 were at the original locations where the clocks were when we did the purple light thing and we were stationary
• The other 2 are at either end of the rod as it is moving in the frame of reference. 1 is at one end, the other is at the other end.

"Current kinematics" means "Newtonian" or "Classical" physics. And, what current kinematics says is: the length of the rod is the length of the rod, it doesn't matter if it's moving or not.

In the paragraph above, the rod has to be l long-- or we just don't know anything about anything and can't get started.

Here Einstein is saying, "I haven't told you why it's going to be different when things are moving. But, it will be."

It's the science version of dramatic foreshadowing.

In (b) instead of being exactly sure where the endpoints are, you are sure when the end points are.

He is going to go into a big explanation (and there will be lots of pictures) about how we are exactly going to do this.

But, let's think ahead. If you know that a light beam is turned on at exactly the when of the beginning of the rod and you know that after a certain amount of time the light ray reached the end of a rod, you could use the speed of light to find out how long that rod was. For example, if the light was on for a year going from 1 end to the other, you would know that the rod was 1 light year long.

In the way of measuring he talks about in (a) you, the rod, the yardstick, the whole frame of reference-- all of you are moving uniformly (no acceleration).

So, you can't tell if you are moving or standing still and the measurement of the rod is what we expect. It turns out to be l.

The "normal" laws of physics still work on all inertial reference frames.

In other words, you are not accelerating.

Translation: The speed of light is special. It is the same for all observers-- whether they are moving or not.

Einstein makes the thought experiment famous. Since he speaks German, people will also commonly refer to these as Gedankenexperiment -- which is just the German for "thought experiment".

If you have been taking the Hakim classes since the beginning, you may think this sounds familiar. This is very close to one of Euclid's axioms. Today we call it Transitive Property of Equality. It says that if A = B and B = C, then A = C.

He hasn't yet explained why we can't talk about a common time unless we do this. We have to work through this thought experiment to see why he says that we can't.

Observer A knows what time things are happening as long as they are nearby. Why just when they are nearby? He's going to get to that.

Translation: If the point is moving (maybe it is the fly in Descartes' room), then if you want to know where the point is, you need to know two things: (1) What is the function that we can input time and get out the coordinates for this thing? And, (2) What time is it right now?

Let's say this always stays at y = 0 and z = 2. But, x = t. AND the current t = 3. Then, we can see that the point has moved to a new spot.

We will use a clock on the wall. The clock tells us what time it is in our "stationary" frame of reference.

Pay attention to his italics. Simultaneous is about to be redefined!

Translation: We can plot a point in that "stationary" coordinate system.

Just the normal 3-d world you have spent your whole life living in. We're pretending with Einstein that it is stationary. Do you think this means he also invented ironic quotation marks?

Assumption 1: The Earth is no bigger than 529,200 km round. (I converted it from his 3,000,000 stadia.) This means that the radius of this "no bigger than" Earth is 84,225 km.

Again, Archimedes is saying, "You can't say my numbers are too small. I will take the biggest estimate and then make the one I use even bigger."

We know very little about this Phidias. Some think he was Archimedes' actual father and others think he was only a teacher that Archimedes thought of as a father-figure.

King Gelon was the 5th century BCE ruler of Gela) and Syracuse.

Google Earth of Sicily, now a part of Italy. But, in Archimedes' time considered part of "Magna Graecia"

Archiomedes' main reason for writing the Sand Reckoner is not so that he can get an accurate estimate of the number of sands in the Universe. It is so that he can put forth his arguments about the number system the Greeks used at this time and how that system should deal with very large numbers. The Greeks knew about the concepts of zero and infinity and even negative numbers. But, they rejected them as things that should be used in math. Archimedes is making his case for very large numbers.

Zeuxippus was the immediate successor to Euclid as a mathematics professor at Alexandria and a correspondent with Archimedes during his life. From Wikisource.org

"A term for "universe" among the ancient Greek philosophers from Pythagoras onwards was τὸ πᾶν tò pân ("the all"), defined as all matter and all space, and τὸ ὅλον tò hólon ("all things"), which did not necessarily include the void." from Wikipedia.org Image from UCR.edu

Aristarchus of Samos Source: Wikimedia.org

Google Earth link to the Island of Samos.

The book was called On the Sizes and Distances.

Image is fron an excellent article on Aristarchus' theories at AncientGreeceReloaded.com. (Use google translate to translate text to English.)

For my GHF class "Aristotle Leads the Way" notice the use of our friend for the year: proportionality. Image is from a great article on proportionality at ExamRefresh.com

It's really hard to understand what he is saying here. So, most scholars use the description of the math below to figure out what he meant to say here.

Archimedes is saying that he will prove that he has thought up numbers that are bigger than the number of sands in the Universe. Unfortunately, we have lost all copies of his work called "The Principles" that he is referring to. So, we use this work to talk about Archimedes' very large numbers until we find a copy of that original work (if one still exists).

Archimedes' good friend Eratosthenes is the person who has estimated the perimeter of the Earth to be about 300,000 stadia. Eratosthenes actually said that the circumference of the Earth was 252,000 stadia which converts to about 44,452.8 km. The actual value of the polar (north-south) circumference of the Earth is approximately 39,940.6 km. While this yields a 10% error it was a vastly superior estimate than any other we have a record of from a 1,000 years before or after Eratosthenes.

Is Archimedes saying his good friend, Eratosthenes was wrong? Not really. He is saying, "Look, I don't want you to criticize my number as too small. So, let's put the Earth at 10 times the current estimate just to prove my point..."

Eudoxus said that the Universe was made of a system of spheres. Image Credit: Prof. Richard Pogge, Ohio State

What in the world is a Galenist?

Who knew-- meditation!

So, here is a glimmer of a theoretical reason. And, it is fascinating to consider in this context. What shape is the Universe? (PBS video link here.)

Sigh. The Scientific Method still had a way to go...

Compare the tone to Galileo. (References)

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This is always problematic-- evolution

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we do all of this in the class-- New Newton students need links to Aristotle homework

add note later

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Ecphantus the Pythagorean

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scientific method

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Pythagoras-- are there other famous ones he is also referencing here?

The expression on the left hand side is a second derivative and it's a Calculus thing. When he says it would not have a general meaning, he's saying-- "you don't get to use Calculus" over a discontinuous function.

Remember in class when we discussed the definition of radiation. It means more than something that is radioactive. All things what emit light or things in the electromagnetic spectrum radiate. Like this radiant cook top is radiating in the thermal range (and in the visible light range of the spectrum as well-- thus the red light we see).f

We did not cover what entropy is in class. Entropy is discussed in Hakim's Newton at the Center book. Entropy is the tendency of the Universe to move toward more disorder. But, it will turn out in Physics and Chemistry that we can measure entropy, energy and a thing called enthalpy. It takes the 3 of them together to decide if a physical process or chemical reaction can happen in thermodynamics.

A limiting case is extremely useful in science. It is something that will usually give us an absolute max or min. So, Planck is saying here that we can use Wien's equation to form a boundary to at least check that we are in the right zone for the correct answer.

So, what was this Wien's Equation that Planck was improving? Wien had come up with a formula that was really close to the one that Planck would describe. But, it didn't work for short wavelengths and high frequencies. Rayleigh and Jeans had earlier come up with a straight-line formula that worked for larger wavelengths and lower frequencies, but not the higher ones. Planck's was the first to work for both. To review the relationship between wavelength and frequency, see the phet simulation here.

Mass is proportional to weight. But, it is not the same thing.

The definition of mass!!!

OMGoodness! I see why no one went to his lectures. What is he saying is that the scientific community has done some experiments and these experiments give us results that make us think the relationship between density and the "quantity" of a material (the amount of matter, now know as the mass) is directly proportional. Here's how the math works out:

He starts with...

• I have a gas of some density, \(d_{1}\). Using \(d_{1}\) is a fancy =, mathy way to say-- I'm going to talk about a lot of things called d. Just remember, this one is the first one. We don't know what \(d_{1}\) is equal to. But, we don't need to yet. We just need to remember that $$density{1} = \frac{mass{1}}{volume_{1}}$$
• Now, if you have a new gas whose density (we will be calling that \(d_{2}\) )is equal to \(2*d_{1}\) . But, \(volume_{2} = 2*volume_{1}\).
• How much "quantity" (matter) is there in the new gas? Basically, what is \(mass_{2}\) equal to? Let's find out:

Here are the facts we are given:

$$d{1}= \frac{mass{1}}{volume{1}}$$ AND $$d{2} = 2*d_{1}$$

I will substitute what \(d_{1}\) equals in the first equation into the second equation and I get this:

$$d_{2} = 2*\frac{mass_{1}}{volume_{1}} $$

But, we can also say \(d_{2}\) this way:

$$\frac{mass_{2}}{volume_{2}} $$

So, I substitute that in on the left hand side and we get this:

$$\frac{mass_{2}}{volume_{2}} = 2*\frac{mass_{1}}{volume_{1}}$$

But, he says we have "double space" in the second case. So, I can substitute \(2*volume_{1}\) for \(volume_{2}\). That gives us:

$$\frac{mass_{2}}{2*volume_{1}} = 2*\frac{mass_{1}}{volume_{1}}$$

If I multiply both sides by \(volume_{1}\), I am left with:

$$\frac{mass_{2}}{2} = 2*mass_{1}$$

Multiply both sides by 2 and we get:

$$mass_{2}= 4*mass_{1}$$

Yep, the "quantity" has quadrupled. You can try the next one about the triple in space (volume) for yourself.

Sometimes math makes things so much easier. Today, we call the definition of the density of an object its mass divided by its volume. $$density = \frac{mass}{volume}$$ He's saying-- that other part of density that isn't the volume? Yeah, that's the matter. We now say that mass is "the amount of matter in an object".

Newton begins with the definitions of things like a good philosopher. He is following in the tradition of the big 3: Socrates, Plato and Aristotle.

Oh, the snark! Hooke had done a lot more work on Jupiter than J Pound. But, Newton can't stand to give him credit for anything good!

It is still there today! Stephen Hawking holds the office that Newton did and teaches there now. To me, it always reminds me of Hogwarts. Even their dining hall looks like the one at Hogwarts: And, students have to wear robes to classes. So, I guess if you don't end up getting your letter from Hogwarts, you can always go here for college or grad school as a consolation. (This is from Oxford-- another Hogwarts alternative that also requires academic/wizard robes.)

What a great word! It means using too many words.

During the Scientific Revolution we got the "race to publish" that is still with us today. If you wanted to be credited with coming up with a new idea, you needed to publish it first. Thanks to the influence of Madame Lavoisier, there were new rules to what "publishing" an idea meant. You had to really describe all that you had done so that someone else could reproduce it and verify what you had actually discovered. This was a BIG change from Alchemy-- where alchemists would claim they could turn lead into gold, but never produced the steps so that people could check to see if their claims were true or not.

So nice to see that Newton didn't fight with all of his contemporaries-- like he did with Hooke and Leibniz!! Halley is most famous for predicting the return of the comet that hears his name. I got to see it in my youth in 1986. It will be back in 2061. But, if you can't wait that long, why not catch a glimpse of the dust trail left behind Halley's Comet? We run into that dust trail once a year. It is known as the Orionid meteor shower.

Wow! How right he was. It turns out that gravity-- which seems like a really big deal to us-- is one of the weaker forces in the Universe. Checkout this great series by SciShow on the 4 Fundamental Forces.

Investigating forces is hard-- because they are invisible. Newton is saying that we have to look at how tings move in order to understand these invisible force things.

Hunh. Does he mean the 5 powers of the soul of every living thing that Aristotle talked about? I guess that would make sense for everything to share those 5 powers since Aristotle thought that fossils were evidence of rock trying to turn itself into living things. If this is what he is referring to, the 5 powers are: "(1) The nutritive: This is the power living beings have to grow and take in nourishment. (2) The appetitive: This is the power of desiring. (3) The sensory: This is the power of perceiving things with the senses. (4) The locomotive: This is the ability to move. (5) The reasoning." Quoted from an article on the 5 powers by David Banach

It's kind of a big deal that Newton is not the first person to use this word. Today, we DEFINE force as a mathematical quantity that is equal to mass*acceleration, thanks to Newton. But, people knew about forces in his day as well as in the time of Aristotle. They just weren't quite sure what they were or how they worked.

Wow! This is very different than today! When we study geometry we almost always study shapes and lines that are "static". That is, they are not moving. Imagine how hard geometry class would be if you had to add things moving!!

There are so many things about Newton's Laws of Motion that require an imaginary and perfect world for thought experiments to be done without friction that I am very grateful that he did not stick strictly to Aristotle's idea of "what we observe is all that is real". Plato helped him think of things in a perfect setting so he could figure these relationships out.

For example, the well-known dispute between Plato (rational only) and his greatest student, Aristotle (we must learn from practical observation).

Remember-- what we call science today was a subset of philosophy then. Maybe it still is?

Almost certainly, this is Pappus of Alexandria. He was a great mathematician. But, as the Greeks did not work with number systems as we do today, it is important to note that we might think of him as a great Geometer more specifically. Euclid was not the only Geometry rock star!! One day, I hope to work through all of his writings myself to try to understand math without numbers as the Greeks did. This reference at the beginning of Newton's greatest work is a reminder that Newton was rumored to have only laughed once-- when asked by someone "What use is geometry?" Newton loved geometry and knew the writings of Pappus, Archimedes and Euclid very well.

It is true that Newton studied things like spells and what we would now call the "Occult" in his study of Alchemy. But, here, he is using the word in an older meaning. Occult refers to hidden things. He is saying that they are investigating difficult problems that are not immediately apparent. (At least, that's what I think he's meaning here...)

The Science of Mechanics is the study of motion and forces. It is often taught separately in physics from Electricity and Magnetism. For example, at MIT, the freshmen core usually includes a Fall class-- 8.01 "Classical Mechanics" (It's Classical because it doesn't include relativity.)-- that is separate from the Spring class-- 8.02 "Electricity and Magnetism".

In Newton's original version, this was just called "The Preface". We call it the Author's Preface here because this work is actually a translation by Andrew Motte and he added his own preface and life of Netwon to his translation. Newton wrote his original in Latin-- the language of scholars of the day. The Romans may not have had a great love for the impractical Greek science nonsense, but by uniting Europe with a common language, they helped foster the Scientific Revolution by allowing for more rapid communication across all of Europe. Imagine how hard it would be for you if each of the states or provinces around you each had their own different language and did not understand yours.

He fell for it...

Oh, he got me! When I read this the first time, I thought Galileo was making a mistake here. But...

Fun! New addition to our Aristotle Leads the Way class for the Eratosthenes Homework!

Ha! He's so funny.

New homework assignment for Aristotle Class! Hurrah!

Aristotle famously fought with his teacher, Plato, that we should make observations. Galileo is making the point that the Peripatetics are just quoting Aristotle's writings. They have not understood what he said about how science should be conducted. Aristotle went out into the world to observe things for himself.

No, they didn't have computers. :-) The word computer used to refer to the person who is doing the math-- like in the movie "Hidden Figures" that tells the story of the African American women who did the math for the early NASA missions. For quite a while, many in computer science and astronomy thought that only a woman's brain was suited for being a computer or a programmer.

Still a problem for us today! This is why credible scientific experiments must always be designed to compensate for the "Cognitive Biases" of the experimenters and-- when humans are also the subjects-- of the subjects.

Simplicio has a point here. Copernicus has proposed how things work outside the conditions we have on the Earth. Simplicio is saying-- "Hey, you want us to observe things? When we observe things, stuff doesn't move perpetually without someone moving it on purpose." It turns out that Copernicus was right-- in space, we don't have the issues of air-resistance and some of the other forces that slow moving objects down.

Galileo puts himself forward as the true Aristotelian in several places-- because they were both people who decided to observe first and then decide things.

Galileo will get some stuff wrong about curved motion being the natural motion, but this bit he gets very right. And, it is something that is really hard for people to remember. We somehow "get" that linear momentum is a thing, but angular momentum is still amazingly surprising. Check out this video for an example of that.

Oh-- I need to find this in Aristotle!

Love this!

This concept would resonate with Newton and help us get to his Law of Universal Gravitation.

Thought Experiment on the Principle of Relativity, Part 2.

The Thought Experiment on the Principle of Relativity, Part 1.

For my students who are reading this to find out about the Principle of Relativity (Newton and Einstein classes) and not theories of the Universe (Newton and Aristotle classes), you might want to skip down to my next comment.

What is truly motionless in the Universe? That's a hard one. Currently, we measure all relative movement in astronomy against the Cosmic Background Radiation or the Cosmic Microwave Background. But, we currently believe-- based on the work of Edwin Hubble-- that space itself is expanding. So, there seems to be no absolutely "still" spot in the Universe. Galileo wants to divide the Universe into the moving and the motionless here. But, we now think that there is nothing in the Universe that is motionless.

• This video explains why the Big Bang should be called the "Everywhere Stretch."
• More on the Cosmic Microwave Background

If Jane and Mary have a motion that is "common" to both of them, it doesn't matter to them that they are moving. But, if Bob has a different motion, both Jane and Mary think that Bob is moving. It's obvious. But, they might forget the the two of them are both moving. My laptop and I are moving at a breakneck speed as the earth spins on its axis and orbits around the sun all while the sun and the Milk Way Galaxy its in are also moving. But, I don't think about that most days. And, I would say that we are both sitting still in a coffee shop right now and the Mocha sitting next to me is not going to suddenly spill. It has a motion common to me and my laptop. But, if we came across something that was totally still-- we would notice! In fact, I would think, "Wow! That thing is moving FAST!"

I love this! Galileo's attitude shines through. Aristotle? Really? Didn't any of those people who worked on ships figure this out for themselves? He's saying, "Hey-- your hero worship is showing."

The Peripatetics were a group of philosophers who thought while walking. Sometimes it is used to refer to traveling philosophers or teachers-- kind of like "visiting professor" today. But, here, the speaker is saying that what SALV has said agrees with Aristotle. That is, " This is good, sound doctrine, and entirely Aristotelian." The Peripatetics were Aristotelian.

This is where the discussion of Galileo's Principle of Relativity really gets going.

This is my favorite passage in all of Galileo's writings.

There are some optical aberrations that were illusions of the telescope. But, they really got conspiratorial in their theories about the telescope at this time. It was much like people today who think our images of the Earth are part of a NASA conspiracy.

Interesting! Galileo was a friend of Kepler-- Tycho's best student and the man who disproved his model of the Universe after his death.

But, here, Galileo may give Aristotle too much credit. Science historians currently see the real lack of new experiments to test a conclusion drawn from observations as the principle difference between the science done by people before the Scientific Revolution and the science done during and after it.

Aristotle isn't Galileo's enemy and he acknowledges it here.

Simplicio made the mistake of saying that there aren't any changes because he never saw any. But, he has no means of detecting them! How many of these assumptions are we making today because we haven't invented some new way of measuring our Universe and its many dimensions?

Nerd-burn again! Ha, ha!

:-(

Aristotle might be more right about this than Galileo is-- that is that motion is made up of straight lines. Galileo will make a mistake about "curved motion" to be natural motion. He did not understand-- as we do-- how vector decomposition could explain the parabolic trajectory of a cannonball.

Wow! He is taking on Pythagoras and his secret society as well as Aristotle. His Dad took on Pythagoras and his music theories when Galileo was a child and his father was a famous composer.

So funny! I wish I could have taken classes from him.

This is a problem of scientists even today. If you go ahead to read Day 4, you might wonder if Galileo himself has this problem-- since his explanation of the tides is a bit off.

Ha! Nerd-humor/Nerd-burn!

This is just another ways of saying the problem they are looking at: "When I heat something to a certain temperature, how can I predict the color of the light that will come off of it?"

He wants you to think about a closed system. If you are doing Newton at the Center or you had me for that class, this is what Lavoisier is on about (and why we had the slide with the walls around Paris over and over again).

So, in a perfect vacuum. Because in anything less than a perfect vacuum, the velocity of light is a bit slower. Like when we use a prism of glass or water to separate light out-- we are using the fact that the waves slow down differently in goopier stuff.

And, these things are in some stuff that lets heat flow through freely. Like a perfect vacuum.

He's just saying you have a bunch of them and they all have their own frequency.

But, he knows his results are the better than anyone else has gotten so far. So, he's saying-- "Hey, don't give up before you check out my awesome results at the end."

A Calculus solution would have been more elegant. He's being a bit defensive here, since his "quanta" won't look like a nice Calculus formula.

Constants of Nature are a big deal. For more on them, I recommend this book: The Constants of Nature

This is a big deal. He is saying that his new theory is general. And, it's a relief to us. Because it means that we can plug in values when we see them and that they should always work.

But, how much of the theory did he actually understand himself at this point?

See Newton at the Center for Boltzman's Constant

So, the M&M experiment will tell us that there is none. But, for the purpose of measurement for the experiment here, it acts as if there is a stationary reference frame. Remember our first homework where you built a reference frame to observe the retrograde motion of mars?

He's saying that you have to look at the experiment for a long time compared to the period of the vibration. The period of the vibration is the frequency. So, it's not hard to look at an experiment much longer than that. Basically, he's saying we would expect there to be a whole bunch of random wavelength/frequency thingies showing up over the time we take a measurement. But, that's not exactly what he found.

Monochromatic means one color. The color of light is determined by its wavelength.

Max Planck's 1900 Paper Part B

So, why is this so separated from the last part? In this part, he wants to make clear what is new.

Link to next paper.

• E is Energy
• C is not the speed of light here. It refers to a constant.
• $$\lambda$$ is always the wavelength
• the little e is a special irrational number called Euler's Number).
• And, I'm not honestly sure if it is being raised to c the speed of light or C the constant in question. And, that's OK. Remember-- we don't need to totally digest any of the papers we are reading for this section. You can always go back when you have more information.

This is not the last we have heard of probability. And, remember when we looked at the Bohr model of the atom? We said that those orbital tracks were useful, but not what the atom was really like. Turns out the electrons are in orbitals of probability clouds.

For what a logarithm is see the optional "Dragon Finding with Logarithms" homework on our class page or this fantastic video by Vi Hart.

They turn out not to be so arbitrary. But, I like how we can see how humble he is at this point. He is just plugging things in to the math that work without knowing why they work and without being able to solve the equation exactly and it's bugging him.

If the continuous function thing were valid, then we could always get from S to U and U to S. But, it's not continuous, so we can't.

Remember, the n is important because it means that what he's talking about is in countable buckets and not continuous like Calculus wants.

Some constant goes here. So far, h is working around with Wien's Law and he's saying that "if you take a special case" you will get to the point where the second derivative (see the 2's in the left hand side of this equation?) of the entropy (he called that S earlier) compared to the vibrational energy (he called that U earlier) is equal to some number divided by the vibrational energy (the U on the right hand side of the equation).

The 2 here and the one below look like we are squaring something, but it's more Calculus. The 2 in this case is the second derivative. The derivative is the slope of the line. If the line is curved, the second derivative is how fast the slope of the line changes.

Scientific correspondence is still a big deal.

f(something) is a special notation in math. It looks like it means fsomething. But, it doesn't. It means "an equation where the only thing you have to plug in is the something". And we say that "f is a function of something". So, f(x) means that "f is a function of x". Here is an f(x): xx*x + 3. To know what that is equal to, all you need to know is what x is equal. Here he is saying that there is some equation where you just needed to know U. U above was the vibrational energy. The little n next to the U means that it is the particular vibrational energy for the resonator he was talking about a minute ago. Don't worry about the math other than that. But, there are some \(\bigtriangleup\)'s here. Remember from our "don't look down" lessons that this means the change in something.

Spoiler Alert! There is no Stationary Radiation Field. Michelson and Morley prove that and Einstein will talk about what that implies in his papers.

n is kind of a big deal here. In statistics, it relates to the number of things you are counting. And, what "n identical resonators" means is that there are a countable number-- so it isn't continuous like with Calculus.

Infinitesimal is talking about Calculus. Remember that in class we said that Calculus was giving them the wrong answers. When you add up the infinitesimal stuff he is looking at you get something like this: But, what Planck found out was that the energy comes in distinct buckets. So, it would look more like this in reality:

Remember in class when we talked about the wave on the string simulation? The beads on the string stayed in place and just move up and down. The vibrational energy he is talking about here is the energy the person who holds the end of the string would feel when the wave got to their hand.

To understand more about resonance, take a look at this phet.colorado.edu simulation.

Remember-- Planck is thinking about the Ultraviolet Catastrophe. Physics Girl and PBS Space Time both have great videos on this.