# POSITION LINE BY THE TABULAR METHOD

## 71. CIRCLES OF EQUAL ALTITUDE

A position line is a straight line drawn on a chart somewhere upon which a vessel’s actual position must lie. In coastal navigation, for example, a position line can be obtained by observing the compass bearing of, say, a lighthouse, converting this bearing into a true bearing by applying the compass error, and laying off the reciprocal of this true bearing from the charted position of the lighthouse in a seaward direction. The observer’s vessel must then lie somewhere on this position line; it cannot be determined precisely where on this position line unless a second bearing is observed nearly simultaneously with the first and at least 30º away from it, so that two intersecting position lines can be drawn on the chart; the vessel’s position would then be at the point of intersection called a fix.

A position line obtained from a celestial observation in called an Astronomical position line and as was stated in § 14-17, it is possible obtain an Astronomical position line from an observation or sight of a celestial body in spite of the fact that the computation involves using an estimated Lat. and Longitude which may not be (and usually is not) the vessel’s actual position.

An Astronomical position line is the projection on a chart of part of a circle of equal altitude the centre of which is located at the G.P. of the observed body. At every point on a circle of equal altitude the celestial body has the same altitude, and the greater is the altitude of that celestial body, the greater is the radius of the circle of equal altitude. The radius of a circle of equal altitude is equivalent to the zenith distance of the body, as fig. 18-1 shows and it follows that the great circle arc of the Earth’s surface in miles between the G.P. of a celestial body and any point on its circle of equal altitude is equal to the zenith distance-of the body in minutes of arc.

As we have shown impervious chapters, when the Obs. Alt. of a celestial body is corrected to the True. Alt. and then subtracted from 90º, the zenith distance is obtained Z1 in fig. 18-1. The point Z, however, may-be any point on a small circle of radius Z1 and centre I - the circle of equal altitude.

On the Earth, the observer's position, z., lies on the circumference of a small circle, the centre of which (x) is the celestial body’s G.P. The radius of this small circle zx, is also the zenith distance, and since it is now measured on the surface of the Earth, it can be expressed in nautical miles. This circle is also known as a-circle of equal altitude or more simply, as a position circle. The Astronomical position line is the small arc of this position circle on which the navigator discovers his position to be.

If the zenith distance of a celestial body is small, the circle of equal altitude may be plotted on a Mercator chart as a position circle, the centre of which is the G.P. of the observed body. Fig. 18-2 represents a-portion of a Mercator chart. Suppose the G.P. of the Sun at a certain instant to be at G in position Lat. 08º 00.0 N., Longitude 84º 00.0 E. If an observer, observes at this time the True. Alt. of the Sun to be 89º 00.0, then the zenith distance would be 1º 00 and his position. must lie on a circle of radius 1º or 60' the centre of which is at G. If another observer, observed the True. Alt. to be 88º 00.0 at the same instant of time, so that the zenith distance was 2º 00.0, then his position would lie on another position circle centred at G whose radius is 120 miles.

The Lat. of the G.P. of any celestial body changes and the Dec. of that body changes. The Longitude of the G.P. of any celestial body changes at the rate of change of the hour angle of the body. For the Mean Sun, this rate in 360º in 24 hours, i.e., 900' of Longitude per hour or 15' per minute. For a star, it is slightly greater. The greater the True. Alt. of a celestial body, the smaller its zenith distance and the smaller is the position circle, which is obtained from it. Normally the zenith distance of an observed celestial object will be so great that the position circle cannot be plotted on the chart, since its radius (zx in fig. 18-1) may be of the order of l., 000 miles or so, and the G.P. (x) will seldom be on the chart which the navigator is using for keeping his reckoning. As will be seen later, however, it is not in the least necessary to draw the complete position circle, except when the radius is less than about 100 miles (as in the case of the first observer In fig. 18-2). The part of the position circle, which concerns the navigator, can be found by methods, which confine the necessary plotting to the neighbourhood of the ship’s actual position.

An observer standing on a particular circle of equal altitude and facing in the direction of its centre will also be facing in a vertical plane coinciding with that of the vertical circle through the body. In other words, the direction of the G.P. of a celestial body corresponds to the azimuth of the body at the time of the observation. It follows, therefore, that if the azimuth of the body at the time of observation can be found, the direction of the position circle - that is, a line at right angles to the direction of any of its radii - can also be found.

Any small arc of a normal position circle may be regarded as a straight line, which is tangential to the circle. The direction of this straight line, which is known as a position line, is at 90º to the direction of the bodies G.P. from the observer, in other words at 90º to the body’s azimuth at the time of observation.

In fig. 18-3, an observer at A would observe the body whose G.P, is at G, to be due S. or 180º. The position line obtained from this observation (drawn as a straight line with a single arrowhead at each end) would lie in a 090º ~ 270º direction as shown. An observer at B, however, would lie on a position line the direction of which is 135º ~ 315º because the body’s G.P. bears 045º from him.

The concept of circles of equal altitude, position circles and position lines which we have described above is so important that it is worth repeating with reference to the illustration of the Earth shown in Fig. 18-4. If at any given moment the altitude of a celestial body were observed, corrected, and the true zenith distance deduced from it, a circle of position might be drawn on the Earth’s surface with the G.P. of the body as its centre and the zenith distance as its radius. In fig 18-4. we have assumed a body I who’s Dec. is 20º N. and G.H.A. 2 hours, so that it’s G.P. is therefore Lat. 20º N., Longitude 30º W. Now suppose the Obs. Alt. corrected gives a true zenith distance of 20º then with centre X and radius 20º we have described a circle which is seen to touch the equator and to extend 20º all round X. At every point on that circle the zenith distance of X would be 20º to all observers who observed the altitude of X simultaneously, but the positions of the observers on that circle would be indeterminate from this information above.

If however, from one particular observer, the true bearing of X happened to be N. then that observer would be on the equator and on the same geographical meridian as the body X, namely Longitude 30º W. The position of this observer would be Lat. 0º, Longitude 30º W. because the line of bearing, N. cuts the position circle at a definite position.

Suppose from another observer the body X bore 180º, then that observer would be on a parallel of Lat. twenty degrees due N. of the G.P. of X, i.e., in position Lat. 40º N. Longitude 30º W., because the line of bearing, S., cuts the position circle at a definite position.

Now suppose from yet another observer the body X bore 116º T., as indicated by the dotted line in fig. 18-4, then this line of bearing would cut though position circle at the observer’s position.

If we now conceive a second celestial body X1, who’s Dec. in also 20º N. but whose G.H.A. is 4 hours, then its G.P. would be in Lat 2Oº N., Longitude 60º W. If this second body were observed by the last observer simultaneously with the observation of X, and its zenith distance were 14º and true bearing 225º T., we would describe on the globe a circle of radius 14º (840 miles) with X, as centre. This observer would now have two position circles, one from the body X and another from the body X1.

The precise position of this observer must be at the intersection of these two position circles. The fact that there are two positions at which the position circles intersect is easily resolved because the bearings of the bodies would determine which was the correct one.

It would seem, therefore, that given the G.P. of a body, its zenith distance and azimuth at a particular moment in time, the position of the observer on the Earth’s surface may be fixed relative to that body, and so it can theoretically. The operation has to be modified in practice because it is clearly impossible to represent position circles of such magnitude on the navigator’s working chart with the accuracy necessary to determine a position.

The method in common use today is the Methode du Point Rapproche (the intercept method), credit for the invention of which is given to Captain (later Admiral) Marcq St. Hilaire, an officer of the French Navy endowed with a profoundly acute mathematical mind.

## 72. THE INTERCEPT METHOD OF DETERMINING AN ASTRONOMICAL POSITION LINE

The theory of intercepts is best introduced by using an example from coastal navigation fig. 18-5. Suppose we estimate by eye the distance from a lighthouse A to be 7 miles, its true bearing being 090º T., then we may lay off the bearing (AB) on the chart, and with compasses opened out to 7 miles, plot the position C.

This would only be an E.P. because the range was estimated by eye, but we would know with certainty that the vessel was somewhere on the line of direction AB. If we were to obtain the distance from the-lighthouse by vertical sextant angle or with a rangefinder and found the true distance from the lighthouse to be 5 miles instead of 7 miles, then we would know that C was not the position of the vessel but that she was miles nearer the lighthouse along the line of direction CA.

The observed position of the vessel would be at D, so we could measure CD, 2 miles towards the lighthouse CD, in position line navigation, would be called the intercept, and in this case would be called 2' towards.

Had the distance by observation been 9 miles from the lighthouse, the observed position would have been at E, 2 miles further away from the lighthouse than E.P. C. In this case the intercept CE would be called 2' away.

In nautical Astronomy, the method adopted is much the same. Fig. 18-6 represents the Earth, U being the G.P. of a celestial body X. A Sext. Alt. of the body is observed which, when corrected in the usual way, gives the true zenith distance (T.Z.D.), which, as we have already shown, is the great circle distance in nautical miles of the observer from the G.P. of the body (JU) and the radius of the observer’s position circle (shown in pecked line). It would be impossible to draw a position circle of such magnitude on the navigator’s working or plotting chart, but as will now be shown, this is not necessary. The navigator next chooses any position in the neighbourhood of the position expects his vessel to be. This is called the Assumed Position and we have marked this A in fig. 18-6. The navigator then calculates the zenith distance of the body X for this assumed position for the instant the Sext. Alt. was observed, by solving the spherical triangle PAU which is the equivalent of the celestial triangle PZX. This calculated zenith distance (ZX or AU) is clearly the great circle distance in nautical miles of the assumed position from the G.P. of the body for the instant of observation.

If the navigator now co compares this calculated zenith distance (C.Z.D.) with the true zenith distance (T.Z.D.) obtained by sextant observation, the difference between the two (AJ in fig. 18-6) is called the intercept. The direction of J from A is decided by the azimuth of the celestial body, which is angle PZX in the celestial triangle or angle PAU in fig. 18-6.

If, for example, the C.Z.D. is 50º 00.0, the navigator knows that his Assumed Position 3000' from the G.P. of the body. If, at the same time, his T.Z.D. 49º 55.0, he also knows that his true position is 5' nearer the G.P. than the assumed position and that this true position lies on the circumference of the position circle. If then, a line of bearing in the direction of the body’s azimuth is drawn on the chart from the assumed position, and a length equivalent to 5' measured off along it, the point obtained will be the point J (fig. 18-7) and the line of bearing will coincide with the radius of the position circle for a short distance. Point J is therefore one point, which is at the correct distance from the G.P. of the body, but is not necessarily the vessel’s position. It can only be said that the celestial position lies somewhere on the circumference of the position circle in the neighbourhood of J.

Since the circumference of a circle at any point is at right angles to the radius at that point, and since position circles are so big that the curvature of the circumference may be neglected under normal conditions, the small arc which forms a position line may be drawn in practice as a straight line through J. at right-angles to the line of bearing or azimuth.

Although on rare occasions the C.Z.D. may accidentally agree with the T.Z.D, thus making the intercept nil and point J coincide with the assumed position A in fig. 18-7 it does not follow that the vessel is necessarily in the same position as the assumed position. As above, the vessel’s position lies somewhere on a position line drawn through the assumed position at right-angles to the true bearing of the observed celestial body In Astro-navigation, as in coastal navigation, one observation gives a position line only. At least two observations, giving position lines, which cross, at a reasonable angle, are required for a fix.

The Marcq St. Hilaire or Intercept Method is thus a comparison between a vessel’s observed distance from the G.P. of the celestial body and the calculated distance of some arbitrary assumed position from the same G.P. The first distance is the T.Z.D., and the second the C.Z.D., the difference between them being the intercept. The student should keep clearly in mind that the zenith distance obtained by sextant observation is the actual or true distance of the vessel from the G.P. of the body, and that the calculated zenith distance is the distance the vessel would have been from the body had she been at the assumed position used for the calculation.

The intercept has been shown to be the difference between the true and calculated zenith distances. Since, however, the value obtained from a sextant observation is the altitude, and a further subtraction is required to derive the zenith distance, it is clearly more convenient to use the altitude instead of zenith distance to find the Intercept.

Intercept  = T.Z.D. ~ C.Z.D.

= (90º - True Alt.)  ~  (90º - Calc.Alt.)

= True Alt.  ~  Calc.Alt.

The intercept is thus also the difference between the observed True. Alt. and the calculated altitude for the assumed position, and is a measure of the error in the original assumption of position. This error, in nautical miles, is therefore equal to the number of minutes in the intercept, and the navigator must know whether this error is away from the G.P. of the body or towards the G.P. of the body.

A useful mnemonic for remembering the sign of the intercept is the word G.O.A.T. If the Obs. Alt. is greater than the Calculated Altitude, the assumed position was too far away from the G.P. by the amount of the intercept, which should therefore be plotted towards the G.P. The word Goat sums this up nicely: -

G(reater) O(bserved) A(ltitude) T(owards)

Conversely, if the Obs. Alt. is less than the Calculated Altitude, the -intercept is said to be away. Thus, if the Obs. Alt. is found to be 40º 30’ and the calculated altitude for the assumed position is 40º 25’, the intercept is 5 miles TOWARDS. If the Obs. Alt. is 40º 20’ and the calculated altitude is 40º 25’, the Intercept is 5 miles AWAY. The method of obtaining the calculated altitude for an assumed position is explained later in this Chapter, and the various methods of plotting an intercept and position line on a plotting chart will be explained in detail later in this chapter, but a brief explanation of plotting is given here to enable the student to see where he is going in learning to reduce a sight to an Astronomical position line. By assuming a position and finding the intercept and the azimuth of the celestial body for that position the navigator can now draw to scale on a plotting chart a straight position line on which his vessel must lie. This is shown in (fig. 18-8/18-9) (fig. 18-8) illustrating an intercept away and (fig. 18-9) an intercept towards. In both cases the intercept is shown as 3 miles, and the navigator draws, through his assumed position AP, a line in the direction of the azimuth AZ of the body observed. For an intercept away (fig. 18-8) he measures off, along this line, the intercept of 3 miles away from the direction of the body observed.

Through the point thus found he draws the position line Pl at right angles to the azimuth line. For an intercept towards (fig. 18-9) he does exactly the same thing except that, in this case, he measures the 3-mile intercept towards the body observed. In both cases, his vessel’s position lies somewhere on the position line PL. In describing the Marcq St. Hilaire or Intercept Method above we have referred several times to the navigators assumed position. In practice at sea, of course, the navigator does not know his position, otherwise there would be no need to take a sight at all. He has, however, a rough idea of his position from his dead reckoning (D.R.) and previous observations. It does not matter if the assumed position is wrong (so long as it is not ridiculously wrong) because the Intercept Method depends upon finding the error in an erroneous assumed position.

If. therefore, it convenient for reasons of calculation to choose an assumed position twenty or thirty miles away from the D.R. position, it can be done. Indeed, as will be seen later, it usually is done.