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# 2.16 Counting and Measurement

All of our counting and arithmetic is based on the decimal system. Yet many familiar quantities are divided in ways that do not use factors of ten. Think of the 360 degrees in a circle, or the 60 minutes in an hour, or the 24 hours in a day, or the 16 ounces in a pound, or the 12 inches in a foot, or the 8 pints in a gallon. Where do these units come from? Most ancient people could not read or write, so they preferred systems that made it easy to divide a number (or cows or coins) in many different ways.

The diagram shows the angles formed by the hands of an analog clock. Each minute represents six degrees in a 360 degree circle. Click here for original source URL.

The earliest cultures often used systems of measurement based on 60 — this is called the sexagesimal system. Why use 60? It is the lowest number that can be divided into 2, 3, 4, 5, or 6 equal parts. During the period from 3000 B.C. to 2000 B.C., the Babylonians invented the divisions of angle and time that we still use today. The fact that the unit of angular measurement is a degree is almost certainly based on astronomy. Each day Earth moves about 1° in its orbit around the Sun. To take the viewpoint of ancient people, if you view the sky each night at the same time of night, you find that the stars have shifted position by about 1° compared to the previous night. Over several weeks, this shift in the star positions is quite noticeable. The approximation that the year has 360 days may have reinforced the choice of 360° to define a circle.

Degrees are subdivided into minutes of arc and seconds of arc just as hours are divided into minutes and seconds. You might wonder why the Babylonians gave us our smallest units of time, when there were not even clocks or watches 5000 years ago? Remember that they used the stars to keep accurate time, and knew the length of a year to within a fraction of an hour. Since the Earth is rotating, angles and time are related. With one complete rotation every 24 hours, the Earth rotates 1° in about (24 × 60)/360 = 4 minutes. So any star rises and sets 4 minutes earlier each night.

Comparison of angular diameter of the Sun, Moon and planets. To get a true representation of the sizes, view the image at a distance of 102.6 [= 1 / tan(33.5/60*pi/180)] times the width of the largest (Moon: max.) circle. For example, if this circle is 10 cm wide on your monitor, view it from 10.26 m away. Click here for original source URL.

Astronomy starts with angular measurement. We should distinguish between linear measure and angular measure. Linear measure gives the actual length of something in inches, or meters, or miles. Angular measure by contrast gives the angle covered by an object. It measures the apparent separation between two objects at a particular distance from the observer, in units such as degrees. The verb subtend refers to the angle covered by such an object; for example the Moon subtends ½°. To get a sense of this angle, extend your little finger to arm's length. Your fingernail at this distance subtends ½°.

There is a wide range of angular measures in astronomy. The human eye is an amazing optical system with an angular range of roughly 70°. Cameras come with lenses that range from "wide angle" to "telephoto." The difference between such lenses is merely the angular width of the view, sometimes called the field of view of that lens. A normal lens usually covers an angular field of view of about 40° to 50°, whereas a typical telephoto lens may show about 10° to 20°. At the other extreme, the finest feature that the eye can see subtends an angle of only three minutes of arc (or 3') and many telescopes can measure features as small as one second of arc (or 1").

A schematic diagram of the terms "Azimuth" and "Altitude" as they relate to the viewing of celestial objects. Click here for original source URL.

Description of relations between Axial tilt (or Obliquity), rotation axis, plane of orbit, celestial equator and ecliptic. Earth is shown as?viewed from the Sun; the orbit direction is counter-clockwise (to the left). Click here for original source URL.

Ancient people found it useful to think of the stars as imprinted on a celestial sphere that rotates overhead. The location of every object in the sky can be specified by two angles. Suppose you are looking at a star in the night sky. You can measure the angle of the star above the horizon, called the altitude. You can then measure the angle of that point on the horizon from due north or any other reference point, called the azimuth. These two angles uniquely give the position of an object on the celestial sphere. In a similar way, any point on the Earth's surface can be defined by two angles — longitude and latitude. Of course the altitude and azimuth of a star will vary with time and with position on the Earth. So astronomers use a different pair of angles which are referenced to the celestial equator, which is the circle on the sky that is at every point ninety degrees away from the pole star Polaris. Star maps show how astronomers measure positions on the sky and how you can use the stars to find your way.

The early counting systems like the 60-based sexagesimal system are certainly convenient for dividing things up. Many of our familiar units of measurements derive from multiples of small numbers —inches, ounces, pints and so on. However these systems become very clumsy when dealing with large numbers. For example, the old Roman number for the decimal year 1988 is MCMLXXXVIII (still used in movie credits, for example). Imagine writing large astronomical numbers or doing your taxes in this way! We tend to stick with familiar systems even when they are not the best or the easiest to use.

The decimal system was perfected by the Arabs, who brought it from India around 750 A.D. Note that it requires the invention of a zero to stand for an empty place in a number. Zero is an Arabic word. The decimal system leads naturally to the metric system of measurement, where the units are related by powers of 10. For example, metric length units are kilometers or meters or millimeters and metric weight units are kilograms and grams. As an experiment after the Revolution of 1789, the French even developed a short-lived decimal system of time. However, you can see that our modern word is still littered with archaic units of measurement.