Radians

1 Key excerpts on "Radians"

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  eBook - ePub

  How Round Is Your Circle?

  Where Engineering and Mathematics Meet

  • John Bryant, Chris Sangwin(Authors)
  • 2011(Publication Date)
  • Princeton University Press

    (Publisher)

  Lenses were also applied to read scales more accurately, and the astronomer Ole Römer (1644–1710) used a low-power microscope containing a number of hairlines to remove the problems of two scales wearing against each other. Gascoigne also developed a device consisting of two movable cross-hairs, controlled by a graduated screw. By measuring the number of turns, the distance between the two cross-hairs could be calculated and from this the angle between two objects which were very close together could be calculated. This was used by him to measure the angular width of the Moon and the planets. Of course, the accuracy of this device depends crucially on the accuracy of the threads of the screws, which in the early seventeenth century were extremely difficult and expensive to make by hand. Screw threads were also used to move and position telescopic sights accurately, although early designs by Robert Hooke and others were not particularly successful from a practical point of view. The Astronomer Royal John Flamsteed (1646–1719) complained bitterly that he was ‘much troubled with Mr Hooke who, not being troubled with the use of any instrument, will needs force his ill-contrived devices on us’ (Flamsteed 1725, p. 103).

  5.1 Units of Angular Measurement

  It is well worth pausing for a moment to review mathematical systems for angular measurement. The most commonly used system for measuring angles is the division of a circle into three hundred and sixty degrees, i.e. 360°. Each degree is subsequently divided into sixty minutes and each minute into sixty seconds . An angle is then written as 17° 34′ 12″. Each minute is ≈ 0.0167 of a degree, and each second is ≈ 0.000 278 of a degree. As we have seen, dividing a circle into six equal parts is simple, and can be done accurately. Then subdividing each into sixty equal parts to give degrees and then again into minutes and seconds is a remnant of the ancient Babylonian base-sixty number system. This division into 360° is also convenient from a practical point of view. For example, on a basic school protractor with a diameter of 90 mm a single degree occupies an arc length of about 0.75 mm on the edge. This is perfectly easy to distinguish by eye, and yet still small enough a unit to be practically useful. Since 60 and 360 have so many integer factors, many fractions of a rotation are easy to express exactly as whole numbers of degrees.

  Mathematicians use a different system for measuring angles and it is known as the radian . In this system a circle is divided into 2π Radians. The advantage of this is that many formulae become extremely simple and elegant to express. For example, in a circle of radius r take an arc with angle t , measured in Radians. The arc length is simply rt . The corresponding formula, with t in degrees, is πrt . However, since π is an irrational number there is no longer a whole number of Radians in a whole number of full rotations. From a practical point of view this is a catastrophe.

  Another unit for measuring angles is the decimal degree. In this scheme a right angle is divided into 100 units, and many modern scientific calculators implement these units as grads . For military purposes the artillery divide a full circle into 6400 parts, called mils

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