Clockwise and counterclockwise
A clockwise motion is one that proceeds 'like the clock's hands': from the top to the right, then down and then to the left, and back to the top. In a mathematical sense, a circle defined parametrically by the equations x = r sin t and y = r cos t, where r is the radius of the circle, is traced clockwise as t increases in value. The opposite sense of rotation is anti-clockwise (UK) or counterclockwise (U.S.).
Before clocks were commonplace, the terms 'sunwise' and 'deasil' or 'deosil' (from the Scottish Gaelic deiseil, from the same root as the Latin dexter, "right") were used for clockwise. (Of course, deasil (righthandwards) is only sunwise in the Northern Hemisphere.) 'Widdershins' or 'withershins' (from Middle Low German weddersinnes, "opposite course") was used for counterclockwise.
Technically, the terms clockwise and counterclockwise can only be applied to a rotational motion once a side of the rotational plane is specified, from which the rotation is observed. For example, the daily rotation of the Earth is counterclockwise when viewed from the North Pole, and clockwise when viewed from the South Pole.
Clocks traditionally follow this sense of rotation because of the clock's predecessor: the sundial. Clocks were first built in the Northern Hemisphere, and they were made to work like sundials. In order for an horizontal sundial to work (in the north), it must be placed looking southward. Then, when the Sun moves in the sky (east to south to west), the shadow cast by the sundial moves in the opposite direction, that is west to north to east. That's why hours were drawn in sundials in that manner, and that's why modern clocks have their numbers set in the same way.
Occasionally, clocks whose hands revolve counterclockwise are nowadays sold as a novelty. Historically, some Jewish clocks were built that way, for example in some Synagogue towers in Europe. This was done in accordance with the right-to-left reading direction of Hebrew.
Typically, screws and bolts are loosened (moved towards the observer) counterclockwise and tightened (moved away from the observer) clockwise, in accordance with the right-hand rule. One mnemonic for remembering this is "righty-tighty, lefty-loosey" (right to tighten, left to loosen.) The problem with the mnemonic is that it only works when viewing right and left relative to the top of the circle. When viewing relative to the bottom, the mnemonic becomes "left-tighty, righty-loosy". Another simple to use procedure based on the right-hand rule, is to point the thumb of the right hand for right-handed threads or left hand for left-handed threads in the direction one wants the screw, nut or bolt to move, then the fingers of the hand will curl in the direction one needs to turn the screw, nut or bolt to achieve the desired result.
Sometimes the opposite sense of thread is used for a special reason; a thread might need to be left-handed to prevent the prevalent stresses from loosening it. In a pair of bicycle pedals, for instance, one must be reverse-threaded, or the pedal will fall off. Some gas fittings are left-handed to prevent disastrous misconnection; for example, oxygen fittings are right-handed but acetylene and other flammable gases use left-handed fittings.
In trigonometry, and mathematics in general, plane angles are conventionally measured counterclockwise. In navigation, compass headings increase in a clockwise direction around the compass card, starting with 0° at the top of the card.
Clockwise and counterclockwise distinctions occur throughout nature: see