# Tag Archives: mathematics

## Why are seconds called seconds?

### 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

## Amazing sculptures (or: what I want for Christmas)

Have you seen anything like this before?

Metatron

Universal Clef

These are the work of Bathsheba Grossman, who describes herself as “an artist exploring math and science in sculpture”.

My first reaction to seeing these and her other handheld sculptures was to want them. All of them.

They are beautiful, sometimes complex, and very satisfying. They seem to me to be what the artists who did Celtic knowtwork designs would have produced if they’d had three dimensions to work in and had seen M. C. Escher’s more geometrical drawings.

My second thought was that these shapes are impossible to make: imagine trying to carve one, or to produce a mould to cast one…

My third thought was that she had, however, made them. Otherwise she wouldn’t be offering them for sale. So off I went to her technique page to find out how on earth they’re done.

They are indeed impossible to make by traditional processes. Although Bathsheba will typically make a plasticine model as a starting point, the actual sculpture is produced by first using CAD software to define its precise shape, and then using 3-D printing process to make the actual object. This involves building the sculpture up layer by layer from powdered metal. A few more stages turn this into a solid metal sculpture.

To find out more about these and other works of hers, visit her website. Then buy one of each and send them to me for Christmas…

You’ll also find some remarkable internally-etched blocks of glass, containing such things as a genuinely 3-D map of our nearby stars… Beautiful stuff.

#### Notes

• Photos used by permission. Do not re-use without linking to http://www.bathsheba.com and crediting the artist.
• Bathsheba describes her glass pieces as “modelling three-dimensional data” and mentions here that she’s interested in suggestions for further such pieces–see her site for details.