Max

y12 + x22 = M2 x3/x2 = (M + L)/M x3 = ((M + L)/M) x2 (y3 - y1)/y3 = (M + L)/L y3 = (-L/M) * y1 x32/((M+L)2/M2) + y32/(L2/M2) = y12 + x22 = M2 which is indeed the equation of an ellipse. As the denominator in the x-term is larger than the denominator in the y-term, the long axis of the ellipse is in the x-direction.

So now it's again: 1 - 2*pi*X = x/(1+x) (X+1)*(1 - 2*pi*X) = X X + 1 - 2*pi*X2 - 2*pi*X - X = 0 X2 + X - 1/(2*pi) = 0 X = - 1/2 + 1/2 SQRT (1 + 2/pi) (here only the positive root makes sense) = 0.13965220479 Your estimate was not bad! My attempt was more difficult because I used a general distance "walked" Z instead of 1 mile, and the fact that I made a correction for the difference between the radius of a circle in a plane and measured along a meridian. For circles with radius about 1 mile on Earth the difference is negligible, but not for larger circles. Perhaps I was just too ambitious...

Ah, that makes it definitely less pleasant. Your previous version was a quadratic equation, easy to solve. Now you get a third degree equation. This can be solved, but doing that is as pleasant as a third degree interrogation. You'll find many methods on the Internet, but whatever method you use, the law of conservation of misery applies. Perhaps the easiest solution in practice is to use a numerical method like Newton-Raphson. It does also have its difficulties, but these are not too bad in comparison. I haven't looked further at your equations, as I'm now busy with my own calculations, and I'm already confused enough...

Could you tell me the meaning of your symbols? "1" = "1 mile: , or just "1"? pi2X: 2*pi*X? X = the "overshoot" on the small circle? Is 1 + X miles an approximation? Certainly justified in the case of 1 mile trips, but I'd like to know... At least your equation can be solved if I understand your notation correctly... I tried to be too general in my calculations, resulting in quite complex equations that can't be solved analytically.

I've tried to calculate the second problem, but I get always implicit functions that cannot be solved analytically, only numerically, so I guess this problem has no explicit solution. For generality I've used a variable for the "walked" distance instead of "1 mile".

We're talking here about the geometric north, ignore the magnetic poles.

No, that would be like objecting to Aristotle's paradox by insisting that his wheel is not a good car wheel, it's not in the spirit of the puzzle. It's just the conversion from miles to latitudes etc. We can always assume that we're talking about Jesus or St Francis of Paola, who allegedly could walk on water.

Similar to the solutions around the South Pole in the original puzzle, but now with the small circle with a circumference < 1 mile, such that after traversing 1 mile west you cross the meridian that crosses the larger circle 1 mile west of the starting point.

How many degrees, minutes, seconds, north... I'm not familiar with miles (nautical? statute?).

0.5 mile north of the equator. I'm too lazy to calculate the corresponding geographic location...

I agree, those terms were badly chosen. Further, the idea of a beneficial (let alone "benevolent") universe is a bit of a tautology: man evolved in such a way that he could survive in his environment. It's the anthropic principle again: we shouldn't be surprised that the universe makes intelligent life possible, we wouldn't be there to be surprised if that had not been the case. Bacteria in those hot springs could also wonder that their local universe is so beneficial to them. while they of course evolved in such a way that they could survive in that environment (which would be lethal for humans). But beneficial or not, it won't continue endlessly, one day, when that big asteroid comes, we're finished, probably long before the sun finally kills us. Even the dinosaurs (OK, with exception of the birds) were wiped out, while they had existed for some 100 million years, so they were exceptionally well adapted to their environment, and yet the universe decided one day to be no longer benevolent to them, to borrow for a moment the anthropic view of the universe.

You ignore the possibility that going south and going north can be done on the same line of longitude, while going west between these two displacements.

But the inner track is part of the original problem, as you easily can verify by reading the original text. After all this is Aristotle's paradox and not a "What I Find Intersting About Wheels" discussion. Aristotle wasn't trying to design some new wheel or commenting on the quality of the Greek wheels of his time, this is about kinematics, about circles rolling over tracks, not about dynamics. These may become relevant when constructing functioning wheels, but they are completely irrelevant for solving the paradox. That can be done by purely geometrical/mathematical methods. If you remove that second track, you throw the paradox-baby out with the track-bathwater. That is not the same as solving it.

I think the puzzle originally was supposed to have only 1 solution, until people discovered more solutions. Mum's the word. Well, one suggestion: make a few sketches, that may help.

That's an old riddle, I remember seeing it around 1960. Probably even much older.

I've no idea where this fancy border comes from...

Nobody ignores the fact that a point on the smaller wheel has a lower tangential speed than a point on the larger wheel. And an inner track is given, read the original text. No, it does not cancel out slippage, as I've shown in my proof (you should really read that some time). I've given the exact amounts of the various speeds, instead of your vague conjecture. He is correct, they do rotate with equal velocity, rotation velocity or angular velocity is the vector ω, with magnitude = number of revolutions/ time unit. The tangential velocity = ω x r and the tangential speed is ω*r. With boldface I indicate vectors, instead of trying to put an arrow over the variable (see also below). "x" indicates a vector product. The formatting has gone awry, but the explanation is correct, it is in fact the same as my proof, only using velocity vectors instead of speed scalars+direction. I've reformatted it, avoiding those pesky arrows by indicating vectors with boldface (and "x" for the vector product). vC = V + ω x OC and vC’= V + ω x OC’ [I see that I now somehow have used different fonts, ignore that; quirks of this editor.] if vC = 0 then V = - ω x OC [the translation velocity and the tangential velocity at the contact point cancel when there is no slipping] and vC’ = - ω x OC + ω x OC’ = ω x (OC’ - OC) = ω x CC’ This cannot vanish, unless C = C’. So the assumption that both circles turn without sliding is false. [ vC and vC’ are the velocity vectors of the contact points (6 o’clock positions) of respectively the larger and the smaller circle. vC = 0 is the condition that the large circle rotates without slipping, vC’ = 0 is the condition that the small circle rotates without slipping.]

It isn't rational to just dismiss "some paper" as the product of those awful scientists. The Danish study I referenced followed all children born in Denmark in the period January 1, 1991 to December 31, 1998, a total of 537,303 children followed for a total of 2,129,864 person-years. Read the paper to see how careful this study was set up, how painstakingly and meticulously all kinds of possible factors were taken into account. If you think you can just dismiss the study, you should point out the errors therein. Follow also the references in that study to see the results of other studies that come to the same conclusion. As I said before, "data" is not the plural of "anecdote" and "post hoc ergo propter hoc" is a common fallacy. With many millions of people it is statistically unavoidable that there will be "remarkable" coincidences. How impressive these might seem, in themselves they don't prove anything. Therefore you need large and carefully designed scientific studies, not a collection of anecdotes. In such cases I trust only scientific data. Not that these are automatically correct (far from it!), but at least I have some possibility to check the accuracy and the soundness of the methods used.

We now know that this is the mechanism behind the further rolling, but Aristotle didn't understand it, as I've shown in one of my previous posts. Therefore it is no longer a real puzzle for us, while it was an enigma for those guys in the past. But I'm glad to know that you now have also been converted to the Slipping School.

As I've shown before, cycloids are a completely unnecessary element added by you, allegedly "proving" that both wheels travel the same distance. Well, that they do, Aristotle already knew, you can read that in his text. So in that regard you don't prove anything that isn't already in Aristotle's text. The cycloids are just an irrelevant extra.

No, it isn't. Artistotle wrote: "nowhere does the greater stop and wait for the less in such a way as to remain stationary for a time at the same point" and "the smaller does not skip any point", so he considers only stopping of the large circle and skipping of the small circle as possible explanations for the problem. As he rejects these possibilities, he cannot solve the paradox, because he is not aware of the possibility of slipping (forwards for the small circle, forced by the large one, and backwards for the large circle, forced by the small one), which enables a continuous movement that explains the problem.

But even if you suppose that in only one of 1000 cases vaccination would cause autism, this would show up in the statistics if your sample is big enough. The question is not how many vaccinated children become autistic, but: is there a difference in the percentage of children diagnosed with autism between vaccinated and unvaccinated children? If there is no difference, then there is no evidence for the hypothesis that vaccination causes autism, that is elementary statistics. Now "data" is not the plural of "anecdote", you need a large sample to get reliable results. Such studies have been done, and the conclusion of all of them was that there is no evidence that vaccination causes autism. For example, there has been a large Danish study wherein more than half a million children were followed for 8 years ( https://www.nejm.org/doi/full/10.1056/NEJMoa021134 ). There was found no difference between vaccinated and unvaccinated children. In fact among the vaccinated children there was less autism diagnosed (but the difference was not significant). Further "There was no association between the age at the time of vaccination, the time since vaccination, or the date of vaccination and the development of autistic disorder." This doesn't prove that vaccination cannot cause autism, but it does prove that if that were the case, it would be exceedingly rare and certainly no cause for concern.

You mean they aren't?

These are not solutions. "Solution" 1 is nothing else than a short recapitulation of the paradox: Aristotle: If I move the smaller circle I am moving the same centre, namely Α; now let the larger circle be attached to it [...] it will have invariably travelled the same distance [i.e. case 1: the smaller circle forces the larger circle to travel only the distance of the circumference of the small circle] [...]Similarly, if I move the large circle and fit the small one to it [case 2: the large circle forces the smaller circle to travel the distance of the circumference of the large circle] [...] nowhere does the greater stop and wait for the less in such a way as to remain stationary for a time at the same point [Aristotle doesn't understand how in case 1 the large circle is forced to travel the smaller distance] the smaller does not skip any point [neither does he understand how in case 2 the smaller circle is forced to travel the larger distance] [...]When, then, the large circle moves the small one attached to it, [in other words, when the large circle forces the small one] the smaller one moves exactly as the larger one; when the small one is the mover, [that is, the small circle forces the large one] the larger one moves according to the other's movement. Compare that with your "solution" 1: If the smaller circle depends on the larger one (Case I), then the larger circle forces the smaller one to traverse the larger circle’s circumference. If the larger circle depends on the smaller one (Case II), then the smaller circle forces the larger one to traverse the smaller circle’s circumference. This is the simplest solution. It is clear that this doesn't tell us anything new that Aristotle hadn't written already. That "solution" 2 and "solution" 3 are not solutions, I've already shown in earlier posts. In fact they are also just recapitulations of the paradox.

I'm stubborn in that I keep referring to the original formulation by Aristotle (or whoever it was). That seems to me to be appropriate if we're talking about Aristotle's wheel paradox. And in that original formulation the second track, ΗΘ, is an essential element. To fresh up your memory, from the original text (in translation): So that whenever the one shall have traversed a distance equal to ΗΘ, and the other to ΖΙ, and ΖΑ has again become perpendicular to ΖΛ, and ΑΗ has again to ΗΚ, the points Η and Ζ will again be in their original positions at Θ and Ι. As, then, nowhere does the greater stop and wait for the less in such a way as to remain stationary for a time at the same point (for in both cases both are moving continuously), and as the smaller does not skip any point, it is remarkable that in the one case the greater should travel over a path equal to the smaller, and in the other case the smaller equal to the larger. It's obvious that Aristotle's essential problem was that he couldn't understand how the smaller circle could traverse the greater distance without skipping somewhere. The solution is that the smaller circle is not skipping but slipping, i.e. that the point of that circle in the 6 o'clock position has a translation speed > 0. We now know that a 1-1 mapping from a smaller circle or line segment onto a larger line segment is possible, so that is no longer an objection and with calculus we can give an exact description. The interaction between the circle and its track is the crux of the paradox, removing that track is not giving a solution, but explaining the paradox away. Your argument is like that of someone who "solves" Einstein's twin paradox by saying that there are no twins from whom one travels with extremely large speeds through space, they are just a crutch!