Degree (angle)

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This article describes "degree" as a unit of angle. For alternative meanings, see degree.

A degree (in full, a degree of arc, arc degree, or arcdegree), usually symbolized °, is a measurement of plane angle, representing 1／360 of a full rotation. When that angle is with respect to a reference meridian, it indicates a location along a great circle of a sphere (such as Earth, Mars, or the celestial sphere).

The degree and its subdivisions are the only units in use which are written without a separating space between the number and unit symbol (e.g. 15° 30', not 15 ° 30 ').

History

The 360 degree unit of measure for a circle was first derived from the Babylonian method of calculating the circumference of a circle. The evidence for this statement is based on a 1936 discovery of a particular mathematical tablet excavated some 200 miles from Babylon in the city of Susa. The translation of the tablet was partially published in 1950, and states that the ratio of the perimeter of a regular hexagon ([6r]) to the circumference of the circumscribed circle ([2\pi r]) yields an approximate formula for the ratio π

[\frac = \frac = 0.954929\ldots \approx \frac + \frac = 0.96]

which is a decent approximation (it is too high by about 0.53%). (However, the schoolbook approximation [\pi]=3.14 is a roughly ten-fold better approximation, being too low by about 0.051%.) The Babylonians used a base-60 (sexagesimal) numeral system, rather than our present base-10 (decimal) number system; that is why the denominators in the approximation are powers of 60. For more information, see the article on the Babylonian numerals. Interestingly, the aproximation can be improved significantly; the true representation of the hexagon/circle perimeter ratio in Babylonian numerals is

[\frac = \frac + \frac} + \frac} + \frac} + \frac} + \cdots]

Using 18 instead of 36 for the second Babylonian digit results in 0.955, which is high by only 0.0074% (i.e., 74 parts per million), a roughly 8-fold improvement in the approximation. This suggests that the scribe (or the translators) may have made an error of a factor of two, by using 36 instead of 18 for the second Babylonian digit.

The number 360 was probably adopted because it approximates the number of days in a year. Primitive calendars, such as the Persian Calendar used 360 days for a year. This was most likely due to watching stars revolve around the North Star forming a circle one degree per day. Its application to measuring angles in geometry can possibly be traced to Thales who popularized geometry among the Greeks and lived in Anatolia (modern western Turkey) among people who had dealings with Egypt and Babylon.

Further justification

The number 360 is useful since it is readily divisible: 360 has 24 divisors (including 1 and 360), including every number from 1 to 10 except 7. For the number of degrees in a circle to be divisible by every number from 1 to 10, there would need to be 2520 degrees in a circle, which is a much less convenient number.

For many practical purposes, a degree is a small enough angle that whole degrees provide sufficient precision. When this is not the case, as in astronomy or for latitudes and longitudes on the Earth, degree measurements may be written with decimal places, but the traditional sexagesimal unit subdivision is commonly seen. One degree is divided into 60 minutes (of arc), and one minute into 60 seconds (of arc). These units, also called the arcminute and arcsecond, are respectively represented as a single and double prime, or if necessary by a single and double closing quotation mark: for example, 40.1875° = 40° 11' 15". If still more accuracy is required, decimal divisions of the second are normally used, rather than thirds of 1／60 second, fourths of 1／60 of a third, and so on. These (rarely used) subdivisions were noted by writing the Roman numeral for the number of sixtieths in superscript: 1^I for a "prime" (minute of arc), 1^II for a second, 1^III for a third, 1^IV for a fourth, etc. Hence the modern symbols for the minute and second of arc.

Alternative units

In mathematics, angles in degrees are rarely used, as the convenient divisibility of the number 360 is not so important. For various reasons, mathematicians typically prefer to use the radian. In this system the angles 180° and π rad are equal, or equivalently, the degree is a mathematical constant ° = π/180 . This means, that in a complete circle (360°) there are 2π radians. The circumference of a circle is 2πr, where r is the radius.

With the invention of the metric system, based on powers of ten, there was an attempt to define a "decimal degree" (grad or gon), so that the number of decimal degrees in a right angle would be 100, and there would be 400 degrees in a circle. Although this idea did not gain much momentum, most scientific calculators still support it.

References

• Beckmann P. (1976) A History of Pi, St. Martin's Griffin. ISBN 0312381859

See also

• radian
• gradian
• square degree
• steradian

 

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