On the face of it this might just seem silly, but I wonder if I scribe a circle on a sphere will the ratio of the circumference of such a circle to its diameter would still be $\pi$. If so, is there a way I could prove it or disprove otherwise?
Foreword  This question was inspired by initial mistakes in . I wanted to explore the strange circle with $A&gt;\pi r^2$ and got lost into geometrical jungle.  A  is usually described by it's height $h$, radius of the base $a$ and radius of the sphere $r$. Besides we have a relation $a^2+(r-h)^2=r^2$ so the cap actually only has two degrees of freedom.  When considering a case of circle on a sphere (like the Earth), it's the surface of a spherical cap, but we can't see the height or other classic parameters of this spherical cap. Instead we encounter parameters like area $A$, perimeter $C$ and diameter $d$. I will try to write formulae for these: $$A=2\pi rh$$ $$C=2\pi a$$ $$d=2r\theta$$  Where $\theta$ is introduced with $\sin\theta=\frac{a}{r}$ or $\cos\theta=(1-\frac{h}{r})$.  As the cap only has two degrees of freedom, it should be a good problem with unique solution if only two values are known.  A-C Problem  The simple example: $A$ and $C$ are given. I show this only as example before the other tasks.  We can easily find $a=\frac{C}{2\pi}$ and then solve:  $$A=2\pi rh = 2\pi r (r-\sqrt{r^2-a^2})$$  to obtain  $$r=\frac{A}{2\sqrt{\pi(A-\pi a^2)}}$$  Parameters $a$ and $r$ are now known and it is straightforward to find $h$, $d$, $\theta$ and anything else.  C-d Problem  $C$ and $d$ are given.  First step once again is easy: $a=\frac{C}{2\pi}$. The knowing of $a$ suggests that we might prefer the $\theta$ definition with the sine. We obtain: $$d=2r\theta=2r\arcsin\frac{a}{r}$$  That's where I got stuck. Is there an easy way to solve this? Is there a hard way to solve this?  I tried solving numerically with Wolfram Mathematica and it went into complex numbers. When solving the geometrically equivalent $\sin\frac{d}{2r}=\frac{a}{r}$ I got some roots, but the search seemed very unstable and not finding the root closest to initial value. Besides it was finding a lot of different roots while I am pretty sure that there should only be one $r$ for each pair of $d$ and $a$. What would be the right formulation to solve this problem numerically?  A-d problem  $A$ and $d$ are given. This seems the hardest one. We obtain a system of equations:  $$A=2\pi r h$$ $$d=2r\arccos\left(1-\frac{h}{r}\right)$$  (The first equation suggested solving the problem in $h$ and $r$).  Is this system solvable analytically? Or numerically? My experiments with Wolfram Mathematica never converged.   Of course, I could express $h$ or $r$ from the first equation and leave only the second in one of these forms: $$d = 2r\arccos\left(1-\frac{A}{2\pi r^2}\right)$$ $$d = \frac{A}{\pi h}\arccos\left(1-\frac{2\pi h^2}{A}\right)$$  But I have no idea how to solve these either.
Javascript file uploads SF short story ID, written 40s—60s, primitive aliens who are much smarter than us I'm looking for a story that was probably written somewhere from the 1940s to the 1960s. I think I read it in an anthology, possibly in the 1970s.  A group of human explorers landed on a planet with primitive aliens, living in hunter-gatherer villages. They hoped to make a profit from trading our goods for—well, I'm not clear on what they wanted from the aliens.  The native people learned the human language before the explorers could learn theirs. Then the natives suddenly started producing technological innovations far beyond what they originally used. I remember that wooden sailing ships suddenly replaced their canoes. All the natives had to do was hear a story once from the explorers before copying the technology in the story.  Finally, the explorers realized that the natives were much, much smarter than us and that Earth faced a serious problem.  I'm fairly certain that it wasn't a Heinlein story, but I remember that it had some qualities similar to his work. Is my mobile AMD card being used? I have a new Sony E Series Vaio with an intel i5 and an AMD 7xxx mobile card. Despite making sure sure my app (game) is set to High Performance in switchable gfx and choosing high performance power plan I'm not entirely convinced the gfx card is being used over the CPU's HD4000 chipset. Is there anyway I can prove for sure the AMD card is being used?  thanks for reading  (OS is Win8 64bit) Are upvotes on competing posts counted per question or per answer for Sportsmanship? The definition of the :     Up voted 100 competing answers   doesn't specify whether each upvote on competing answers to the same question counts towards this criteria, or if it's limited to one "point" per question no matter how many additional competing answers you upvote. Can someone explain how it works? Cannot empty trash I am trying to empty my trash, and it keeps giving me an error message that says,      .Trash can't be opened right now because it's being used by another task...     I have tried restarting my computer, running Finder > Secure Empty Trash everything yields the same result and error message from above. Prove that $PSL(2,\mathbb{Z})$ is free product of $C_2$ and $C_3$ Prove that $PSL(2,\mathbb{Z})=C_2 \star C_3$.  Now $C_2 \star C_3=\langle a,b\ |\ a^2, b^3 \rangle$ i.e. the free product.  But how do I show that presentation of $PSL(2,\mathbb{Z})$ is this? Defaulting on credit cards before leaving the country I have a colleague who is going back to his country after staying for a couple of years in US. He has 3-4 credit cards plus a personal loan. His total debt would come to around $10,000. Some days ago, he told me he is going to default on the accounts. He figures, since he is not planning to come back to US, it should not matter to him.  Question is, is this a bad move on his part, and if it is, how can I convince him? Deny access to a webpage using web.config How does the bounty system work? A Basic Question about Gravity, Inertia or Momentum or something along those lines This has just dawned on me that if you're writing an operating system then what are you writing it on? I ask this as I am reading a microprocessor fundamentals book from 1980 and this question popped in to my head:   How was the first microprocessor chip programmed?  The answer may be obvious but its bugging me. Predict() - Maybe I'm not understanding it Who paid for Harry's Nimbus 2000? The Nimbus 2000 was the fastest broom at the time of the Harry's first year at Hogwarts.  Professor McGonagall wanted to win the Quidditch Cup so she spoke with Dumbledore and got an exception from the rule that no first-years could have a broom for Harry.  My question is who paid for the broom?  It couldn't have been a cheap broom.  They wouldn't have taken gold out of Harry's Gringotts account without permission.  It wouldn't be right for the school to pay for it and I doubt McGonagall paid for it out of her own pocket.  So who paid for the broom? Can I travel to the UK unaccompanied on an accompanied child visa now that I am over 18? What happens if you use the 32-bit int 0x80 Linux ABI in 64-bit code? Why does the Hilbert curve fill the whole square? Castellation metal loop connector Here's a module with castellated pads attached to an eval board:  (source: ) Looking closely at the wires interfacing with the castellated pads, they're clearly soldered to the eval board. However, they don't look like they're soldered to the module, but rather they're being pressed against the pads with the tension in the wire. It also looks like the free end of the loops are protruding down into a channel cut in the PCB. Do these kind of connectors/metal loops have a name? Extracting only the BibTex records relevant to the paper My colleague asked me to write a page for a paper he writes. I wrote the page in LyX, compiled it to LaTeX, and sent him the TeX files. Then he reminded me that I forgot to send the .bib file with all the references.  But, my .bib file is huge - it contains over 1000 references (I have a global BiBTeX file for all my papers). The page I wrote uses only about 20 of them.  Is there an automatic way to extract from the bib file, only the records relevant to the current paper? I'm trying to solve    So far I've done the first part, evaluating the summation ;  where a is just n. I'm not sure where to go from here or what it even means deduce the second summation. I understand that the summation of simply $$\sum\limits_{k=0}^n\cos(kx)$$ is derived from looking at the real part of  $$\sum_{k=0}^n{\rm e}^{ikx} $$I'm guessing they want me to see how the second summation in the question is defined by 'playing around' with the original? 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