# A Simplified Guide to Small Marine Craft Navigation.

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## Chapter 3 (v.1) - Measurement of Angles in Navigation

Submitted: December 02, 2016

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Submitted: December 02, 2016

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Measurement of Angles in Navigation Illustration 1

“Angles” are formed by rotation, so using a straight line from O to A on illustration 1, and starting from a fixed position on OA, to rotate about the point O in the direction indicated by the arrow, and suppose it to take up the position indicated by OB. In rotating from OA to OB an “angle” AOB is described. Thus we have the conception of an “angle” as “formed by the rotation of a straight line about a fixed point”. If any point C is taken on the rotating arm, it will clearly mark out the arc of a circle CD.

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Using illustration 2, suppose that the rotation from CA to OB is continued until the position OA is reached, in which OA' and OA are in the same straight line. The point C will have marked out a “semicircle”, the pecked line on illustration 2, and the angle formed AOA' is sometimes called a "straight angle".

Illustration 2

If the rotating arm continues to rotate, in the same direction as before, until it arrives back at its original position on DA, it will then have made a complete rotation, and the point C on the rotating arm will have marked out the “circumference “of a circle, as indicated by the pecked line in illustration 3. Illustration 3

The conception of the formation of an “angle” by rotation leads to a convenient method of measuring “angles” in which we imagine the complete rotation to be divided into 360 equal divisions; thus we get 360 small “equal angles”, each of these being called a “degree” and denoted  as 1°.

Since any point on the rotating arm makes out the circumference of a circle, there will be 360 equal divisions of this circumference corresponding to the 360 degrees. If these divisions are marked on the circumference and the points of division joined to the center of the circle, the 360 “equal angles” could be shown and numbered and used for measuring any given “angle”. This is the principle of the circular protractor and of the compass card, both of which will be described later. Illustration 4

Illustration 4 represents a complete rotation such as was shown in illustration 3. If the points D and F are taken halfway between C and E in each semicircle, then the circumference will be divided into four equal parts and the straight line DF will pass through O. The angles COD, DOE, EOF and FOC, each being one-fourth of a complete rotation, are termed “right angles” and each contains 90°. The circle is divided into four equal parts called “quadrants”. When the rotating line has made half a rotation, the “angle” formed, the “straight angle”, must contain 180°, half of 360°.

In Navigation, where great accuracy is required, a degree, small as it appears to be on a protractor, is far too large a unit of measurement. This is because we sometimes have to deal with some very large circles such as the circumference of the Earth itself - a circle which has a diameter of nearly 8,000 miles, so that a degree measured at the center of the Earth would measure 60 miles across on the Earth’s surface. Therefore, a navigator must be able to measure “angles” with much greater precision than a “degree”, and for this purpose a degree is subdivided into 60 equal parts called “minutes”. The symbol for a minute is ' so that we would write thirty five degrees forty six minutes as, 35° 46'

Even this is not accurate enough for navigational purposes and the minute is further subdivided into another 60 equal parts called “seconds”. The symbol for a second is " and we would write one hundred and twenty degrees fifty-three minutes and twelve seconds as, 120° 53' 12"

Remember the basic facts of angular measurement because they are extremely important in the practice of navigation: There are 360° in a complete circle, and there are 180° in a semi-circle or a straight line. There are 90° in a quadrant, or "right angle", and there are 60 minutes in one degree, and 60 seconds in one minute.

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