Or a diagram in which the error is shown by its particular number of degrees east or west of the true north line may be drawn and the variation likewise properly shown east or west of true north. If the error is to the left of the variation the deviation is west and if to the right the deviation is east.
CHAPTER VIII
Longitude
The longitude of any position on the earth is its distance east or west from the meridian of Greenwich, which has been chosen as the meridian of origin. Longitude is measured on the equator eastward and westward through 180°, completing in this way the whole circumference of the earth.
The circumference of every circle comprises 360°, whether it is a great circle of the earth or any of the parallels which range in size from a point at the poles to a great circle at the equator. There are always 360° but the length of each degree is determined by the size of the circle. Thus a degree of longitude on the equator is 60 miles, while on the 50th parallel of latitude it is only about 39 miles, owing to the decreasing size of the parallels of latitude. A minute of longitude on the equator, like a minute of latitude, is equal to one mile, but the difference between the meridians in actual distance decreases toward the poles gradually lessening the linear value of a degree of longitude. Thus it will be seen that when it is desired to represent a difference of longitude in distance, it must be done in terms of departure (miles) corresponding to the particular parallel of latitude of the position.
The sun apparently moves around the earth in its diurnal motion, covering 360° in 24 hours, whether the declination is north or south, and a little simple division shows that in one hour he passes over 15° of longitude, whatever the latitude. This reduced shows that 1° is passed over every 4 minutes. As the standard time, the world over, is reckoned by the movements of the sun, it is plainly seen that when considering longitude, a definite relation exists between time and arc (°-´-´´). Owing to this relation, time and arc become interchangeable by a simple process of conversion.
So it follows that if we have the time at Greenwich by a chronometer, and through a trigonometrical calculation we determine the local mean time at the ship, the difference in time between Greenwich and the ship’s meridian represents the longitude in time, which is readily converted into arc.
The calculation involved in determining the local mean time is the solution of the astronomical triangle, or in other words it is a problem in spherical trigonometry. This triangle has its apex at the pole with one side as the polar distance (90° - declination of the observed body), another side the co-latitude (90° - dead reckoning latitude) and the third side the zenith distance (90° - the corrected altitude of the body).
It is one of the principles of trigonometry that with any three elements given in a triangle any of the remaining elements may be computed; that is, any angle or side is obtainable. The solution of the astronomical triangle for various elements includes the finding of the zenith distance and from this the altitude, which forms the main feature of the problem involved in the New Navigation. It also provides us with the angle between co-latitude and the zenith distance, which is the azimuth of the body, by which the mariner is able to ascertain the error of his compass.
The most important feature of the astronomical triangle is the angle at the pole, known as the hour angle, which when found secures for the navigator his local time. The problem presents itself in the form of three sides being given to find one angle. It is found by the time sight formula, which is too well-known to need any discussion here.