We may now construct the figure on which the problem is based,

and explain the numerous symbols involved in the formula

served star, and A a' a the circle of altitude projected by means of the zenith-distance S A (i.e., 90°-alt.); then,

X is the approximate point (point rapproché) to be determined. Join the points S and E to the point X by arcs of a great circle; E X is, as before shown in Fig. 10, perpendicular to the circle of altitude; S X is also perpendicular to the same circle; now, E X and S X, being both perpendicular to the circle of altitude at the point X, are consequently on the same arc of the great circle.

In the investigation of the problem; A and A" = the two true alts. derived from the observed; A' and A"" = the two estimated alts. by computation;

L = true lat. ; L'

=

estimated lat. by dead reckoning;

M = true long.; M' = estimated long. by dead reckoning; d L' = diff. lat.; d M'

Z

=

diff. long.;

estimated azimuth of A'; Z' estimated azimuth of A"; p = diff. between A and A'; p = diff. between A" and A".

Note on the time and hour-angle.-The Greenwich time of observation is such as is supposed to be shown by a good chro

nometer whose error and rate are known,-date and time expressed astronomically. Ship time is not used, but the longitude by dead reckoning is. An object's hour-angle as a westerly meridian distance from Greenwich is obtained as follows:—

For a star, planet, or the moon ;-To the M.T. at Green. add the mean sun's RA; this gives Green. sidereal time, from which subtract the object's RA; the result is the object's westerly hourangle from Greenwich, and will be indicated by G.

For the sun;-To the M.T. at Green. apply the equation of time; the result is apparent time at Greenwich, and also the sun's westerly hour-angle from Greenwich; this will also be indicated by G.

Note this, that west, whether for hour-angle, longitude, or azimuth, is positive or +; that east is negative or —.

COMPUTATION OF THE APPROXIMATE POINT X.

For the data of the problem we have; an altitude A observed at a certain Greenwich date and hour T; we have also L' and M' the latitude and longitude by dead reckoning. The altitude may be carried, if required, to the horizon of a second estimated point E'; it would be necessary in the case of the sun.

With the hour T at the first meridian, compute, from the Naut. Alm., the sidereal time at the first meridian; also the right ascension RA and declination D, of the star, planet or moon. But for the sun we only require the equation of time and declination. Reverting to Fig. 11, draw the meridians P E and PS; in the triangle PE S we know

The object's hour-angle (G) for Greenwich has already been explained (p. 210); for the hour-angle at the place of observation, we have S PE: = GM'.

We can now compute E S and the angle PE S; we have, moreover, E XES - XS = ES (90° - A).

The arc E X is never so great but that it can be treated as a loxodromic curve.

Now, if the estimated, were the true, point, then the zenith distance already found by the observer would be E S; hence we

may call ES the estimated zenith distance; and for the same reason, the angle at E the estimated azimuth; we will put z for E S, and Z for the angle at E. To compute z, we do so by means of 90°2= A', which we may call the estimated altitude.

As a matter of fact our computations are those of an altitude and a time-azimuth; the former comes through the latter, and this through an auxiliary arc o, by the aid of three fundamental equations of spherical trigonometry—

The formulæ are absolutely general, and may be employed in

all cases.

Obs. 1.-In the computation, (G M') is always to be taken less than 180°; when it exceeds that quantity, take it from 360°, and change the sign of the remainder.

Obs. 2.-The ambiguity of is of no moment; it is always taken between the limits 90° and +90°.

Obs. 3. The azimuth is reckoned from the elevated pole; also

Obs. 4.-If (G M') is positive, the azimuth is also positive, and reckoned towards the west; if (GM') is negative, the azimuth is also negative, and reckoned towards the east.

Reverting to the expression E X = ES-XS=ES- (90° — A); we now know from the computation that E S = 2 = 90° — A'; and as X S = 90° A, we also know that EX = A A'; by substituting p for E X, we have p = A A'; and p will be positive when A is greater than A', otherwise, negative. Thus much for the approximate point X.

It remains to compute A", Z' and p', in respect to the other star, or the second observation of the sun, and to determine another approximate point X'; the method is similar, and can be understood from what has already been said.

We will suppose it done, and we now have :

As before said, p is so small that it may be considered as a distance made in the direction of the azimuth Z, if p is positive; but in an opposite direction if p is negative. The same remarks apply to p' and azimuth Z'. It is very important to attend to these precepts, if we end the problem by a projection, as we easily can do.

Thus, Fig. 12, from the estimated point E set off p (in miles) in the direction of, or opposite to, the azimuth, as required; this determines the position of the approximate point X. In a similar way set off p' (in miles) to determine the second approximate point X'. From X and X' erect perpendiculars; and the crossing point of the two perpendiculars fixes O, the place of observation, i.e., the latitude and longitude of the ship, nearly.

1

This projection can, by the aid of a protractor, be made on the chart as readily as can Sumner's method. It is as well also to remark that though p and p' are arcs of great circles, the substitution of straight lines for curves produces but little effect on the result determined; in fact their length must extend to 63 miles in lat. 60°, and 280 miles in lat. 5° to cause an error of one mile in the approximate point X or X'.

Some of our readers may have noticed that we said the ship's position is determined, nearly; we said so advisedly, for there yet remain the errors of altitude, of the course and distance in the interval, of currents, &c., of which we have taken no account; these appertain to the "New Methods of Navigation," no less than to Sumner, and the old methods. But our continental neighbours are great in projection and nice calculations; they have what may be called "squared paper," that is, paper so ruled as to be covered with an immense number of very small squares; on a sheet of this paper having projected the point O, as in Fig. 12, they rectify it for every conceivable error, including also the chronometric curve for error and rate,-indeed they undertake to correct, by some of the new methods, the error and rate of the chronometer at sea.

But we may be permitted to doubt whether they really obtain a better result, or one of more practical value, than do some of our own able shipmasters, we mean men who observe well, understand the methods of navigation and the errors to which they are liable, make all the necessary corrections so as to have as few imperfect data as possible, look carefully after the instruments-the compass no less than the sextant and chronometer, and who use Sumner's Method under the restrictions we have pointed out, viz., that it gives a zone or space, not a point.

Having shown the method of projecting the last part of the problem, it may not be amiss to indicate how the same result is to be obtained from p, p', Z and Z', by the use of the Traverse Tables. To find the diff. lat. (d L') and thence the latitude (L);

we have, d L' =

and hence L =

P sin Z' p' sin Z
sin (Z' - Z)

L' + d L'.

;

For the diff. long. (d M), and thence the longitude (M);

p cos Z' p cos Z
sin (Z' - Z)

hence d M' = dep. x sec L'; and M = M' + d M'

When using the Traverse Tables in connection with this part of the problem, strict attention must be given to the law of signs, as we have the + or of p and p', as well as for west and for east. Hence the following table will serve for reference :

It may appear to the reader that, in the "New Method," a single altitude is not available for the purpose of navigation as it is in Sumner's method; but such is not the case. As before said, p may be treated as a loxodromic curve. With p, and the azimuth (Z) appertaining to it, enter the Traverse Tables according to the following precepts :