Talk:Degree (angle)/Archive 1

Radians and Gradians
Surely this is a bit pointless, having two subsections on the two major alternative units? I will edit it into one section with a slight introduction to each and then following through by suggesting the user click the links, because surely they have their own pages? Help plz 15:59, 25 June 2006 (UTC)


 * Edited it, it's rough, but imo more on topic, your thoughts? Help plz 16:06, 25 June 2006 (UTC)

Degrees in SI system
I'm pretty sure that Degrees don't exist in the SI system... not even as a derived unit. I'll remove references to that unless someone protests. Fresheneesz 09:01, 1 December 2005 (UTC)
 * They were an acceptable unit (and were listed under "Other units from ISO 1000" in ISO 2955 / 1983), but that's not part of the SI anymore. -- Jokes Free4Me 02:19, 10 August 2006 (UTC)

Does the Babylonian material belong here?
Hi, I edited the Babyloninan history additions, which are cool. I can't help but wonder, though, whether some of that belongs under Pi instead of here. It's probably more relevant to just say that the sixty-fold divisions of the degree relate to the Babylonian numeral system, and that the Babylonians were the pioneers in careful astronomical measurements; everyone later built on or generalized their work.

The tablet picture does not correspond to what's in the text. Can we find a picture of the real tablet or at least a reference number?

As an aside, the Egyptians certainly knew that the year had >360 days; they even had a cute myth about the discrepancy. WillowW 10:42, 27 June 2006 (UTC)


 * I agree; I don't see how the pi information has anything to do with the number of degrees in a circle, except for the fact that Babylonians used a base-60 numbering system. At the least, I think the bit about the "number of days in the year" should lead the history section, with the pi business moved later in the article (if not removed).  I started making the change myself, but I have no idea how to segue into the pi stuff, so I'll leave it alone lest I drift into Speculation.  I'm guessing that people stuck with 360, rather than 365, because of the Babylonian base-60 numbering system; again, I would rather not add that without any proof, which I do not have.
 * (Oh, and do tell about that cute myth; sounds like it would be a good addition to the page.) --ScottAlanHill 23:08, 3 September 2006 (UTC)

Astronomical explanation
I revised the wording somewhat. If this is too OR-ish then some suitable source should be found. Crum375 20:44, 21 September 2006 (UTC)

360 derives from the sexagesimal base
My opinion is that the angle of the Equilateral Triangle tool was a reference angle because however you handle the tool you get the same angle so it makes it a handy tool.

So the reference angle was accorded a number of degrees of 60decimal = 10sexagesimal, as it is today, equal to the base of the Sexagesimal numeration system in use by the ancient Mesopotamian Civilisations and that still today pattern the minute and second arc divisions.

The advantatge of this base is the large number of divisors, facilitating calculations as stated in the wikipedia article Sexagesimal.

How many times an equilateral triangle angle fits in a circle ? Exactly six (since the sum of the triangle angles totals half a circle), which multiplied by the reference angle gives you 360 ! Griba2010 19:34, 4 January 2007 (UTC)


 * Sounds good, except WP is based on published reliable sources. We need a published source that says that, or else it's all original research. Crum375 19:56, 4 January 2007 (UTC)


 * It is published: st-andrews.ac.uk - Babylonian_numerals >>one theory is that an equilateral triangle was considered the fundamental geometrical building block by the Sumerians. Now an angle of an equilateral triangle is 60 degrees so if this were divided into 10, an angle of 6 degrees would become the basic angular unit. Griba2010 11:16, 5 January 2007 (UTC)


 * I read through your reference (both 'numerals' and 'math'), all I find is speculations (no archeological or historical references for Babylonian geometry), and it doesn't tell us anything (that I can find) about the origin of one degree, which is the subject of this article. Crum375 12:48, 5 January 2007 (UTC)


 * My personal guess is that the angle of the equilateral triangle was the real standard. It is the easiest angle to reproduce with fidelity, so a good standard.


 * Any angle was expressed in terms of it following the sexagesimal system. Later in time, the first sexagesimal part became the main standard as many angles were less than the reference one, and a discrete number is always better to express, when talking, than a fractional one. Moreover, a written quantity less than 1 was easily read as integer since there was no sign for the preceding 0 and the meaning had to be taken from the text context. Later on, with finer tools, came the minutes and seconds accuracy. Griba2010 23:18, 6 January 2007 (UTC)


 * That sounds like a logical guess, but we need better sourcing for it. The fact that the stars in the sky seem to advance by 1 degree every evening around the celestial pole (with ancient calendars using 360 days per year) is also convincing. With the Babylonians not even having a '0', just a 'space', the whole subject of their supposed expertise in sexagesimal math seems shaky to me - since 0 digits would show up periodically, sometimes in groups, while doing measurements. A space is woefully inadequate to represent more than one 0 - when you get to 2 or 3 togther, it's really impractical. My fear is that we may be reading more into it than what was really there. (I do realize that 0 'density' in sexagesimal numbers is 6 times less than decimal ones). Overall we are very short on sources here. Crum375 23:27, 6 January 2007 (UTC)

Definition
What is the ancient and modern definitions of the degree? I have seen the following: 1/60 of an angle from an equiangular triangle. 1/90 of a right angle. 1/180 of a straight angle. 1/360 of a circle. 1/360 of a complete rotation. 180/π radians. 360/2π radians.

Although they are all equal, only one can be the definition at a time in order to prove the other are equivalent. Zginder 21:26, 23 April 2007 (UTC)

Proposed WikiProject
Right now the content related to the various articles relating to measurement seems to be rather indifferently handled. This is not good, because at least 45 or so are of a great deal of importance to Wikipedia, and are even regarded as Vital articles. On that basis, I am proposing a new project at WikiProject Council/Proposals to work with these articles, and the others that relate to the concepts of measurement. Any and all input in the proposed project, including indications of willingness to contribute to its work, would be greatly appreciated. Thank you for your attention. John Carter 21:01, 2 May 2007 (UTC)

Other Units I
there seem to be other units as well.

I recently examined a 1904 pattern uglomer (lining plane) sight in the St Petersberg Museum of Artillery, Engineers and Signals. This circular sight was divided into 600 dividions and there was no secondary or micrometeer sclae to divide these units. Obviously this is a source of the Russian 6000 mils.

Next in Sweden I again examined sights, on a 1930s vintage howitzer the azumuth scale used 6300 mils, however, the elevation scale used a degrees symbol and each 'degree' had four divisions, the three 'dividers being marked '12', '24' and '36', impying a degree divided into 48 subunits.

The 6400 mil circle seems to have been a French invention about 100 years ago. Nfe 01:50, 23 October 2007 (UTC)


 * Anatolia is wrong. It should say Ionia 82.204.98.14 (talk) 21:04, 3 June 2008 (UTC)

Equilateral Chords
The caption under the diagram in the History section is troubling me. It says "A circle with an equilateral Chord (geometry) (red). About one fifty-seventh of this arc is a degree. 2&pi; such chords complete the circle". I always thought that the chord was the straight line (as marked in red on the diagram), that one-sixtieth of that angle was one degree, and that it took exactly 6 of those to complete the circle. I'm not sure what the name for the shape would be but the diagram should highlight an arc of the same length as the radius for there to be 2&pi; of them in a full circle. --ClickRick (talk) 11:22, 27 May 2009 (UTC)


 * You are absolutely right. Some half-wit had edited the article and messed this up.  I've now reverted it back to the text that was there before, which said pretty much exactly what you write above, that the chord is the straight line (as marked in red on the diagram), that one-sixtieth of that angle is one degree, and that it takes exactly 6 of those to complete the circle.


 * Thanks for ringing the alarm! Jheald (talk) 15:36, 27 May 2009 (UTC)

Contradiction in the article
The history section is using circular logic. It claims that 360 was chosen because of the number of days in a year, and then goes on to claim that there were 360 days in a year because there were 360 degrees in a circle. 63.228.4.70 22:25, 11 April 2006 (UTC)


 * Basically it is not known why 360 degrees was originally introduced. The use of 360 degrees implies that certain important angles like the right angle and the angle of a equilateral triangle can be expressed as an integer. This may be a reason that 360 degrees has not died out long ago. Entropeter (talk) 10:25, 12 September 2010 (UTC)

Scientific Calculators and decimal degree
Do calculators seriously support this? i've never seen it. It would be very useful to place an approximate year on that peice of information, such as "Whilst this idea did not gain much momentum, most scientific calculators made after 1934 still support it. (Note: I made up that year) — Preceding unsigned comment added by Fresheneesz (talk • contribs) 07:59, 1 December 2005 (UTC)


 * I certainly don't ever remember seeing a scientific calculator (e.g. a calculator that actually supports trig) that didn't offer the grad as an option for its trig functions. but then i'm only 20. Plugwash 01:45, 19 December 2005 (UTC)


 * Here's a phrase from Grad (angle): In the '70s and '80s most scientific calculators offered the grad as well as radians and degrees for their trigonometric functions, but in recent years some offered degrees and radians only. -- Jokes Free4Me 02:04, 10 August 2006 (UTC)


 * The only big exception I've seen is the ubiquitous TI-8x graphing calculators, which don't offer the grad even though TI's cheaper scientific calculators do. Krimpet 14:19, 23 January 2007 (UTC)


 * In the calculators TI-83, TI-84, and Voyage-200 from Texas Instruments one can choose between degrees, radians, or grads as default. The successor TI-89 has a function that converts degrees to gradians but only degrees and radians can be choosen as default.Entropeter (talk) 10:40, 12 September 2010 (UTC)

Circle vs Revolution
I think I'm going to be alone on this one, but I've always had a problem with people stating that there are 360 degrees in a circle.

Here's why:

People say "there are 180 degrees in a triangle". What they mean is the sum of the internal angles of a triangle is 180 degrees. Similarly, there are 360 degrees in a rectangle.

In general, there are 180 * (number of sides - 2) degrees in a polygon.

Now here is where it starts getting debatable. I see a circle as a polygon of infinite number of sides.

Hence, as the number of sides tends to infinity, the number of degrees also tends to infinity.

So there are not 360 degrees in a circle, otherwise it would look like a square :-P

It's more correct to say: "There are 360 degrees in a revolution.", which obviously doesn't relate to the sum of the internal angles.

I heard the question "How many degrees are there in a circle?" on The Weakest Link and I was upset that they allow questions which have, at the very least, debatable answers. What do you think? — Preceding unsigned comment added by Igitur za (talk • contribs) 20:22, 15 July 2007 (UTC)


 * The Strongest Link would have answered instantly at 400 words a minute, "What do you mean? There are two possible answers: 360 degrees and infinitely many degrees, depending on what you mean.  So what do you mean?  Or are you the Weakest Link?"  --Vaughan Pratt (talk) 04:03, 5 April 2011 (UTC)

Angles and their measures
A definition of an angle would be that an angle is the union of two rays that have the same endpoint. The sides of the angles are the two rays, while the vertex is their common endpoint. stands for an angle. You can put it in front of three letters which represent points. The first and third letters represent points on each of the rays that form one of the sides. The middle letter represents the vertex. As you can see in the diagram, each point is represented in the written form. The letters can go either way - that is, first and last letter are interchangeable. So, that angle could either be ABC or CBA. Since B is the vertex, it is always in the middle of the two letters. You can also name an angle by just the letter of its vertex. So, for the example in the picture, the angle could also be labeled B. That's only if there are no other angles that share the same vertex. There is a third way to label angles. In the third way, each angle is designated with a number, so the example could be labeled 1 or 2 or whatever you wanted.

Angles are measured in degrees. The number of degrees tell you how wide open the angle is. You can measure angles with a protracter, and you can buy them at just about any store that carries school items. Degrees are marked by a ° symbol. For those of you whose browsers can't interpret that, a degree symbol looks like this:. I tend to just write it out instead of using the symbol because it's quicker on the computer. There are up to 360 degrees in an angle. As you can see in the picture below, the 360 degrees form a circle.

There are a few more basic things you should know about angles. First of all, the space inside an angle of less than 180 degrees, is a convex set, while the space outside of one is a nonconvex set. The opposite is true for an angle of more than 180 degrees (but less than 360 degrees). The side of an angle that is started at would be called the initial side, and the side that an angle ended at would be called the terminal side. The measure of ABC is written mABC.

When measuring angles, you usually go counterclockwise, starting where the 3 would be on a clock. That would be called a zero angle because there is nothing in it - just a single ray going directly to the right. The next important type of angle is called the acute angle. An acute angle is an angle whose measure is inbetween 0 and 90 degrees. An example would be the 45 degree angle in the picture. The next important type of angle is the right angle. This is probably the most important type of angle there is because of all the spifty things that you can do with one. I won't go into all of them here. (I have to save something for later articles!) A right angle is an angle whose measure is exactly 90 degrees. Continuing around the circle, next is the obtuse angle. An obtuse angle is an angle whose measure is inbetween 90 and 180 degrees. The 135 degree angle in the diagram is an example. The last major kind of angle is the straight angle. A straight angle is an angle that measures exactly 180 degrees. Thus the name - the two rays form a straight line. A negative angle is also possible. This just means that you go clockwise instead of counterclockwise.

A lot of geometry teachers don't go beyond that, at least at first. There isn't much else left to explain, but I'll give it a shot. After straight angles, there aren't any more special angles that you need to know about. A 360 degree angle is an angle that does a full circle. It looks just like a zero angle, but instead of having no degrees, it has 360 of them. (Duh. You can't get more basic than that!)

It is possible to have an angle with more than 360 degrees. To find out what it looks like, all you do is subtract 360 from it until you have an angle less than or equal to 360. (What?! You want an example? C'mon, you people...) For example, if you have an angle that is 546 degrees, you subtract 360 from 546 to get 186. Thus, the angle is the equivalent of a 186 degree angle.

There are a few more terms that you should also know. Supplementary angles are two angles whose measures combined equal 180 degrees. Complementary angles are two angles whose measures combined equal 90 degrees. Two non-straight and non-zero angles are adjacent if and only if a common side is in the interior of the angle formed by the non-common sides. A linear pair is a pair of adjacent angles whose non-common sides are opposite rays. Vertical angles are two angles that have a common vertex and whose sides form two lines. is a bisector of DAC if and only if is in the interior of DAC and mDAB = mCAB  —Preceding unsigned comment added by 199.126.187.152 (talk) 03:09, 9 October 2008 (UTC)
 * You wrote "The letters can go either way - that is, first and last letter are interchangeable. So, that angle could either be ABC or CBA." By "interchangeable" you seem to be saying here that the angle ABC and the angle CBA are equal.  You also wrote "There are up to 360 degrees in an angle."   And you defined an angle to be "the union of two rays that have the same endpoint."  Since these three statements taken together are inconsistent, it would appear that you need to explain what you meant by at least one of them.  Since you're right about complementary and supplementary angles, one imagines you can find and fix the apparent inconsistency here.  --Vaughan Pratt (talk) 04:34, 5 April 2011 (UTC)

Crum785's Edit from 27 October 2009
Crum785 changed the text of the alternative units section changing the equation ° = π/180 to 1° = π/180, under the description "more accurate". However, I'd argue this misses the whole point of the way the section was previously phrased. The way it was previously written was the somewhat provocative, but arguably correct statement that the degree symbol is itself a mathematical constant, in the same way as π or e. To be honest, this is how I always thought of it, and I know at least one prominent professor at my university who feels the same way. The new statement is not at all elucidating, and really doesn't show that treating the degree sign as a mathematical constant works in all cases.

Can you find a source saying that your way is more correct? I couldn't find any that explicitly say ° = π/180 is incorrect, and for the above cited reasons, I'd prefer if we could use this instead. Obviously, we don't want to be spreading false information, but as long as ° = π/180 is not false, it seems preferable. I agree the notation is not commonplace, but that doesn't make it incorrect. 129.15.127.237 (talk) 06:50, 8 March 2010 (UTC)
 * What's at stake here? I learnt back in, what, 2nd grade perhaps, that multiplying anything by 1 gave back the same result, for example 1 times 7 equals 7.
 * Are you questioning whether 1° = ° ? Or are you saying that 1° is not as syntactically correct as ° ?  --Vaughan Pratt (talk) 04:45, 5 April 2011 (UTC)

What's the Translative Equation between Degrees and Time?
So far this is what I've been able to come up with.

16 Directions (022.5 Degrees = 45 Minutes) — Preceding unsigned comment added by Arima (talk • contribs) 08:27, 9 January 2010 (UTC)
 * 360.0 = (12:00)
 * 022.5 = (12:45)
 * 045.0 = (01:30)
 * 067.5 = (02:15)
 * 090.0 = (03:00)
 * 112.5 = (03:45)
 * 135.0 = (04:30)
 * 157.5 = (05:15)
 * 180.0 = (06:00)
 * 202.5 = (06:45)
 * 225.0 = (07:30)
 * 247.5 = (08:15)
 * 270.0 = (09:00)
 * 292.5 = (09:45)
 * 315.0 = (10:30)
 * 337.5 = (11:15)


 * That's for a 12-hour clock, whose hour hand runs at 30 minutes per minute. For the more usual 24-hour clock in this translation the hour hand only goes 15 minutes per minute, though in both cases the minute hand goes at 6 degrees per minute or 6 minutes per second and the second hand goes at 6 degrees per second. --Vaughan Pratt (talk) 05:03, 5 April 2011 (UTC)

Removing the Pi approximation
I agree with the above discussion that the entire Pi related section does not belong in the history of the degree, or to the degree in general. I plan to remove it wholesale unless someone objects and can explain why it should remain here. Crum375 21:22, 21 September 2006 (UTC)


 * There is also a serious lack of reliable sources for virtually everything in this article and apparently over-abundance of original research till proven otherwise. That really needs to be fixed ASAP. Crum375 21:34, 21 September 2006 (UTC)


 * I found this reference in an online forum from 1995, which seems to be the source (copyright issues?) for the PI-derivation explanation for the degree that was included in the article. But at this point, I just don't see it more than speculation and I don't see a good quality source, and to me it's really not very convincing. I think the astronomical derivation, from the amount the stars seem to advance in the sky every night in their annual trek around the celestial pole (about 1 degree per evening, with an error of less than 1.4%), and the fact that some old calendars actually counted the year as 360 days is very convincing and logical. We do need better sources, though. And I would accept the PI-derivation method (copyright addressed) if someone could explain better how it derives the degree (the 1995 message doesn't do for me) and provide a better source. Crum375 22:58, 22 September 2006 (UTC)


 * Presumably the only point here is that the radian is a common measure of angle competing with the degree. &pi; enters into this competition. This is a rare exception to the general rule that unit conversions between conservative and liberal units are by definition rational, for example an inch is an integer number of microns, the velocity of light is an integer number of meters per second, a pound is an integer number of micrograms, Avogadro's number is on the road to becoming an integer, etc., etc. --Vaughan Pratt (talk) 05:36, 5 April 2011 (UTC)

Should we delete this?
I am tempted to delete the following from the lead:


 * "When that angle is with respect to a reference meridian, it indicates a location along a great circle of a sphere, such as Earth (see Geographic coordinate system), Mars, or the celestial sphere. "

Is there any reason why we mention this, as opposed to any of numerous other circumstances in which angles might be measured? Does it have some special importance here that I'm missing? (If retained it needs rewording because it is not clear what "it" refers to.) 86.181.169.8 (talk) 04:15, 20 November 2011 (UTC)