113 Matching Annotations
  1. Dec 2017
    1. 1.

      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.

    2. I suppose the diameter of the sun to be about 30 times that of the moon and not greater.

      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."

    3. Pheidias

      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.

    4. King Gelon

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

    5. Sicily

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

    6. Again there are some who, without regarding it as infinite, yet think that no number has been named which is great enough to exceed its multitude.

      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.

    7. Zeuxippus

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

    8. universe

      "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

    9. Aristarchus

      Aristarchus of Samos Source: Wikimedia.org

    10. Samos

      Google Earth link to the Island of Samos.

    11. a book consisting of some hypotheses

      The book was called On the Sizes and Distances.

    12. His hypothese

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

    13. proportion

      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

    14. We must however take Aristarchus to mean this: since we conceive the earth to be, as it were, the centre of the universe, the ratio which the earth bears to what we describe as the 'universe' is the same as the ratio which the sphere containing the circle in which he supposes the earth to revolve bears to the sphere of the fixed stars.

      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.

    15. some exceed in multitude the number of the sand which is equal in magnitude to the sphere referred to

      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).

    16. some have tried, as you are of course aware, to prove that the said perimeter is about 300,000 stadia

      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.

    17. ten times the size that my predecessors thought it

      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..."

    18. Eudoxus

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

  2. Feb 2017
    1. radiation

      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

    2. entropyS

      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.

    3. a limiting case

      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.

    4. Wien’s Equation

      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.

  3. Jan 2017
    1. for it is proportional to the weight

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

    2. It is this quantity that I mean hereafter everywhere under the name of body or mass.

      The definition of mass!!!

    3. air of a double density, in a double space, is quadruple in quantity

      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.

    4. The quantity of matter is the measure of the same, arising from its density and bulk conjunctly.

      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.

    1. Mr. Pound's of the proportion of the diameters of Jupiter

      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!

    2. Cambridge. Trinity College

      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.)

    3. prolix

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

    4. publishing

      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.

    5. Mr. Edmund Halley

      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.

    6. for I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of bodies

      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.

    7. the phænomena of motions to investigate the forces of nature

      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.

    8. five powers

      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

    9. forces

      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.

    10. it comes to pass that geometry is commonly referred to their magnitudes, and mechanics to their motion

      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!!

    11. if any could work with perfect accuracy

      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.

    12. rational, which proceeds accurately by demonstration: and practical

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

    13. philosophy

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

    14. Pappus

      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.

    15. occult

      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...)

    16. the science of mechanics

      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".

    17. The Author's Preface

      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.

    1. Such people have the same kind of mind as do those who deny the antipodes on the grounds that one cannot walk with his head down and his feet attached to the ceiling

    2. but as to the first, I think I could clear that up myself.

      He fell for it...

    3. The first was that if it were true that the sun and other stars did not rise over the eastern horizon, but the eastern side of the earth sank beneath them while they remained motionless, then it would follow that after a short time the mountains, sinking downward with the rotation of the terrestrial globe, would get into such a position that whereas a little earlier one would have had to climb steeply to their peaks, a few hours later one would have to stoop and descend in order to get there.

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

    4. The other was that if the diurnal motion belonged to the earth, it would have to be so rapid that anyone placed at the bottom of a well would not for a moment be able to see a star which was directly above him, being able to see it only during the very brief instant in which the earth traverses two or three yards, this being the width of the well. Yet experiment shows that the apparent passage of such a star in going over the well takes quite a while--a necessary argument that the mouth of the well does not move with that rapidity which is required for the diurnal movement. Hence the earth is motionless.

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

    5. You wonder that there are so few followers of the Pythagorean opinion, whereas I am astonished that there have been any up to this day who have embraced and followed it.

      Ha! He's so funny.

    6. So take a sheet of paper and the compasses; let this page be the enormous expanse of the universe, in which you have to distribute and arrange its parts as reason shall direct you. And first, since you are sure without my telling you that the earth is located in this universe, mark some point at your pleasure where you intend this to be located, and designate it by means of some letter.

      New homework assignment for Aristotle Class! Hurrah!

    7. They, as most reverent and most humble slaves of Aristotle, would deny all the experiences and observations in the world, and would even refuse to look at them in order not to have to admit them

      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.

    8. fault of the computer

      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. https://youtu.be/WnZRw8juTsQ

    9. In the long run my observations have convinced me that some men, reasoning preposterously, first establish some conclusion In their minds which, either because of its being their own or because of their having received it from some person who has their entire confidence, impresses them so deeply that one finds it impossible ever to get it out of their heads.

      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.

    1. Nor in this case is there any use in Copenicus saying that this motion, because it is natural to the earth and not constrained, works contrary effects to those of forced motions; and that things which are given impetus are destined to disintegrate and cannot long subsist, whereas those made by nature maintain themselves in their optimum arrangement.

      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.

    2. Aristotle's arguments are drawn mostly from the things around us, and he leaves the others to the astronomers. Hence it will be good, if it seems so to you, to examine those taken from earthly experiments, and thereafter we shall see to the other sort.

      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.

    3. Then why are two circular motions not contrary? Being made upon the surface of the land or sea, which as you know is spherical, these motions become circular. Do you know what circular motions are not contrary to each other, Simplicio? They are those of two circles which touch from the outside; one being turned, the other naturally moves the opposite way. But if one circle should be inside the other, it Is I . impossible that their motions should be made in opposite directions without their resisting each other.

      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.

    4. And I question whether Aristotle entirely understood it when selecting it from some good school of thought, and whether he has not, by altering it in his Writings, made it a source of confusion among those who wish to maintain everything he said.

      Oh-- I need to find this in Aristotle!

    5. I should think that anyone who considered it more reasonable for the whole universe to move in order to let the earth remain Fixed would be more irrational than one who should climb to the top of your cupola just to get a view of the city and its environs, and then demand that the whole countryside should revolve around him so that he would not have to take the trouble to turn his head.

      Love this!

    6. For a motion which is perceived only, for example, in the moon, and which does not affect Venus or Jupiter or the other stars, cannot in any way be the earth's or anything but the moon's.

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

    7. For a simple movable body there can be but a single motion, and no more, which suits it naturally; any others it can possess only incidentally and by participation. Thus when a man walks along the deck of a ship, his own motion is that of walking, while the motion which takes him to port is his by participation; for he could never arrive there by walking if the ship did not take him there by means of its motion.

      Thought Experiment on the Principle of Relativity, Part 2.

    8. it does not act, and is as if It did not exist. Thus the goods with which a ship is laden leaving Venice, pass by Corfu, by Crete, by Cyprus and go to Aleppo. Venice, Corfu, Crete, etc. stand still and do not move with the ship; but as to the sacks, boxes, and bundles with which the boat is laden and with respect to the ship itself, the motion from Verflice to Syria is as nothing, and in no way alters their relation among themselves. This is so because it is common to all of them and all share equally in it. If, from the cargo in the ship, a sack were shifted from a chest one single inch, this alone would be more of a movement for it than the two-thousand-mile journey made by all of them together.

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

    9. ?

      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.

    10. having divided the universe into two parts, one of which is necessarily movable and the other motionless

      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.

    11. it is operative only in the relation that they have with other bodies lacking that motion

      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!"

    12. I should have thought it somewhat older.

      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."

    13. Peripatetic

      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.

    14. Motion, in so far as It is and acts as motion, to that extent exists relatively to things that lack it

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

    1. I cannot without great astonishment -- I might say without great insult to my intelligence -- hear it attributed as a prime perfection and nobility of the natural and integral bodies of the universe that they are invariant, immutable, inalterable, etc., while on the other hand it is called a great imperfection to be alterable, generable, mutable, etc. For my part I consider the earth very noble and admirable precisely because of the diverse alterations, changes, generations, etc. that occur in it incessantly. If, not being subject to any changes, it were a vast desert of sand or a mountain of jasper, or if at the time of the flood the waters which covered it had frozen, and it had remained an enormous globe of ice where nothing was ever born or ever altered or changed, I should deem it a useless lump in the universe, devoid of activity and, in a word, superfluous and essentially nonexistent. This is exactly the difference between a living animal and a dead one; and I say the same of the moon, of Jupiter, and of all other world globes.

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

    2. for an illusion of the telescope

      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.

    3. nor have I ever set much store by Tycho's verbosity

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

    4. What you refer to is the method he uses in writing his doctrine, but I do not believe it to be that with which he investigated it. Rather, I think it certain that he first obtained it by means of the senses, experiments, and observations, to assure himself as much as possible of his conclusions. Afterward he sought means to make them demonstrable. That is what is done for the most part in the demonstrative sciences; this comes about because when the conclusion is true, one may by making use of analytical methods hit upon some proposition which is already demonstrated, or arrive at some axiomatic principle; but if the conclusion is false, one can go on forever without ever finding any known truth -- if indeed one does not encounter some impossibility or manifest absurdity. And you may be sure that Pythagoras, long before he discovered the proof for which he sacrificed a hecatomb, was sure that the square on the side opposite the right angle in a right triangle was equal to the squares on the other two sides. The certainty of a conclusion assists not a little in the discovery of its proof -- meaning always in the demonstrative sciences. But however Aristotle may have proceeded, whether the reason a priori came before the sense perception a posteriori or the other way round, it is enough that Aristotle, as he said many times, preferred sensible experience to any argument. Besides, the strength of the arguments a priori has already been examined.

      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.

    5. I declare that we do have in our age new events and observations such that if Aristotle were now alive, I have no doubt he would change his opinion.

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

    6. You say that alterations which may be seen near at hand on earth cannot be seen in America because of the great distance. Well, so much the less could they be seen in the moon, which is many hundreds of times more distant.

      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?

    7. But if you have to content yourself with these visible, or rather these seen experiences, you must consider China and America celestial bodies, since you surely have never seen in them these alterations which you see in Italy. Therefore, in your sense, they must be inalterable.

      Nerd-burn again! Ha, ha!

    8. nothing will do but circular motion or rest


    9. And from this he infers it to be necessary and proper that all simple motions are confined to these three kinds; namely, toward the center, away from the center, and around the center.

      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.

    10. in order to prevent the things they admired from being exposed to the slander and scorn of the common people, the Pythagoreans condemned as sacrilegious the publication of the most hidden properties of numbers or of the incommensurable and irrational quantities which they investigated. They taught that anyone who had revealed them was tormented in the other world.

      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.

    11. I do not even understand, let alone believe, that with respect to legs, for example, the number three is more perfect than four or two

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

    12. a certain Peripatetic philosopher whose greatest obstacle in apprehending the truth seemed to be the reputation he had acquired by his interpretations of Aristotle

      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.

    13. These men indeed deserve not even that name, for they do not walk about; they are content to adore the shadows, philosophizing not with due circumspection but merely from having memorized a few ill-understood principles.

      Ha! Nerd-humor/Nerd-burn!

  4. Dec 2016
    1. energy is distributed over the vibrations of the resonator and overthe various of the radiation present in the medium, and what will be thetemperature of the total system

      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?"

    2. bounded by reflecting walls.

      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).

    3. ight velocity c

      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.

    4. diathermic medium

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

    5. Nof frequencyν(per second),N0of frequencyν0,N00of frequencyν00, ...,with allNlarge number

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

    6. the solution of the problem

      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."

    7. arbitraryand complicated

      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.

    8. one constant of nature only

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

    9. without knowing anything

      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.

    10. thereal core of the theory

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

    11. Mr. Boltzmann

      See Newton at the Center for Boltzman's Constant

    12. the same stationaryradiation field

      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?

    13. time intervals long compared to the period of one vibration

      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.

    14. monochromatic

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

    15. Max Planck's 1900 Paper Part B

    16. On the Theory of the Energy Distribution Law

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

    17. radiation
    18. E=Cλ−5ec/λT−1
      • 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.
    19. probability

      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.

    20. Sas a logarith-mic function ofU

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

    21. ompletely arbitrary expressions

      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.

    22. Uitself must also be known for this

      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.

    23. nidentical processesoccur independently

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

    24. const

      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).

    25. 2

      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.

    26. written to me

      Scientific correspondence is still a big deal.

    27. f(Un)

      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.

    28. a stationary radiation field

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

    29. n

      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.

    30. the evaluation of an infinitesimal increase

      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:

    31. vibrational energyU

      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.

    32. linear resonator

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

    33. Spectrum

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