What Degree Is North on a Compass
Version of 7 June 2008.
Dave Barber's other pages.
Thanks to Gerard Mittelstaedt, whose web page contains valuable historical information, especially a table from Bowditch.
The terms north, east, south, and west, along with their equivalents in other languages, have been used for millennia to denote directions along the surface of the earth. Because a greater precision has often been required, especially by mariners on the open seas, added in early times were the directions northeast, southeast, southwest, and northwest. Northeast is halfway between north and east, et cetera.
An eight-point compass card or compass rose shown below displays the eight directions mentioned so far; also included are bearings in degrees, which are often preferred in modern navigation. Like many charts, this one observes the custom (but not requirement) of showing north at the top. The diagram employs the abbreviations N, E, S, and W with the obvious meanings. See also a pdf version, in higher resolution.
Improvements in navigational equipment and maps eventually made worthwhile a finer specification of direction, with 16 points. To form the new direction names, a prefix of north, east, south, or west was added to northeast, southeast, southwest, or northwest. By this stage a standard appeared: the name of a direction does not contain both north and south, nor does it contain both east and west -- these exclusions will continue with the finer compass cards discussed later. Here is a chart summarizing the sixteen directions introduced so far:
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A 16-point compass card is shown below. (pdf version)
The compass card with 32 points is the most detailed version in wide use.
To form the symbol for a direction not found in the 16-point compass, one of the letters x or b and one of the letters N, E, S or W are suffixed to a term of the 8-point (not 16-point) system. Each suffix adjusts the direction by adding (moving clockwise) or subtracting (moving counterclockwise) 11.25 degrees. This angle is termed one point, in contrast to meaning of that word above, where point simply means any named direction.
| Suffix | In the northeast quadrant | In the southeast quadrant | In the southwest quadrant | In the northwest quadrant |
|---|---|---|---|---|
| xN | −11.25° | - | - | +11.25° |
| xE | +11.25° | −11.25° | - | - |
| xS | - | +11.25° | −11.25° | - |
| xW | - | - | +11.25° | −11.25° |
A direction such as SWxW is read "southwest by west", and is equivalently written SWbW or SW by W. A symbol such as SSWxW is unambiguous, but it equals the more concise SWxS -- this exemplifies why the symbols of the 8-point system are sufficient as bases for the suffixes, and the full set of 16-point symbols need not be used.
Although a famous motion picture is entitled "North by Northwest", such a direction is not found in the mariners' compass; absent also is "South by Southwest", the name of a music festival in Texas. Further, there is no obvious way to extend traditional terminology to give meaning to these directional names.
The 32 points are listed in this next table, which also includes the bearing angle notation often used by surveyors.
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A 32-point compass card is shown below; to the 16-point card above have been added intermediate points and a numerical point scale. (pdf version)
The compass card with 128 points is the most detailed to be found, as navigational calculations requiring greater precision are more conveniently reckoned in degrees. Complicating matters is that the angular measure known as a point is still 11.25 degrees, even though the angle between named directions is now only 2.8125 degrees.
To form a symbol for one of the additional directions, one of the fractions ¼, ½ or ¾ and one of the letters N, E, S or W are suffixed to a term of the 32-point system. Each suffix adjusts the direction by some fraction of a point, but for clarity the table below speaks of adding (moving clockwise) or subtracting (moving counterclockwise) the appropriate number of degrees. Other fractions are possible, but quite rare.
| Suffix | In the northeast quadrant | In the southeast quadrant | In the southwest quadrant | In the northwest quadrant |
|---|---|---|---|---|
| ¼N | −2.8125° | - | - | +2.8125° |
| ½N | −5.6250° | - | - | +5.6250° |
| ¾N | −8.4375° | - | - | +8.4375° |
| ¼E | +2.8125° | −2.8125° | - | - |
| ½E | +5.6250° | −5.6250° | - | - |
| ¾E | +8.4375° | −8.4375° | - | - |
| ¼S | - | +2.8125° | −2.8125° | - |
| ½S | - | +5.6250° | −5.6250° | - |
| ¾S | - | +8.4375° | −8.4375° | - |
| ¼W | - | - | +2.8125° | −2.8125° |
| ½W | - | - | +5.6250° | −5.6250° |
| ¾W | - | - | +8.4375° | −8.4375° |
This next table enumerates the 128 points, many of which have two equivalent names. Bearing angles are omitted, but are calculated the same as in the previous table.
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The table below gives three approaches toward choosing between the two possible names for fractional points. The first column is from Bowditch (American Practical Navigator, 1916) giving United States Navy usage. The second column, also from Bowditch, gives an alternate usage of "some mariners". The third column offers an arguably simpler approach, avoiding any direction with the word by in its name. (Inadequate is the obvious criterion of choosing the shorter name because in some cases, such as NNW¼N versus NxW¾W, they are the same length.) Other quadrants can be generated by substituting south for north, or west for east.
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A 128-point compass card is shown below, as an augmentation of the 32-point card above. (pdf version)
Historical note. The symbols that resemble quote marks to represent minutes (as in 33′) or seconds (15″) are in fact derived from Roman numeral superscripts, and represent how many times a degree is to be divided by sixty to obtain the subunit. Hence the angle written above as 357° 11′ 15″ could have been written 357° 11I 15II. This sexagesimal numeration goes back several thousands of years, and it applies similarly to the hours, minutes and seconds of time.
The table below shows how several example angles could be written. The precision in the approximations is carried to 16 decimal places or 9 sexagesimal places, because 1016 very nearly equals 609. If the number of degrees is rational, the approximation will have a repeating pattern, although it may involve more digits than appear below. The values are truncated rather than rounded to make any repetition clearer.
| Exact Angle | Decimal approximation | Sexagesimal approximation |
|---|---|---|
| 360° ÷ 7 | 51.4285 7142 8571 4285° | 51° 25I 42II 51III 25IV 42V 51VI 25VII 42VIII 51IX |
| 360° ÷ 11 | 32.7272 7272 7272 7272° | 32° 43I 38II 10III 54IV 32V 43VI 38VII 10VIII 54IX |
| 360° ÷ 19 | 18.9473 6842 1052 6315° | 18° 56I 50II 31III 34IV 44V 12VI 37VII 53VIII 41IX |
| 1 radian | 57.2957 7951 3082 3208° | 57° 17I 44II 48III 22IV 29V 22VI 22VII07VIII 32IX |
Using one radian for instance, the sexagesimal form is spoken as "57 degrees, 17 minutes, 44 seconds, 48 thirds, 22 fourths, 29 fifths …".
Sometimes the word prime is used instead of minute, and this term survives in a non-sexagesimal context of mathematical analysis. If f represents some function, then f′ (spoken "f prime") denotes its derivative, f″ ("f double prime") its second derivative, etc. This notation (which can be traced back to Joseph Louis Lagrange) reverts to Roman numerals for higher derivatives.
Although decimal numeration is not intrinsically simpler than sexagesimal, it is certainly more familiar, and this is why much modern usage involves decimal fractions of a degree. Similarly, the time unit known as the second is almost never divided into thirds, fourths, etc; but rather into milliseconds, microseconds, etc.
What Degree Is North on a Compass
Source: http://tamivox.org/dave/compass/index.html
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