19

I read in trigonometry class that 1 minute is equal to 1/60 degrees. So, 'minute' is an angular unit. But also 'arcminutes' are used to measure seperation between celestial objects and also equals to 1/60 degrees. Are they any similar? If not, then what's the difference?

Tanya
  • 301
  • 2
  • 7
  • 5
    "Minute" is a length of time. Your trig teacher meant to say arcminute. – Daddy Kropotkin Jun 01 '21 at 17:00
  • 21
    @DaddyKropotkin: "Minute" is not just a length of time: it can also mean the same as "arcminute"; this is confusing, but not incorrect. – psmears Jun 02 '21 at 11:19
  • 6
    @DaddyKropotkin To follow up on psmears comment: Minute, as used in measurement, is a magnitude adjustment just like "hekto", "mega" or "deci" - it specifically means one sixtyeth. It is also used figuratively: a minute adjustment, meaning "tiny". It is more than a measure of time. – Stian Jun 02 '21 at 12:45
  • 3
    @psmears and @ Stian Yttervik thank you both for clarifying – Daddy Kropotkin Jun 02 '21 at 12:49
  • 10
    @StianYttervik: Note that "minute" as in "a minute adjustment" is a different word (it's an adjective rather than a noun, and it is pronounced differently too). – psmears Jun 02 '21 at 12:57
  • @psmears You are correct, I was pointing to their common appearance in writing. It would be imprecise to claim that the (written) word "minute" has one meaning, it has at least these 3. – Stian Jun 02 '21 at 12:59
  • @StianYttervik How big are minutemen then? – Mast Jun 02 '21 at 13:26
  • 1
    Not only the terms are confusing, but also the symbols. For time, the only acceptable symbol is min (with no period after), while for angle, the symbol is ′. However, some people use them interchangeably, which is improper use. The same goes for seconds of time, symbol s, and seconds of arc, symbol ″. Note also that ′ and ″ (which are straight angled) are not the same as ' and " (which are straight vertical) or as ’ and ” (which, depending on the font, are curly). – Pierre Paquette Jun 02 '21 at 17:49
  • @StianYttervik, you got it the wrong way around. Etymologically, a minute is a small part, and you can see the same root in the adjective “minute”. And a second refers to the second subdivision into parts. – Carsten S Jun 03 '21 at 12:23
  • @CarstenS Yes - but that is what I am saying? It all came from the sexagesimal system the babylonians used to count. Their astronomy, and thus time was based on 12/60. The partitioning into 60 sized lots got absorbed into the latin language, making the word for "small parts" over time to mean both a measurement of time, and a measurement of angle. But its root is the sexagesimals. – Stian Jun 03 '21 at 12:31

4 Answers4

35

This can get a bit confusing, because "arcminute" and "minute" are both sometimes used in celestial coordinate systems but mean two different things.

An arcminute is 1/60th of a degree, and an arcsecond is 1/60th of an arcminute. That's simple enough, and when talking about small angular distances, it's often much handier to refer to something as being, say, 140 arcseconds across, rather than 0.0389 degrees. So you're likely to see angular sizes or scales quoted in degrees, arcminutes and arcseconds.

If you're trying to state the position of an object on the sky, things get a little more complicated, thanks to the commonly-used equatorial coordinate system, which states an object's position on the celestial sphere in terms of its declination and its right ascension. The declination of an object is usually given in degrees, arcminutes and arcseconds. Its right ascension, on the other hand, is usually given in hours, minutes and seconds. Here, one "hour" corresponds to 1/24th of a circle, or 15 degrees. One minute is then 1/60th of an hour, and one second is 1/60th of a minute. So as units of angular separation in this context, an arcminute is different from a minute, and an arcsecond is different from a second.

HDE 226868
  • 36,553
  • 3
  • 123
  • 201
16

Your trigonometry book isn't wrong: both "minute" and "arcminute" can refer to $\frac1{60}$ of a degree. It's certainly a very good idea to use the term "arcminute" when referring to $\frac1{60}$ of a degree, but it's not essential if there's no ambiguity, eg, in a static geometry problem where there's no mention of time.

The term "arcminute" is relatively new. According to Google Ngrams, "arcminute" and "arcsecond" started becoming popular around 1970-1980. FWIW, I went to high school in the 1970s, and I can't remember any of my books or teachers using those terms.

From the arcminute Ngram

arcminute Ngram

Here's the arcsecond Ngram

arcsecond Ngram

The results for "arc minute", "arc-minute", etc, are similar.

However, the terms "minutes of arc" and "seconds of arc" were quite popular before that era, and it appears they have mostly been displaced by "arcminute" and "arcsecond".

minutes of arc

minutes of arc Ngram

seconds of arc

seconds of arc Ngram

As the Wikipedia Sexagesimal article mentions, people have been using the base 60 system to represent fractional quantities since the 3rd millennium BC. The Chaldeans (Babylonians) inherited it from the Sumerians, and they used it to record celestial positions.

The Babylonian Astronomical diaries span 7 centuries, and Babylonian data was one of the sources used in Ptolemy's Almagest. The trigonometry tables in the Almagest use base 60 both for the angles and for the values of the trig functions (Ptolemy trig tables used chord length, which is closely related to the sine function).

European mathematicians continued to use sexagesimal for recording and computing with fractional quantities up to the late 17th century, but it was gradually displaced by decimal fractions.

PM 2Ring
  • 13,836
  • 2
  • 40
  • 58
  • 4
    The word in latin was minuta/minutus, which means "made small". Likely it was convenient for them to define these 1/60ths as "small parts" and here we are - same root for the time division and the angle division, same measurement name. pars minuta prima, the first part made small. It is then fairly clear to see where also the second came from - dividing again into a "second" part. – Stian Jun 02 '21 at 12:52
1

One minute has two definitions. One is 1/60th of an hour, or 60 seconds, while the other is 1/60th of a degree. Typically, one minute is shorthand for "arcminute." You can tell between the two by replacing the word with "60 seconds." If it makes sense, then it means that. If not, then it is one arcminute.

An exception is the use of "minute" as an adjective (means tiny), but it is obvious that when used in a sentence, you can distinguish its meaning.

WarpPrime
  • 6,653
  • 1
  • 24
  • 61
  • 13
    I don't see how your proposed test can tell between the two - a minute can be replaced with "60 seconds" for both meanings of the words: [ 1 arcminute = 1 minute (of arc) = 60 seconds (of arc) = 60 arcseconds ] vs [ 1 minute (of time) = 60 seconds (of time) ] – Steve Jun 02 '21 at 08:51
  • 3
    @Steve Exactly, I don't know why does this have so many upvotes. – User123 Jun 02 '21 at 18:44
-2

The four N-gram traces of @PM 2Ring's answer

Put into perspective, it becomes clear that the traditional terms are still competitive. I would prefer them.

Rainald62
  • 135
  • 4
  • 1
    This seems very similar to @PM2Ring's answer. What makes yours different? – WarpPrime Jun 04 '21 at 12:34
  • The difference is the common scale, whereas theirs have no scale at all. The difference in noise at peak level made me suspect that the scales could be very different. It no longer "appears they have mostly been displaced by arcminute and arcsecond." – Rainald62 Jun 07 '21 at 17:58