What Degree Is North on a Compass

Traditional mariners' compass points.
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:

Name Degrees,
minutes
Decimal
degrees
north 0° 00′ 0.0°
north-northeast 22° 30′ 22.5°
northeast 45° 00′ 45.0°
east-northeast 67° 30′ 67.5°
east 90° 00′ 90.0°
east-southeast 112° 30′ 112.5°
southeast 135° 00′ 135.0°
south-southeast 157° 30′ 157.5°
Name Degrees,
minutes
Decimal
degrees
south 180° 00′ 180.0°
south-southwest 202° 30′ 202.5°
southwest 225° 00′ 225.0°
west-southwest 247° 30′ 247.5°
west 270° 00′ 270.0°
west-northwest 292° 30′ 292.5°
northwest 315° 00′ 315.0°
north-northwest 337° 30′ 337.5°

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.

Points Designation Degrees,
minutes
Decimal
degrees
Bearing
angle
0 N 0° 00′ 0.00° N 00.00° W
or
N 00.00° E
1 NxE 11° 15′ 11.25° N 11.25° E
2 NNE 22° 30′ 22.50° N 22.50° E
3 NExN 33° 45′ 33.75° N 33.75° E
4 NE 45° 00′ 45.00° N 45.00° E
5 NExE 56° 15′ 56.25° N 56.25° E
6 ENE 67° 30′ 67.50° N 67.50° E
7 ExN 78° 45′ 78.75° N 78.75° E
8 E 90° 00′ 90.00° N 90.00° E
or
S 90.00° E
9 ExS 101° 15′ 101.25° S 78.75° E
10 ESE 112° 30′ 112.50° S 67.50° E
11 SExE 123° 45′ 123.75° S 56.25° E
12 SE 135° 00′ 135.00° S 45.00° E
13 SExS 146° 15′ 146.25° S 33.75° E
14 SSE 157° 30′ 157.50° S 22.50° E
15 SxE 168° 45′ 168.75° S 11.25° E
Points Designation Degrees,
minutes
Decimal
degrees
Bearing
angle
16 S 180° 00′ 180.00° S 00.00° E
or
S 00.00° W
17 SxW 191° 15′ 191.25° S 11.25° W
18 SSW 202° 30′ 202.50° S 22.50° W
19 SWxS 213° 45′ 213.75° S 33.75° W
20 SW 225° 00′ 225.00° S 45.00° W
21 SWxW 236° 15′ 236.25° S 56.25° W
22 WSW 247° 30′ 247.50° S 67.50° W
23 WxS 258° 45′ 258.75° S 78.75° W
24 W 270° 00′ 270.00° S 90.00° W
or
N 90.00° W
25 WxN 281° 15′ 281.25° N 78.75° W
26 WNW 292° 30′ 292.50° N 67.50° W
27 NWxW 303° 45′ 303.75° N 56.25° W
28 NW 315° 00′ 315.00° N 45.00° W
29 NWxN 326° 15′ 326.25° N 33.75° W
30 NNW 337° 30′ 337.50° N 22.50° W
31 NxW 348° 45′ 348.75° N 11.25° W

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.

Points Ascending
quarters
Descending
quarters
Degrees,
minutes,
seconds
Decimal
degrees
0 N 0° 00′ 00″ 0.0000°
N¼E NxE¾N 2° 48′ 45″ 2.8125°
N½E NxE½N 5° 37′ 30″ 5.6250°
N¾E NxE¼N 8° 26′ 15″ 8.4375°
1 NxE 11° 15′ 00″ 11.2500°
NxE¼E NNE¾N 14° 03′ 45″ 14.0625°
NxE½E NNE½N 16° 52′ 30″ 16.8750°
NxE¾E NNE¼N 19° 41′ 15″ 19.6875°
2 NNE 22° 30′ 00″ 22.5000°
NNE¼E NExN¾N 25° 18′ 45″ 25.3125°
NNE½E NExN½N 28° 07′ 30″ 28.1250°
NNE¾E NExN¼N 30° 56′ 15″ 30.9375°
3 NExN 33° 45′ 00″ 33.7500°
NExN¼E NE¾N 36° 33′ 45″ 36.5625°
NExN½E NE½N 39° 22′ 30″ 39.3750°
NExN¾E NE¼N 42° 11′ 15″ 42.1875°
4 NE 45° 00′ 00″ 45.0000°
NE¼E NExE¾N 47° 48′ 45″ 47.8125°
NE½E NExE½N 50° 37′ 30″ 50.6250°
NE¾E NExE¼N 53° 26′ 15″ 53.4375°
5 NExE 56° 15′ 00″ 56.2500°
NExE¼E ENE¾N 59° 03′ 45″ 59.0625°
NExE½E ENE½N 61° 52′ 30″ 61.8750°
NExE¾E ENE¼N 64° 41′ 15″ 64.6875°
6 ENE 67° 30′ 00″ 67.5000°
ENE¼E ExN¾N 70° 18′ 45″ 70.3125°
ENE½E ExN½N 73° 07′ 30″ 73.1250°
ENE¾E ExN¼N 75° 56′ 15″ 75.9375°
7 ExN 78° 45′ 00″ 78.7500°
ExN¼E E¾N 81° 33′ 45″ 81.5625°
ExN½E E½N 84° 22′ 30″ 84.3750°
ExN¾E E¼N 87° 11′ 15″ 87.1875°
8 E 90° 00′ 00″ 90.0000°
E¼S ExS¾E 92° 48′ 45″ 92.8125°
E½S ExS½E 95° 37′ 30″ 95.6250°
E¾S ExS¼E 98° 26′ 15″ 98.4375°
9 ExS 101° 15′ 00″ 101.2500°
ExS¼S ESE¾E 104° 03′ 45″ 104.0625°
ExS½S ESE½E 106° 52′ 30″ 106.8750°
ExS¾S ESE¼E 109° 41′ 15″ 109.6875°
10 ESE 112° 30′ 00″ 112.5000°
10¼ ESE¼S SExE¾E 115° 18′ 45″ 115.3125°
10½ ESE½S SExE½E 118° 07′ 30″ 118.1250°
10¾ ESE¾S SExE¼E 120° 56′ 15″ 120.9375°
11 SExE 123° 45′ 00″ 123.7500°
11¼ SExE¼S SE¾E 126° 33′ 45″ 126.5625°
11½ SExE½S SE½E 129° 22′ 30″ 129.3750°
11¾ SExE¾S SE¼E 132° 11′ 15″ 132.1875°
12 SE 135° 00′ 00″ 135.0000°
12¼ SE¼S SExS¾E 137° 48′ 45″ 137.8125°
12½ SE½S SExS½E 140° 37′ 30″ 140.6250°
12¾ SE¾S SExS¼E 143° 26′ 15″ 143.4375°
13 SExS 146° 15′ 00″ 146.2500°
13¼ SExS¼S SSE¾E 149° 03′ 45″ 149.0625°
13½ SExS½S SSE½E 151° 52′ 30″ 151.8750°
13¾ SExS¾S SSE¼E 154° 41′ 15″ 154.6875°
14 SSE 157° 30′ 00″ 157.5000°
14¼ SSE¼S SxE¾E 160° 18′ 45″ 160.3125°
14½ SSE½S SxE½E 163° 07′ 30″ 163.1250°
14¾ SSE¾S SxE¼E 165° 56′ 15″ 165.9375°
15 SxE 168° 45′ 00″ 168.7500°
15¼ SxE¼S S¾E 171° 33′ 45″ 171.5625°
15½ SxE½S S½E 174° 22′ 30″ 174.3750°
15¾ SxE¾S S¼E 177° 11′ 15″ 177.1875°
Points Ascending
quarters
Descending
quarters
Degrees,
minutes,
seconds
Decimal
degrees
16 S 180° 00′ 00″ 180.0000°
16¼ S¼W SxW¾S 182° 48′ 45″ 182.8125°
16½ S½W SxW½S 185° 37′ 30″ 185.6250°
16¾ S¾W SxW¼S 188° 26′ 15″ 188.4375°
17 SxW 191° 15′ 00″ 191.2500°
17¼ SxW¼W SSW¾S 194° 03′ 45″ 194.0625°
17½ SxW½W SSW½S 196° 52′ 30″ 196.8750°
17¾ SxW¾W SSW¼S 199° 41′ 15″ 199.6875°
18 SSW 202° 30′ 00″ 202.5000°
18¼ SSW¼W SWxS¾S 205° 18′ 45″ 205.3125°
18½ SSW½W SWxS½S 208° 07′ 30″ 208.1250°
18¾ SSW¾W SWxS¼S 210° 56′ 15″ 210.9375°
19 SWxS 213° 45′ 00″ 213.7500°
19¼ SWxS¼W SW¾S 216° 33′ 45″ 216.5625°
19½ SWxS½W SW½S 219° 22′ 30″ 219.3750°
19¾ SWxS¾W SW¼S 222° 11′ 15″ 222.1875°
20 SW 225° 00′ 00″ 225.0000°
20¼ SW¼W SWxW¾S 227° 48′ 45″ 227.8125°
20½ SW½W SWxW½S 230° 37′ 30″ 230.6250°
20¾ SW¾W SWxW¼S 233° 26′ 15″ 233.4375°
21 SWxW 236° 15′ 00″ 236.2500°
21¼ SWxW¼W WSW¾S 239° 03′ 45″ 239.0625°
21½ SWxW½W WSW½S 241° 52′ 30″ 241.8750°
21¾ SWxW¾W WSW¼S 244° 41′ 15″ 244.6875°
22 WSW 247° 30′ 00″ 247.5000°
22¼ WSW¼W WxS¾S 250° 18′ 45″ 250.3125°
22½ WSW½W WxS½S 253° 07′ 30″ 253.1250°
22¾ WSW¾W WxS¼S 255° 56′ 15″ 255.9375°
23 WxS 258° 45′ 00″ 258.7500°
23¼ WxS¼W W¾S 261° 33′ 45″ 261.5625°
23½ WxS½W W½S 264° 22′ 30″ 264.3750°
23¾ WxS¾W W¼S 267° 11′ 15″ 267.1875°
24 W 270° 00′ 00″ 270.0000°
24¼ W¼N WxN¾W 272° 48′ 45″ 272.8125°
24½ W½N WxN½W 275° 37′ 30″ 275.6250°
24¾ W¾N WxN¼W 278° 26′ 15″ 278.4375°
25 WxN 281° 15′ 00″ 281.2500°
25¼ WxN¼N WNW¾W 284° 03′ 45″ 284.0625°
25½ WxN½N WNW½W 286° 52′ 30″ 286.8750°
25¾ WxN¾N WNW¼W 289° 41′ 15″ 289.6875°
26 WNW 292° 30′ 00″ 292.5000°
26¼ WNW¼N NWxW¾W 295° 18′ 45″ 295.3125°
26½ WNW½N NWxW½W 298° 07′ 30″ 298.1250°
26¾ WNW¾N NWxW¼W 300° 56′ 15″ 300.9375°
27 NWxW 303° 45′ 00″ 303.7500°
27¼ NWxW¼N NW¾W 306° 33′ 45″ 306.5625°
27½ NWxW½N NW½W 309° 22′ 30″ 309.3750°
27¾ NWxW¾N NW¼W 312° 11′ 15″ 312.1875°
28 NW 315° 00′ 00″ 315.0000°
28¼ NW¼N NWxN¾W 317° 48′ 45″ 317.8125°
28½ NW½N NWxN½W 320° 37′ 30″ 320.6250°
28¾ NW¾N NWxN¼W 323° 26′ 15″ 323.4375°
29 NWxN 326° 15′ 00″ 326.2500°
29¼ NWxN¼N NNW¾W 329° 03′ 45″ 329.0625°
29½ NWxN½N NNW½W 331° 52′ 30″ 331.8750°
29¾ NWxN¾N NNW¼W 334° 41′ 15″ 334.6875°
30 NNW 337° 30′ 00″ 337.5000°
30¼ NNW¼N NxW¾W 340° 18′ 45″ 340.3125°
30½ NNW½N NxW½W 343° 07′ 30″ 343.1250°
30¾ NNW¾N NxW¼W 345° 56′ 15″ 345.9375°
31 NxW 348° 45′ 00″ 348.7500°
31¼ NxW¼N N¾W 351° 33′ 45″ 351.5625°
31½ NxW½N N½W 354° 22′ 30″ 354.3750°
31¾ NxW¾N N¼W 357° 11′ 15″ 357.1875°

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.

Points US Navy Others No "by"
N¼E N¼E N¼E
N½E N½E N½E
N¾E N¾E N¾E
NxE¼E NxE¼E NNE¾N
NxE½E NxE½E NNE½N
NxE¾E NxE¾E NNE¼N
NNE¼E NExN¾N NNE¼E
NNE½E NExN½N NNE½E
NNE¾E NExN¼N NNE¾E
NE¾N NE¾N NE¾N
NE½N NE½N NE½N
NE¼N NE¼N NE¼N
Points US Navy Others No "by"
NE¼E NE¼E NE¼E
NE½E NE½E NE½E
NE¾E NE¾E NE¾E
NExE¼E NExE¼E ENE¾N
NExE½E NExE½E ENE½N
NExE¾E NExE¾E ENE¼N
ENE¼E ExN¾N ENE¾E
ENE½E ExN½N ENE½E
ENE¾E ExN¼N ENE¾E
E¾N E¾N E¾N
E½N E½N E¾N
E¼N E¼N E¾N

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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