Category Archives: science/maths

Why are seconds called seconds?

Posted on Apr 2, 2009 | 5 comments

Minute minutes?

I still have a few snatches of memory from childhood about learning to tell the time, and learning how it was divided up. In particular I remember when I first learnt how long a second actually was (considerably longer than I expected) and that there were sixty of them in a minute.

I also learnt that minute wasn’t spelt minnit or anything like that. And I already knew that minute meant “very small”, which seemed odd, since really it was the seconds that were small, not the minutes. And I half-remember thinking it was strange that minutes weren’t called firsts. Why not?

I didn’t know, but it was fun that the words were like that. Evidently I’ve been interested in language for a very long time.

A prime example

In my teens, I got interested in reading popular mathematics books, such as Martin Gardner’s collections from his “Mathematical Diversions” page in Scientific American. (That started quite early too: I remember being excited in my last year at junior school, which translates as age 10, when our class teacher got us to make flexagons. These are like a sort of hexagonal origami conjuring trick which make an appearance in one of his books. I think the one we made was the hexahexaflexagon. Sadly if I tell you about them now it’ll be too much of a digression from this post.)

Sometimes in maths you’ve been using a symbol—say the letter a—to represent something, then find yourself wanting to represent a similar-but-different thing. One traditional way is to simply add a little mark to the symbol: a becomes a′, then maybe a′′ and so on.

From the popular maths books I learnt, somewhat to my surprise, that whereas at school we very logically called these symbols a-dashed and a-double-dashed, having added little dashes to them, the American books called them by the rather strange names a-prime and a-double-prime. What a strange word. How had they been primed? They didn’t have anything to do with prime numbers. How odd.

A degree of confusion

And there was another intriguing thing: when I learnt geometry—specifically, angles—it was apparent that it wasn’t just hours which were divided up into minutes and seconds: degrees were, too. Which was interesting, but the notation was puzzling: 33 degrees, 12 minutes and 3 seconds was written 33° 12′ 3′′ .

“How confusing!” I thought, “Surely 12′ 3′′ means 12 feet and 3 inches? It’s bad enough making them sound like times without also making them look like distances!

So what on earth is going on?

These questions niggled me for years, because although they were intriguing I never quite got round to looking them up.

The revelation

The answer appeared out of the blue about a year ago, and everything fell into place. Very neatly and satisfyingly. (Except it would be more satisfying if a foot had sixty inches in, but never mind.)

Thirty years or so after first wondering about minutes and seconds, I was reading a fascinating book about early mathematics. [1] Among other things it talked about the Babylonians who, as you probably know, were the ones who divided a day into 24 hours, an hour into 60 minutes and a minute into 60 seconds. In fact, they did all their calculations in base 60. (By the way, they were able to solve quadratic equations in 1700 BC, knew Pythagoras’ Theorem many centuries before Pythagoras even lived, and were able to calculate square roots so as to use it).

The Babylonians were the only people who had a decent system for representing fractions. For us, 1:23:45 means an hour, 23 minutes and 45 seconds; for them, the equivalent in their writing meant the number 1, plus 23 sixtieths, plus 45 sixtieths of sixtieths, and they’d have happily gone on adding smaller and smaller divisions, like we do with our decimal places.

The astronomer Ptolemy also featured in the book. He used some ingenious geometry to work out a trigonometry table in half-degree steps. [2] In his introduction he commented that by far the best system for representing fractions was the Babylonian one and that he’d therefore adopted it.

And now comes the Great Revelation. Ptolemy himself wrote in Greek, but once maths like his started appearing in Latin, what did people call their fractions of a degree? The answer turns out to be:

  • “the first small part”: pars minuta prima
  • “the second small part”: pars minuta secunda!

Look at that for a moment. Isn’t it beautiful? All my questions answered in those two short phrases. It’s obvious, but let’s spell it out anyway, and enjoy it all making sense:

  • A minute of time is the first small part, or  pars minuta prima, of an hour.
  • A minute of arc  is the first small part, or  pars minuta prima, of a degree.
  • A second of time is the second small part, or pars minuta secunda, of an hour.
  • A second of arc  is the second small part, or pars minuta secunda, of a degree.
  • The little mark you use for marking a minute—a pars minuta prima—is called a prime.
  • To mark a second [small part] you use two of them: ′′.  Presumably if we used sixtieths of seconds, we’d call them thirds and mark them ′′′.
  • Feet and inches are also first and second small parts of something, so they too get labelled with ′ and ′′.

So all those years ago, I was right. Minutes are “minute”. Seconds do come second! Minutes were called “firsts”, but in Latin.

In a way it’s a shame about the feet and inches, because they don’t quite fit the scheme. An inch isn’t a sixtieth of a foot. On the other hand, isn’t a fathom five feet (sixty inches)? or is it six? I can’t remember.

So I don’t quite know about the feet and inches. But I was stunned when I came across those two short phrases which made everything else fall into place. Isn’t language amazing?

Notes

[1] Asger Aaboe, Episodes from the Early History of Mathematics, Cambridge University Press, 1997. Back
[2] In our terms, what he calculated was twice the sine of half a given angle. Back

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Imagining the unimaginable

Posted on Mar 23, 2009 | 3 comments

NARRATOR: The Hitch-Hiker’s Guide to the Galaxy is a truly remarkable book. The introduction starts like this: ‘Space,’ it says, ‘is big. Really big. You just won’t believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it’s a long way down the street to the chemist, but that’s just peanuts to space. Listen . . .’ And so on.
. . . After a while the style settles down a bit and it starts telling you things you actually need to know.

Douglas Adams, The Hitchhiker’s Guide to the
Galaxy: the original radio scripts, Pan Books, 2003.

This post was going to be another instalment of my thoughts on science on religion—about the religious implications of living in a rather big universe—but it’s turned into an unashamed attempt at expressing the sheer size of space. The approach used here is the only one so far that’s worked for me; I hope it might work for you too.

By the way it does not involve describing things as “the size of a football field”, “the thickness of a human hair”, “the length of a London bus”, or “the size of a blue whale”. So if your heart was already sinking in anticipation of those, please feel free to cheer up again. It does however involve an orange. I apologise for that.

Everything’s too big

Imagining big things is difficult. Beyond a certain size, the numbers all blur into “too big to imagine”. So the task I’ve set myself is rather ambitious. For example, astronomical distances are measured in light years—the distance a beam of light would travel in a year. Even the distance it travels in a second is uncomfortably big for the imagination: 186,000 miles, 23 times the diameter of the earth, or two thirds of the way to the moon. Alarmingly, a light year is over 30 million times as far as that. Our nearest neighbour star, Alpha Centauri, is about 4 light years away. These distances are not small.

I do know, though, what a mile looks and feels like. It’s a little less than the distance from here to the bank; I walk it in about 16 minutes. And I know that a train journey to London takes around 3 hours. People who fly regularly know how long it takes them to get to their destinations. This gives us some inkling, at least, of the size of the Earth: Its diameter is about the same as 8,000 walks to the bank, or 1½ flights to America.

Another way to get a sense of how big the Earth is is too go to the beach, stand at the water’s edge and look out at the horizon. Assuming you’re about six feet tall, the horizon is about two miles away. It’s effectively a six-foot-high bulge created by the Earth’s curvature. Now try to imagine what’s over the horizon. More sea, curving away downwards. Mentally double the distances: 4 miles, 8 miles, 32 miles. Feel the earth’s gravity holding you on the ground. Try to get a grasp of what an 8,000 mile sphere looks like.

My task is to somehow relate the distances of space to the everyday ones we have a grasp of. My tool is the Ordnance Survey map: the one you might use when out walking in the country, or the equivalent map if you live outside the UK. I’m going to use the old, inch-to-the-mile maps, printed at a scale of 1:63,360 (the number of inches in a mile). If you don’t remember them, well, the current 1:50,000 maps are the same idea, but blown up a little bit.

Make it smaller!

OK, here goes.

Step 1

Scale: 1:126,720,000 (1 inch to 2000 miles)

Shrink the Earth to the size of an orange. (Fruit is traditional and compulsory in these comparisons.) In other words, make a model of the Earth on a scale of 2000 miles to the inch. At this scale:

  • the Earth is a squishy ball of not-very-solid rock, 4 inches across.
  • alarmingly, the nice solid crust, on which we live, is less than a thousandth of an inch thick.
  • The Sun is a very hot ball of fire, ¾ of a mile away and 36 feet across.
  • Alpha Centauri is, I’m afraid, still 200,000 miles away: a bit less than the actual distance to the moon.

Well I can imagine an orange and I can imagine a ¾-of-a-mile walk, but if I’m honest, 200,000 miles still doesn’t mean much to me. Time for the OS map . . .

Step 2

Scale: 1:8,028,979,200,000 (1 inch to 126.72 million miles)

Let’s make a map of the scale model, at a scale of an inch to the mile—that is, shrink everything down again. The new situation is:

  • the Earth is now about 1/16,000″ across, or 1/600 of a millimetre. You can see what a millimetre looks like by looking at an ordinary ruler or tape measure.
  • the Sun is ¾ of an inch away, and 1/150″ across, or 1/6 of a millimetre.
  • our friendly neighbourhood star, Alpha Centauri, is now 3.1 miles away—not quite in our neighbourhood, but walkable in under an hour.
  • Our galaxy, The Milky Way, is about 70,000 miles across.

That is, an inch-to-the-mile map of an inch-to-2000-miles map of the Galaxy needs a sheet of paper 70,000 miles across, the Sun is just about big enough to be seen with the naked eye, and the Earth can only be seen with a microscope.

Step 3

Scale: 1:508,716,122,112,000,000 (1 inch to 1.36 light years)

Let’s make another map: an inch-to-the-mile map of our inch-to-the-mile map of our inch-to-2000-miles scale model of our bit of the universe. It turns out it still needs a pretty big sheet of paper, but at least it’s not an unimaginably big one. On this new map:

  • the Earth is almost exactly 10-9 inches across, or 1/40 of a nanometre; ¼ ångstrøm. In other words, it’s smaller than a hydrogen atom, the smallest atom in nature.
  • the Sun is 1/84,000 inches away, and 10-7 inches across.
  • Alpha Centauri is just over 3 inches away.
  • The Milky Way is just over a mile across. It contains about 1011 stars; you could imagine it as a mile-wide snowstorm, with the stars scattered like snowflakes (but considerably tinier).
  • Our neighbour galaxy Andromeda, in the so-called Local Group (ha!), is 26 miles away.

However, we can’t stop there. What really brought home to me the appalling vastness of space was a map of approximately a million galaxies (just the ones brighter than a certain value) in a small section of the sky. It presented a lumpy grey appearance; the grey was simply the visual result of printing a dot for each galaxy. Not only were the Sun and Earth utterly tiny and insignificant among the 1011 stars of our galaxy; the Milky Way itself was insignificant among the many millions of galaxies in the observable universe.

Think what this means. To find the Earth, you first have to find the right galaxy–the correct dot on the lumpy grey map. That’s not exactly easy. Then, having scaled this tiny dot up so that it’s a mile across, you have to find something smaller than a hydrogen atom, too small to see even with an electron microscope. You scale this up to a decent size, and all of humanity is living on a thin layer less than 1/1000 of an inch thick . . .

The size of space is, in itself, still unimaginable; what I’ve done is to shrink some of it to an imaginable size, by what I hope are imaginable steps. If I’ve succeeded, maybe you now at least have a sense of how unimaginable it is.

I said above that this means we’re totally insignificant. Actually, that’s not the only way to see it. I’ll go into that in another post. But it seems clear that we can’t claim to be any more significant than any other populated planet in the unimaginably big universe. Claiming that it’s all centred on us seems so impossibly arrogant that it simply doesn’t make sense.

At this point a lot of people would look at the world’s religions, which make rather a lot of the importance of humanity to God, and say “Well that proves it. Religion is impossible.” What they really mean is that religion which puts human beings, or a particular tradition’s ideas, at the centre of the Universe is impossible. But that is only one kind of religion . . .

More on that in another post.

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Plutonymics

Posted on Mar 17, 2009 | 8 comments

An exercise in astrolexicography

Plutoids and plutinos . . .

When the former planet Pluto was demoted to the status of “dwarf planet” fairly recently, two new words were defined by the International Astronomical Union: plutoid and plutino. If you ask me, these would be damn good words whatever they meant: they belong to that group of words which seem to exist as much because they’re fun to say as because they’re needed.

Several weeks ago one of my contacts on Twitter, @Exoplanetology, came up with the word exoplutoid, meaning a plutoid in a planetary system other than our own.

Should you wish to know, a plutino is an object which, like Pluto, orbits the Sun twice for every three orbits made by Neptune. (This is called a 2:3 resonance, and the object remains trapped in that orbit.) A plutoid, roughly speaking, is simply a dwarf planet which orbits the Sun further out than Neptune does.

I suppose an exoplutoid might be a dwarf planet in another star system, further from its star than the last convincing planet.

Nice words. Are there more?

Plutonyms in the dictionary

Let’s proceed with caution. A look at the dictionary reveals that a number of pluto- words already exist. Furthermore, not all of them are anything to do with Pluto. Plutocrats, being plutocratic in a plutocracy, get their name from the Greek word ploutos, which means wealth.

In geology, plutonic relates to rocks which have solidified from a molten state at the fiery depths associated with the god Pluto and his underworld, and a pluton is a “body of instrusive igneous rock”. Geology also uses the word plutonism in this connection.

In chemistry, the element plutonium has nothing to do with plutonism; the elements uranium, neptunium and plutonium take their names (rather nicely) from Uranus, and Neptune and Pluto, which were all planets at the time.

Plutogenous neologisms

Given the existence of all these words already, are we to conclude that Pluto has contributed all it can to the English language? I think not!

There are still plenty of Pluto-related situation requiring words. Some of the situations are more “serious” than others. But all need words, and it is my pleasure to present them to you. They are grouped by function rather than alphabetically. Use and enjoy.

plutaceous:
similar in material or structure to Pluto.
exoplutoid, exoplutino:
a body in another planetary system analogous to a plutoid or plutino in ours.
plutogenous:
originating from, or generated or caused by, Pluto and its status. For example, plutogenous fisticuffs might result from a heated discussion about its classification. See plutonym, below.
plutectomy:
removal of Pluto or a Pluto-like object, e.g. from a list of recognised planets or (as a more advanced engineering project) from a planetery system
plutogenic:
relating to the creation of Pluto-like objects, i.e. to plutogenesis.

plutonym:
a word created with reference to Pluto and its status; that is, one which enters the language as a plutogenous neologism.
plutonymics:
the study of plutonyms.
plutolexicography:
the creation of a dictionary or glossary of plutonyms
plutamnesia:
an inability to remember what Pluto is officially classified as these days.
plutamnesic:
suffering from or relating to plutamnesia.
plutamnesiac:
someone who suffers from plutamnesia.
paraplutosis:
1. condition of accidentally using the wrong plutonym, e.g. calling a plutoid a plutino or describing plutogenous situation as plutogenic. The corresponding adjective is paraplutotic.
2. erroneous identification of an object as Pluto.

Got any more? Post them here and I’ll do the plutolexicographer’s job of gathering them together, time and energy permitting. Especially if they’re good.

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An excellent find

Posted on Mar 9, 2009 | Leave a comment

Yesterday I exchanged a few messages on Twitter about the relationship between music and language (a relationship which I also mentioned recently in my post speculating about background noise).

What should I see in the library today, while waiting for a computer to be free, but a book by Steven Mithen called The Singing Neanderthals: the origins of music, language, mind and body (Weidenfeld & Nicolson, 2005)!

It’s substantial. And looks very interesting. I’ve taken it out of the library.

A review may follow . . .

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Believing in God and in science: some beginnings

Posted on Jan 16, 2009 | 6 comments

I was asked a while back to say something about my religious beliefs. It’s hard to know where to start, so I thought I’d start somewhere that’s particularly important to me and which relates to things I’ve already been blogging about…

A lot of people believe there’s a fundamental incompatibility between science and religious belief. I believe that they’re fundamentally wrong 😉

Is there a conflict?

There are scientists who reject religion. I suppose the most famous of these is Richard Dawkins, who has almost made attacking religion into a religion of his own. And there are religious people who reject science: for example those who treat the Bible utterly literally and insist that the world was created in six days as (supposedly) described in Genesis 1.

Cleearly there can be a genuine conflict. Someone who believes God does not exist, and someone who believes God created the world in six days, will never agree with each other. There is a fundamental disagreement between them. But is that the only kind of believer and the only kind of scientist? No–it’s an extreme variety of religion and only one kind of scientist. In fact there is no reason why scientific thinking has to reject God, or why religious belief has to reject the scientific understanding the earth’s history and of our origins. I think the debate typically takes place between people one of whom understands science but not religion, and the other understands religion but not science. And sometimes, I fear, there are religious people who don’t understand religion… though that might be a bit more contentious.

My starting point

In my first year at university, startled by my first encounter with biblical literalists, I made a conscious decision which I’ve followed ever since: anything which I believe as a consequence of my religion must be compatible with what I believe as a consequence of science.

There is only one reality, whether you’re looking at it through religious or scientific eyes. Science and religion both try to discover some truths about it. Truth can’t contradict itself; so if they do discover truth, it must be consistent. It’s no good to believe during the week that we eveolved by natural selection, only to believe on Sundays that we were specially created out of the blue 6,000 years ago. Science and religion must both live in the same real world. Theology and science must both adapt in response to known evidence, as we make more sense of the world we are in. Otherwise we’re disconnecting ourselves from the world and our beliefs are simply attractive ideas which have nothing to do with reality.

Do we want reality, or fantasy? I think that if we’re basing our lives on it, we should go for reality. Or at least, the closest we can get to reality.

Some misconceptions…

A number of misconceptions seem to be lurking in the background whenever science and religion come into conflict. So here are some things I don’tbelieve:

… about religion

  • Religion claims infallible truth
  • Religion is a set of beliefs
  • Scripture is an infallible, divinely dictated book containing those beliefs
  • All religious people see it that way, or should do
  • All religious people reject science and rational thinking
  • Faith is intellectual acceptance of [impossible] ideas despite evidence
  • Religious ideas are arbitrary.

… about science

  • Science claims infallible truth
  • Science works by proving things true
  • All true scientists are atheists and reject religion
  • Science is merely opinion
  • Scientists seek to control the world
  • Science starts out with a particular view of things, which it then seeks to justify in a biased way.

… about both

  • Religion and science are based on conflicting “facts” (e.g. the claim that the world was made 6,000 years ago, versus the scientific evidence that it is much older).

Sometimes some of the misconceptions are agreed on by both sides, and then the trouble starts. Copnsider a scientist and a Christian fundamentalist who both think it’s essential for a Christian to believe in six-day creation. They will argue forever over whether the world was created in six days. They’ll almost certainly never question the assumption that it’s an essential part of religion. So they’re doomed never to get anywhere…

Some definitions of my own

To answer all those misconceptions properly would turn this blog post into quite a long book chapter (last time I checked it was over 1500 words long as it is), so forgive me if I don’t do that in detail just yet. Instead, here are some attempted definitions which reflect my approach to it all:

Religion
Religion is the response of human beings to the divine.
Theology
Theology is the attempt to make sense of that response and produce a logically consistent set of ideas: about the encounter, and about what we’re encountering.
Science
Science is the attempt to make sense of the physical world by testing ideas against careful (ideally repeatable) observation.
The Bible
The Bible is a set of writings, accumulated over many centuries, providing a record of around two thousand years of religious experience and reflection on it. The experience was that of human nature encountering God and the world; the reflection is influenced by how writers at the time saw the world, and is expressed in many different genres.

It should be fairly obvious that the things on my Misconceptions List are incompatible with those ideas. I’m worried about the length of this post so I won’t go into that in detail now–maybe in another post if needed. Instead, here are

Some consequences

Religion as a response

What is a reasonable response to being loved by someone, or falling in love with them? Is it to come up with a set of rigid beliefs and theories about them, and put all your effort into intellectually accepting those theories? No–your response is “Wow!” or “I want to be with this person” or to love them back or to want to join in with their activities. Similarly with our response to God: it’s not a set of ideas, and it probably can’t even be put into words because God is so far beyond what our language can describe. But after a while we feel the need to understand what’s going on, and that’s where theology comes in, so we try to describe it anyway. The beliefs aren’t the starting point.

Similar and different

Theology and scientific theorising are in some ways very similar activities. Both try to make sense of human experience. In the case of science, this is the experience of doing certain experiments and getting certain results; in the case of religion, it’s our subjective, yet shared, experience of being conscious beings, of relating to the world, and of relating to what we perceive to be its creator. Science has a distinct advantage in its area, because it deals as much as it can with things which can be made objective and measurable and repeatable.

Yet science can’t handle God at all, for a very good reason. The only way we can experience God is subjectively, in our consciousness, within ourselves. Yet the whole idea of science is to remove everything subjective and personal as far as we can, in order to be objective and repeatable. It works by letting us stand back from what we are studying. (The physicist Schroedinger expressed this well; I’ll try to find the quote.)

I believe that good theology must behave in a similar way to good science. It must take account of the real world we live in, and the real evidence we see. Its job is to make sense of the world and our religious experience as they are, not as we say they should be. It’s not a matter of taking some pre-existing belief in, say, the infallibility of the Bible and forcing ourselves to believe all the consequences; it’s about taking what we see and experience and trying to fit it all together.

Also it seems clear to me that neither theology nor science is in a position to claim absolute knowledge of the truth. They’re each a search, hoping to get nearer to the truth as they progress. Both need humility and the willingness to change if a new piece of evidence comes in. Their “truths” are always provisional: the best we can come up with so far, but open to change and refinement.

The Bible

OK, this is the bit which you won’t like if you’re a fundamentalist…

What’s special about the Bible is not that “God wrote it”, but that it contains all those centuries of experience and reflection. Human nature is universal. God is universal. So, if the biblical writers encountered God, they encountered the same God we do. They sometimes interpreted the encounter differently from us; and sometimes had some odd ideas. For example a lot of the Old Testament assumes that God’s love for us must mean God hates our enemies and wants to wipe them out. The idea of God loving them too didn’t seem to occur to the writers. Yet even that horrible and blatantly unchristian idea came from the belief that the God they had encountered was a loving one. Just not one whose love extended to other people too… And certain aspects of the encounter are consistent through all those centuries of experience; we connect with them in our experience too.

This is all scene-setting, really. I’ve not even started on basic things like what sort of God I believe in! But I hope it helps you to see my starting point.

A plea

I know that if you’re a particular kind of atheist, or a fundamentalist Christian, you’ll disagree strongly with what I’ve written. That’s fine–but please respect what I’m doing here: I’m simply setting out my beliefs for some people who’ve asked about them, and I haven’t the energy to launch into heated debate. Gentle disagreement is OK though 🙂

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