UK readers, this one's urgent and time-constrained. Please use one of the templates provided to write to your MP.
Maybe they should have reconsidered which photo to accompany this story with…
Eww, girls! EWW, period!
Welcome to the 14th century.
Or the mind of a ten year old boy.
The problem of the soulless twin, solved!
Please sign this petition.
Seriously, sign it, I beg you.
The Angry Mob: Consequences. Make sure to read comment #2. Spot on!
Newton TV. Short science documentaries.
What does the Catholic church consider a worse crime than paedophilia?
Treating women as equals.
The ongoing travesty of justice known as "the twitter trial," in gory detail.
Christian parade … erm … not banned, and gays and Muslims … erm … not to blame, the Daily Fail … erm …doesn't quite get 'round to saying.
Must-watch BBC documentary: The Dolphins Of Shark Bay
I don't normally do adverts, but this one's in a good cause. It's for a book called Aftershocks: Stories from the Japan Earthquake, and all of the purchase price of the e-book goes to the Japan Red Cross. A worthy cause, I'm sure you'll all agree.
Hat tip: Amy
I love a good rant, as some here may have noticed, and this is a really good rant.
The Streets of Ashkelon By Harry Harrison.
What happens when true innocents are exposed to religion.
Forty Shades Of Grey puts some common sense into politics
Sometimes, Godwin ≠ Fail
Richard Littlejohn. Is he actually human?
I keep getting told thatatheism is a faith…
Evangelicals' free school would include creationism on science curriculum
Prayer: How to do nothing and still convince yourself that you're helping
A stunning collection of pictures of snowflakes
A haaaaandbag!? Actually it's a tale of what happens when bureaucracy and security join hands, but, like the author, I couldn't resist the Lady Bracknell-ism.
Create your very own Daily Fail headlines at the click of a button, with the Daily Mail-o-matic. It has a fairly limited vocabulary, but then, that only goes to make it more realistic.
British education makes a massive leap backwards
Anyone who's perused right-wing christian blogs and boards will have often seen references to the 'Gay Agenda'. Quite what this agenda is, is not usually explained, which leaves folks like me rather puzzled. Finally, someone's come clean! Folks, I present to you, in all its glory, the extremely radical Gay Agenda!
Don't click plan to do anything else for the next half-hour at least.
UNESCO has a program for the next 20 years to make half the world population homosexual, says Cardinal
possibly the funniest Wikiquote page in existence
It would be cruel to laugh…
I'm an atheist, so how come I seem to spend more time defending Muslims than criticising? Martin Robbins has the answer.
More of the 'War On Christmas', from Jesus & Mo
Five Chinese Crackers neatly turns the tables on the tabloids, in the 'War On Christmas'
Catholic priest tries to hire a hitman to shut up rape-victim
Don't panic! The end of the Earth has been postponed
Call them lifts, or call them elevators, they're probably a lot more interesting than you thought
Now I need a new irony meter
Free time-saving link for anyone who regularly converses with fundies. Just bookmark it and deploy as needed.
The Five Chinese Crackers Tabloid bullshit of the month award for November goes to…
Macer Hall at the Express!
'Rockstars of Science' should be 'Scientists of Rock'
Martin Robbins says
"GQ's patronising attempt to link scientists to rock stars shows a fundamental failure to understand which are cooler"
And he's right
Charlie Brooker is, without doubt, one of the funniest writers of opinion pieces on the planet.
Wile E Coyote faces charge of attempted murder
The Scale Of The Universe.
Much more fun than it sounds!
Conservapedia on quantum tunneling. (Quantum mechanics
for by dummies.)
Pinkydead vs God examines
13th of November. It's Robert Louis Stevenson's birthday
The Daily Fail,
home of sleaze
The first tiny step towards a Grand Unified Theory? Maybe. There's a long way to go, though, and I don't expect to be able to follow the maths if they do get there. Even so, it's a nice thought that somebody could.
NASA survey suggests Earth-sized planets are common
Cecil Adams on the
pros and cons of sushi
Dan & Dan
The Daily Mail Song
And Dan & Dan on
A sound-bite worth bookmarking
You'll know when to link to it…
Mark Twain hilariously tears James Fenimore Cooper apart
The Incredible Human Journey
Alice Roberts. Mmmm
No, I won't spoil the surprise. Just click it, already!
The Simpsons are Catholics! Says the Vatican
Every now and then, I like to browse a page or two of Conservapedia. Not only does the laugh do me good, but it's nice to know that no matter how little I may know, I'll never ever be as cluless as this.
Ben Goldacre on inadequate journalistic caveats
Technically, the "Thou shalt not covet thy neighbour's ass" joke shouldn't work in British English.
Okay, but I find stuff like this interesting!
Rape within marriage should not be crime,
says leading UK cleric
If you can watch this dry-eyed, I pity you.
World Of Science
Some good, if rather abbreviated, chronologies
The Haynes Big Book Of Lies
The scary thing is, most of it is true.
I watched a TV show the other night touching on some aspects of the Census Of Marine Life, and it prompted me to check out their website. It's great stuff. While you're there, be certain to visit the image gallery. Stunning!
Ancient theological problem solved
by yours truly!
Some of the crazier items that appear in the in-box at
A New Pope
Physical activity can occasionally lead to injuries!</sarcasm>
Talk about non-news! Sheesh
A private space-program
Well okay, 'edge of space.' Bloody pedants!
Neatly combining the recent zombie theme and the also recent Easter item is
this little play
on the idea of transubstantiation
The real meaning of Easter*
The creationist worldview.
It's jaw-dropping stuff!
Eleven-year old girls forced to cover their faces.
Later note: This story appears to have originated in the Daily Mail, and therfore may be exaggerated or even untrue.
Some thoughts from India
on the degeneration of democracy into mob rule.
God, according to the omnipotence hypothesis, is a monster, both by human standards and by the standards he apparently sets for us, according to the Bible. Every day thousands of people die in accidents and natural 'act of God' disasters which an all-powerful god could easily prevent. Earthquakes, tyre blow-outs, typhoons, cancer—the list is endless.
If a human being were to cause such events, or were known to be able to prevent them but didn't, we'd be horrified at them. They'd be pilloried from every parliament, podium and pulpit in the world. Even God, in his simple instruction, "Thou shalt not kill" tells us that this person would be clearly in the wrong. Yet, according to most religions, God is that person. He can prevent that suffering and death, but doesn't. Indeed, if he's the all-powerful creator of everything, he's responsible for it. After all, an all-powerful perfect creator must, by definition, have been able to create a world with no suffering, no sin, and no need for pain or Hell.
This one's a guest-post from Fisherman. Fishy's a keen golfer, fresh-water fisher, quiz enthusiast and, most importantly for us here, a chemistry teacher.
I've been after him for a while now to do a piece on kitchen chemistry; something that parents could do with their kids, maybe, or those of us who never grew out of that childhood fascination with the world that makes us gasp with delight when, with some ordinary, everyday bits & bobs, we realise we can discover some quite astounding things.
Anyway, enough from me, already! Here's Fishy:
It's amazing what small, provincial and humourless minds many fundies have. I suspect most people who regularly browse atheist blogs will have seen the video of an Australian spoof TV advert advocating the banning of religion. Well it seems some humourless god-botherers actually think it's a real proposal. I kid ye not!
First off, here's the video to watch. It's only a few minutes, and quite amusing.
Thought it was about time I produced a new post, (i) because I really ought to, and (ii) because It gets a tad borin' copy-pasting old posts and reformatting them. Due to lack of time, though, I haven't actually written anything.
Time for a quiz, then! Having been a member of a quiz group—hosting live quizzes in a chat-room—for years, I have quite a few kicking about.
Not moving house, but moving the blog. This site started as a static website, and ended up doing service as a blog. So it makes more sense to move to a proper blog space, where I can take advantage of the fact that they do a lot automatically for me, that I've had to do manually here.
Anyway, I decided to move to Wordpress. I've posted a brief introductory bit over there, going into a bit—though not much—more detail than I have done here, and I'll be spending a few days migrating what I think is the most important and/or best-written stuff over there. If there's anything folks think I should include, please let me know. (I know Amy, for instance, likes my more ranty pieces :-) ). Any comments regarding the move, please leave them on the I've Moved article at the new blog.
01 Sept 11
I'd been thinking of doing something like this for a while, and, as Nat over at Forty Shades Of Grey recently accused me of reading her mind by making all her points before she could, I thought I'd turn the tables and do it just after she just had, thus showing that it seems to be a two-way street. And—imitation being the best form of flattery—I reorganised my sketchy notes on it and nicked her ordering and headings, too. :-)
It concerns, if you didn't click the link, the arguments from the opposition that we see over and over again. While some are actually meant seriously, the vast majority of times you see them, they're from trolls; people who aren't actually trying to make a serious case rather than just muddy the waters.
Of course, Nat did it from a feminist perspective, where I've gone for the gnu-atheist perspective, so there's bound to be some differences. Indeed, if there were none, I'd have just linked to her article. As it is, the two are similar enough to lead one to the conclusion that a conservative troll of little imagination is still a conservative troll of little imagination, no matter their specific area of conservatism, whilst being dissimilar enough for me to add something new.
[Added at the last minute. Talking of Forty Shades Of Grey, Facebook users really need to read Nat's latest post. Words fail me, they really do. On the lighter side, I have a substitute YouTube link.]
Right then, let's be about it…
1. "Prove it"
Most commonly, "Prove God doesn't exist."
The onus is on the person asserting the existence of something (in this case, gods) to provide evidence for that existence.
(Thus far, the only 'evidence' I've ever seen has amounted to an argument from ignorance; "I can't see how such-and-such happened without intelligent input, therefore goddidit." Not only does this not prove anything, but the problem of how we get from 'a god did something' to 'this particular god, championed by this particular religion/sect did something' never seems to get addressed. But I digress.)
But if we have to play this game, maybe these people should be asked to 'prove' the non-existence of every other god bar the one they worship. Here's a handy list to start with, though it's by no means even close to being complete.
2. "Stop being so angry, sweary and rude"
Nat uses one word; 'No'. I prefer two; 'Fuck off'.
Really, address my arguments, not my tone. If you really can't see past a few rude, possibly insulting, words; if you're that prudish that a few pixels on a screen can make you all fainty and weepy, then stop reading and piss off and find a comments board with Mary Poppins nannyware that blocks such uncouthness. Which is related to…
3. "You're using ad hominem arguments, so I'm not going to listen to you!"
If I really do utter an ad hominem, please feel free to return the favour. Inventively, if possible. As Nat points out though, nearly all such accusations are based on a confusion between an insult and an ad hominem phallacy.
To make an ad hominem is to attack the person rather than the argument: "You're wrong about x because you're an egotistical, Bible bashing, fundy areswipe" is an ad hominem.
"You're wrong about x, and/therefore you're an egotistical, Bible bashing, fundy areswipe" is an insult. Okay, it might not further the conversation much, but it can be fun for its own sweet sake, and it's not a phallacy. Whether to explain why the arsewipe is wrong about x depends on whether Andrex-head seems capable of understanding the explanation, or—given that we're mostly talking about trolls here—whether it seems they'll actually bother to read the explanation.
Actually, feel free to return insults, too. I love the smell of a good internet slanging-match in the morning. Smells like… victory.
4. "What about men?"
Well, obviously, this one is more applicable to feminist topics than to purely atheistic ones. Given that religion is probably the most common cause and/or exacerbating factor in organised campaigns to restrict women's rights, though, those of you who don't regularly peruse the atheist, and especially the gnu atheist, web-o-sphere might be surprised how often feminism rears its interesting head. Nat covers two of the three topics that usually bring out the worst of the anti-feminist trolls on atheistic as well as feministic boards. The third is female genital mutilation, sometimes known as female circumcision.
For those who've never come across this particularly disgusting practice, here's a brief description from the World Health Organisation's page on it.
Female genital mutilation (FGM) comprises all procedures that involve partial or total removal of the external female genitalia, or other injury to the female genital organs for non-medical reasons.
And, from a bit further on:
FGM has no health benefits, and it harms girls and women in many ways. It involves removing and damaging healthy and normal female genital tissue, and interferes with the natural functions of girls' and women's bodies.
Immediate complications can include severe pain, shock, haemorrhage (bleeding), tetanus or sepsis (bacterial infection), urine retention, open sores in the genital region and injury to nearby genital tissue.
Suffice to say that nearly all FGM is performed for religious reasons. Anyway, back to the trolls…
You can almost guarantee that at some point, usually sooner rather than later, someone will say something like "But you're ignoring male circumcision!" Well, no. Male circumcision is frequently discussed, along with all other religion-based barbarities. It's just that today we're discussing FGM. And no, male circumcision is not 'just as bad', though any unneeded surgery (leaving aside cosmetic) is a bad thing, and performing such on people without their consent, or on minors who by definition can't give consent, should be classed as a human rights violation regardless of the shape of the genitalia involved. FGM, though, is much much worse and can have much worse consequences, and deserves special consideration.
And, as Nat also says, if you're that bothered about the subject you're talking about, whatever it is, do something about it. Constantly derailing discussions on separate but related issues won't get anything done about your particular hobbyhorse, and alienates people who would most likely have wanted to share ideas and resources, given the similarities between the two topics.
I'll skip Nat's next, as it isn't really applicable to atheists. (Please note, if anyone's comparing the two; this throws the numbering out of whack.) So we come to…
5. "She's an atheist, you should support/agree with her"
Sorry, no. Unlike the religious, who are taught from an early age to follow authority, us atheists don't think that because someone is right on some things, that they must be right on all things. This seems to be the commonest misconception of atheists by theists; that if they can prove Dawkins, Hitchens, whoever wrong on one thing, that, by extension, we'll cease to trust anything that person says. It just ain't so. Their words are read with as much of a critical eye as anyone else's, and each statement stands or falls on its own merits, not those in a previous chapter or book. Of course, someone who proves to be wrong most of the time is less likely to be listened to, and the opposite is also true, but scepticism is applied when we do listen, regardless of source.
(Personal note: As it happens, the 'celebrity atheist' I find myself agreeing with most often isn't one of the so-called four horsemen at all, but PZ Myers.)
6. "You're censoring free speech!"
Utter rubbish. Criticism and debate isn't censorship. And neither is it persecution, the other last-ditch objection used by theists who can't think of a better argument for telling atheists to shut up, or to not look sceptically at religious claims.
On the other hand, advocating the illegal removal or vandalism of atheist billboards, the removal of atheist books from bookshop shelves, the burning of a child's painting depicting human evolution (how fucking petty can people get?); these things most definitely are censorship.
7. "Here's a link to Answers In Genesis that shows you why you're wrong!"
Oh, just fuck off.
And now I'll round it out to ten with some of my own that, obviously, aren't applicable to feminism.
8. "Darwinism led to Nazism"
No, it didn't. The Nazis may have used a skewed version of the theory of evolution to justify some of their beliefs, but that didn't make them right. And also, so what? Germ theory isn't wrong because biological weapons have been developed. E=mc2 is still true, no matter how nasty nuclear weapons are. The consequences, whether real or imaginary, of an idea have no bearing on the factualness of that idea. And talking of Nazis; may's well get all the Godwins out of the way at one fell swoop…
9. "Hitler, Lenin and Mao were all atheists, and look at what they did!"
Well, Hitler may or may not have been a Catholic, but let's let that slide. Again, so what? Hitler, Lenin and the scary-looking bloke in Sparks all had moustaches. Does that mean all facial hair is evil? And, anyway, all three (Mao being the third, not the chap in Sparks. Ahem) acted on other ideologies than religion; their atheism had nothing to do with it, any more than Churchill's love of painting led him to oppose Hitler.
I could add, and indeed am doing so, that a better example would be modern Sweden; one of the most atheistic, according to polls, countries in the world. Please note: This does not prove that atheism necessarily leads to more civilised behaviour. It does, however, prove that atheism doesn't necessarily lead to uncivilised behaviour, contrary to the claims of most hard-line theists.
10. "Atheism and/or science is a religion/faith"
Atheism is nothing more than the statement that there is no evidence fore the existence of gods and other supernatural beings and realms. See №1 for more details.
Science is the very antithesis of faith, and I've never really worked out why fundies keep throwing this one at us. (Unless it's meant as a smear, but if so, why would they see accusations of religiosity as a smear?)
Science is not a belief system, but a method for taking personal bias and 'faith-based' ideas out of the loop. It's a way for individuals and groups to systematically check their ideas against the real world, and for making sure that others can check the results.
Contrast this to faith-based belief in ancient myths as codified by almost as ancient desert goat-herders…
26 Aug 11
We've all seen this argument: 'Science can't define what 'good' is. Religion does define it. Therefore we need religion.' Well, as you may already suspect, I have several problems with that…
Firstly, let's look at how religion definesit. Is it a 'good' definition?
Religion doesn't, it seems to me, give any real definition of good. Something is good, says religion, because God says so. But that doesn't really answer the question of 'what is good'; it merely shifts it to a higher stage. (For now, let's ignore the fact that gods have never been shown to exist, and take that existence as a given.)
Is it good because God says so, or does God say so because it's good? If the latter, then God is not the all-powerful free agent he's usually made out to be; he's constricted to making only 'good' pronouncements. If the former, then invoking God does nothing to define good unless we presuppose that God would never lie or make self-serving pronouncements (and if you believe that, I urge you to check out the first four of the ten commandments).
Both ways of stating the problem, therefore, make presuppositions about the nature of God that aren't supported by fact, or even by religious text. More importantly, for this discussion, neither actually defines 'good'; rather they leave that definition for God to sort out—if he can—and tell us to do nothing but submit to his authority. Anyone wishing to do so, I'd advise reading the Old Testament before deciding. Don't know about you, but the actions of the god portrayed therein don't strike me as being particularly good.
Secondly, does the want of, or even the emotional need for, the existence of gods, mean that they exist? Don't be silly. Just because some people want a heavenly authority to tell them how to behave correctly doesn't magically conjour that being into existence, any more than my ardent wish for a Brough Superior will make one suddenly appear in my garage. (There wouldn't be room, anyway; Carl Sagan left his bloody dragon in there. I wouldn't mind, but it's getting hard to find enough princesses to feed the damn thing on.)
And thirdly, it should be bloody obvious that the whole thing is a false dichotomy. 'Good' is a word, and it's not a scientific principle but a philosophical one, so the people to ask aren't scientists, but linguists and philosophers. It's a concept, not an object, like politeness or egotism. And, to be honest, it's not even a difficult word to define.
An action which is morally good is one which promotes happiness and/or health. It's as simple as that. And the converse, a bad action, would be one which promoted unnecessary pain or other suffering. Of course, the application can be complicated, with much weighing up of relative outcomes—which is where the grey areas of 'better' and 'worse' slip into the discussion—but the basic principle is simple and clear-cut.
Clear-cut, that is, to those adults among us who've actually thought about it but not clear-cut, maybe, to those who prefer to stay in perpetual childhood, and want a father-figure to hand down rules so that they don't have to think for themselves on the consequences of their own actions.
10 Aug 11
[EDIT: I've added a section after the post-script, in response to an enquiry about some trigonometry I used to work out the length of a chord, in a minor aside when talking about the definition of a degree. This also led to me correcting the figure I used for the diameter of the circle; also in the aside. I could have made it a new post, but decided that, as it partly involved a correction of this one, it properly belonged here.
Had a conversation with a work-mate the other day, about our remembrances of school lessons. It seems we both left with much the same impression; that much of our learning was based more on effect than cause. (We're both of a similar age, and I have no idea if this is still the case in UK schools.) In history lessons, for instance, we learned a lot of events, much of them almost by rote, but very little about the causes of those events. We were just taught what happened. In science subjects we were taught a lot of laws and facts, but not the underlying causes. For instance we were taught that water expands and becomes less dense as it cools from 4° to 0°, but not why it does so. (Turns out it's to do with the crystalline structure of ice—it has gaps between the molecules. It also turns out that water's not quite as unique in this as we were led to believe. Gallium, bismuth, antimony and silicon behave in the same way as they freeze.)
Most damningly, though, neither of us were taught, or made to find, proofs of some quite basic mathematical concepts; which really is a pity, as some of them are easy enough to be set as an exercise for quite young students, and encouraging youngsters to look for proofs and evidence would really help them develop a sceptical, 'scientific' frame of mind, rather than accept 'facts' on authority, which would be of enormous benefit in later science lessons, as well as in 'real life'.
Anyway, when the conversation turned to maths, my workmate mentioned a few that had puzzled him slightly, and I, having been puzzled enough by the same things as a kid, had worked a couple out on my own (albeit with a hint or two from my maths teacher, when I explained my problem to him), so I managed to explain them to him. The first, involving opposite angles, I managed verbally; the other, involving the angles of a triangle, had to wait for a few hasty back-of-an-envelope diagrams in the canteen at lunch time—which gained a small audience of three or four others who'd apparently had much the same experience of school that we remembered (much to my embarrassment—I began to feel as if I were showing off, somehow!).
All of which got me thinking that maybe there'd be an audience for a blog post or two on the interesting bits of basic maths that we were never taught at school. So here's a couple I puzzled out for myself as a kid, possibly to be followed by more of the same if enough people find it interesting. I've tried to go back to basics as much as possible, so that even those for whom maths lessons were nothing more than a chance to catch forty winks should stand a chance of following it. Those who did stay awake may want to skip this next bit. :-)
First off, the terminology.
Imagine a circle with 360 equally spaced marks around the circumference. Now imagine a line is drawn from each mark to the centre. What you should now have in mind is a circle divided into 360 segments. The 'width' of each of those segments is 1 degree. (Usually written as 1°.)
What's important to note is that a degree isn't a measure of distance. Each segment gets wider, the further from the centre you travel. If the circle has a circumference (the distance around the edge of the circle) of 360 yards (it would, if you want to know, have a diameter of 114.57 yards, give or take a gnat's tadger), then a person could step one yard along the circumference and have moved one degree. Another person, who's unlucky enough to be on the edge of a circle of 360-mile circumference (114.57-mile diameter), would have to walk a mile to cover the same fraction of the circle's edge (assuming she doesn't cheat by cutting across in a straight line, of course, in which case she'd save a tad under
So when an angle is said to be so-many degrees, what's being said is that if the point at which the two lines meet is considered to be the centre of a circle, then the segment (shaded in the diagram) of the circle between them covers so-many 360th-sized segments of the circle. A right-angle, for instance (by definition), sweeps out one quarter of a circle, so it's
For those who are interested, the minute hand of a clock sweeps out a degree every ten seconds. To walk one degree along the Earth's equator, you'd have to travel a tad over sixty-nine statute miles. (A nautical mile, on the other hand, is defined as
If we're talking about angles, we need a way of referring to them in such a way that we're all looking at the right place. What we do, as most will remember from school maths lessons, is give every point of interest, such as the ends of lines and places where lines meet or cross, a letter to identify it. We can then say that we're referring to line AB, or angle AEC (shaded) and we'll all know that we're all on the same page. AEC, for instance, is the angle formed at E by the lines AE and CE. I could even say 'Now draw a line CB,' and talk about angle EBC, and you'd all know what angle I meant. Or we could add a point, F, somewhere on the line ED, and talk about the angle AFE or BFD, and we'd still all know what was happening; though if many additions like that were planned, it might help to have put them in the diagram in the first place. Anyway, now you know the basics, here's the meat of the essay…
Opposite angles are always equal.
Well, I was taught that they're called opposite angles. You may know them as vertically opposite, or as vertical angles. Whatever they're called, though, what this means is that angles AEC and BED in the diagram will be the same as each other, and that CEB and AED will be the same as each other. We all learned that at school. I don't remember ever being taught the proof though, or—more importantly—being asked to prove it. Maybe my experience wasn't typical (though the fact that this essay was inspired by someone else not knowing does show that I'm not alone), but what a pity that is. Going beyond maths, this would have been an easy introduction to sceptical thinking; and it really is easy.
Right, here we go. Angle AEB is, obviously, 180°, being a straight line. Therefore angles AEC and CEB have to add up to 180.
If AEC is, for instance, 20°, then CEB has to be 160°.
CED is also a straight line, and therefore 180°, and as we already know that CEB is 160°, then BED has to be 20°, and following the same reasoning, DEA has to be 160°.
So it's really just a corollary of the obvious fact that adjacent angles on a line have to equal 180° when added together; which itself is important for our next point, that…
The internal angles of a triangle always add up to 180°.
I wasn't taught the proof of this one, either, but I remember puzzling it out because it troubled me. After all, if I walk around a triangle, starting in the middle of one of the lines, I'll, obviously, have turned through 360°, not 180°; or else I'd not arrive back at my starting point facing in the same direction I started out in. Hmm, maybe, I thought, it's the external angles I need to look at. Nope, a couple of minutes disproved that one; they always add to 900°. Well, so I thought. A minute or two more got me the answer, and it turns out it is the exterior angles, though not by the definition I was using, though I'd eventually figured it out using my own nomenclature anyway. Time for another diagram.
Turned out that what I'd been thinking of as the exterior angle—for instance the PacMan-like grey-shaded angle at ABD—isn't the exterior angle, contrary to what common sense might say. (I don't know what it is called, but it ain't the exterior angle.) The exterior angle is defined as that which is adjacent to the interior angle, if the line were extended beyond the length of the side, like the blue-shaded angle at EDA, which is adjacent to BDA. And when you stop to think of it, you'll see that if I was walking along the line, that's the angle I'd turn through, when making the corner. To turn through the grey-shaded angle, I'd have to have been walking backwards along AB, then have turned (the long way around) to walk forwards along BD. John Cleese might get away with it; the rest of us, I think not. Anyway, I want to avoid using Xs and Ys and such, so lets plug some numbers in.
As you can see, the interior angles do indeed add up to 180°. (Hurriedly checks arithmetic: Yep!) What about the exterior angles, though? Remembering that adjacent angles always add up to 180°:
Add 'em together and they do, indeed, make 360°.
All of which proves that the interior angles can add up to 180° while the exterior angles add to 360°—which is nice for those of us who don't want to end up walking backwards round triangles, as well as being the problem I briefly wrestled with in my youth; doubtless while I was supposed to be doing actual homework—but doesn't prove that they have to add up to 180°. For that, we need to briefly consider some parallel lines.
It's a property of parallel lines that if they cross a third line, they both cross it at the same angle. (Indeed, if all you have is a protractor and a straight-edge, that's a perfect way to test whether two lines are parallel; draw a line across them and measure the angles they make with it.) Since we know that opposite angles are equal, and that the parallel lines make the same angles where they meet the bisecting line, we know that angles a, b, c and d are equal (and all the angles adjacent to those labelled are equal to each other). What does this have to do with triangles, though?
Here's a triangle drawn with the base parallel to a line which touches the apex, or, to put it another way which should ring bells; at which the two 'upright' lines would cross if they were extended. As with the previous example, we can now spot angles that are equal, which I've shaded for ease of discussion. The reds are equal to each other, as are the blues. It should be pretty obvious, also (but I'll point it out anyway, 'cause I'm kind like that), that the three angles at the top—one red, one blue, and the internal angle at the apex (shaded black)—add up to 180°.
Feel free to applaud…
It's not just triangles though, though I don't intend to try to prove this. All polygons' interior angles have a fixed number of degrees which their interior angles add up to:
The rule for working it out is to count the number of sides (n), take 2 from that figure and multiply by 180°.
In short, that's;
If the shape is regular (all sides and angles are equal) and you want to know what each individual angle is, divide the result by the number of sides, so:
P.S. Came across a feature of Google calculator when I tried to figure out how much distance the person cutting across the circle would save, in my 'introduction to degrees'. I bisected the triangle and prepared to use trig for the first time since I left school—but, hey, I knew all the angles, so it was just a case of finding a sine or cosine, followed by a bit of multiplication. As a check, I knew that the answer had to be less than 1, that being the long distance around the circumference; the chord had to be shorter than that. So imagine my surprise when, [i] the chord came out as seemingly longer than the curve, and [ii], using sin and cos gave different answers. Had I misremembered something as basic as sohcahtoa? Was I just going potty? Nope. Using Windows™ calculator got me a believable answer straight off, as well as providing the comforting sight of sin
Additional: This section added, 11 Aug 11.
Had an email asking if I could run through the trigonometry by which I worked out the length of the chord that our cheating walker could have followed, as the enquirer's remembrance of trigonometry was fuzzy. Since I'd deleted the notepad document I made the calculation on, I had to redo it; which led to the rather embarrassing finding that I'd made a couple of errors, based on a numerical typo. I've corrected the figure I gave in the text for the length of the diameter in light of this, but not the distance she saved in her walk, which turns out to be even less than the 1⁄10 of a mile that I gave.
I'd like to mention that I'm in no way trying to make myself appear some sort of maths wizard. All this is based on stuff I remember from school, and I barely scraped an O-Level. Where I seem to differ from many people I meet isn't in my small ability at basic maths, but more in the fact that, through reading popular-science and science fiction, I've had the occasional refresher, which has kept what little I do know from fading through disuse.
Anyway, here's how I worked it out (but—hopefully—correctly this time):
Let's start with the basics, then: How do we work out the radius of the circle? Well, first we work out the diameter, then halve that to get the radius. Pretty simple:
The length of the circumference equals pi times the diameter. (I'm using pi to the number of decimal places offered by Windows calculator, which is way more accurate than needed, but I'll be doing the same with all other numbers, then rounding off at the end, so it makes sense to do the same with pi.)
Or, in short, c=πd
To solve for d, we just divide both sides by π, so:
Divide that by 2 to get the radius and:
r=57.2957795130823208767981548141. So that's the length of the two legs of the triangle in the diagram above (which, though you probably guessed this, is not meant to portray the angles accurately. A triangle sweeping out 1°, at that scale, would hardly look more than a line). What we want to know is what the length of the base (the chord that our intrepid walker would follow) is. This is where the trigonometry comes in.
From here on in, though I'll still be showing my working, I'm going to assume that anyone who's stuck with me will have at least a vague recollection of using sines, cosines and tangents, and is more in need of a memory-jog than a complete explanation of how to find the cosine or whatever of an angle, and such. Oh, and I'm old enough that when I learned trig at school, we looked them up in books of tables like this [pdf], which also contained tables of logarithms. Ah, you youngsters today, you don't know how easy you have it, etc etc etc…
Ahem. Where was I?
First we work out the angles of the triangle. We already know that BCA is 1°, and the other two are equal, since this is an isosceles, so they have to be
Hands up, who remembers 'sohcahtoa'? Our maths teacher told us it was the name of a Red Indian chief, but I'm not too convinced of that. Still, it helped get it embedded in our young minds, so he achieved his purpose. It breaks into three syllables:
Sine = Opposite over Hypotenuse.
Cosine = Adjacent over Hypotenuse.
Tangent = Opposite over Adjacent.
All of which means that, given the length of two sides, we can work out either of the two unknown angles; or, more pertinently for us, given either non-right-angle and a side, we can work out any other side. We have a choice; we can either use the sine of BCD, 0.5°, or the cosine of DBC, 89.5°. Let's go with the former:
∴ DB=0.49999365382791892382807286970969 miles
∴ AB=0.999987307655837847656145739418 miles.
Let's round that off to 0.99999 miles, shall we?
Or 1,759 yards, 29.6 inches.
She'd save a whopping great 6.4 inches! If she followed a chord between every degree-mark, she'd shave 64 yards off her 360-mile journey. Or she could book into in a nearby hotel, relax for a week and a half, sneak back to the starting point while no one's looking, and announce that she's knackered but proud…
20 July 11
Well, today's the 20th and we all know what happened in 1969 on that date, yes?
Constellation, the NASA project to take us back to the moon by 2020, was cancelled by President Obama back in February, thus ending what I suspect was Geedubbya's attempt to be remembered for something more than starting wars and mangling the English language. Whatever, it looks like being the end of that particular dream, for some time at least.
Anyway, I decided I couldn't face the rather gloomy job of going on about past glories that I really want to see repeated but am unlikely to, so I put together a playlist of my favourite moon songs. Well, not my absolute favourtites; I made sure to include some songs set on the moon or about trips to it, even though they weren't in my first-choice list. There's twelve in all—one for each person that's stood there. (Actually I had eleven planned, in honour of Apollo 11, but thought of another one at the last minute. Good recovery though, eh?) What you won't hear is the song that gave me the title of this post, 'cause I think it's a shite record. Mind you, I have a feeling that most folks would say the same for much of this playlist. Each to their own. They're very much in keeping with the types of music that have formed the core of my tastes. Hopefully you'll find one or two you like.
The technical stuff: An oddity of this player is that you can't see the list 'til you've pressed play. With that in mind I've made sure to make the first number a nice gentle one, so that no one gets their ears blasted before getting a chance to adjust the volume. Also, the file-hosting site where the mp3s are stored doesn't allow simultaneous downloads, so if nothing happens straight away, it probably means somone else is listening to it. Shouldn't be a huge problem as this isn't the most heavily trafficked site on the intertubes, and I've made the files small; just 32 bit (which also lowers the quality, but it's good enough for most rock and pop), so the download time is a few seconds.
Enjoy! (Or run away in disgust.)
If you think this site deserves more exposure to an unsuspecting world, please add the link http://myweb.tiscali.co.uk/thedixieflatline/ wherever you see fit. Many social networking sites have a space for your favourite site on your profile… :0)
Click play to start, and to view the playlist.
Due to financial restrictions, I'm relying on free storage for music files, which means no simultaneous downloads, so if the player or songs are slow to load, it's because someone else is also listening to it. I've spread the files around several different hosting-sites, so some should generally play if you hit the Next Track control a few times. Other than that, I can only suggest patience or trying again later.