time, by means of preconcerted signals, from two or more stations on shore, the bearing of the observer at sea with some fixed objects. Theoretically, this method is the most accurate, but practically, it is found that even well concerted signals cannot always ensure simultaneous observation. As

the times of observation must moreover be registered at all the stations as well as at sea, a single error in the series arising from unseen signals leads to constant misapprehension, and can scarcely be rectified by a subsequent comparison of the different field-books, if the series embrace many observations. Independently of these objections, others present themselves in the form of a greater consumption of time, and a necessity for an increased number of experienced observers.

The third method consists in measuring from the boat, or vessel, or rock, by means of the sextant, the angles subtended by three or more objects on shore, the positions of which are given,-from these data the position of the observer is determined.

Theorem

The mutual distance of three remote objects being given, with the angles which they subtend at a station in the same plane, to find the relative place of that station.

Let the three points A, B,

and C, and the angles ADB and B DC which they form

B

at a fourth point D, be given:

to determine the position of D.

* LESLIE's Trigonometry, page 385, et seq.

X

DI

B

First suppose the station

A

D to be situated in the direction of two of the objects, A and C.

All the sides A B, AC, and B C of the triangle ABC being given, the angle B A C is found, and in the triangle ABD, the side A B with the angles at A and D being given, the side A D is found, and consequently the position of the point D is determined.

Secondly. Suppose the three objects A, B, C to lie in the same direction.

A

E

B

C

Describe a circle about the extreme objects A, C, and the station D; join D A, DB, and DC; produce DB to meet the circumference in E, and join AE and CE. In the triangle AEC, the side AC is given, and the angles E A C and ECA, being (Euclid III. 21) equal to C D E and ADE, are consequently given; wherefore the side A E is found. The triangle A E B, having thus the sides A E and A B, and their contained angle E A B or BDC given, the angle ABE and its supplement A B D are found. Lastly, in the triangle A B D, the angles A BD and ADB, with the side A B, are given, whence BD is found. But since the angle ABD and the distance BD are assigned, the position of the station D is evidently determined. Thirdly. Let the three objects form a triangle, and the station D be either within or without it.

B

join E A, and CE.

Through D, and the points A and C, describe a circle: draw BD cutting the circumference in E, and

1. In the triangle A E C, the side A C, and the angles ACE and CA E, which are (Euclid III. 21) equal to AD B or its supplement, and to BDC or its supplement, being given, the side A E is found.

2. All the sides of the triangle ABC being given, the angle CAB is found.

3. In the triangle B A E, the sides AB and A E are given, and the contained angle E AB, (being

B

E

either the difference or the sum of CA E and C AB,) is also given, whence the angle ABE or ABD is found.

4. In the triangle DA B, the side A B and the angles ABD and ADB being given, the side AD or BD is found, and consequently the position of the point D, with respect to A and B, is determined. By a By a like process the relative position of D and C is deduced; or CD may be calculated from the sides A C, AD, and the angle ADC, which are given in the triangle C A D.

It is obvious that the calculation will fail, if the points B and E should happen to coincide. In fact the circle then passing through B, any point D whatever in the opposite arc ADC will answer the conditions required, since the angles AD B, and DB C, being now in the same segment, must remain unaltered.

This third case, in which the three objects form a triangle, involves the conditions under which the problem has in general to be solved, the first case in which two of the objects, and the second in which the three objects are in a line, occurring but rarely. The reader will, however, doubtless have been impressed with the extremely laborious nature of the solution which this case involves, demanding no less than four separate trigonometrical calculations before the required answer is obtained. It would evidently there

fore, be a most tedious process, and one but little suited to practical purposes; other means have therefore been devised of solving the problem which are better suited to practice.

On A B (Euclid III. 33) describe a segment containing an angle equal to that subtended by the objects A and B, and on B C describe another segment BDC, containing an angle equal to that subtended by the objects B and C; the point D, where the two circumferences intersect, will evidently mark the station required. Should the two circles have the same centre, their circumferences must obviously coincide, and therefore every point in the containing arc will answer the conditions required, in which case the problem becomes indeterminate.

A

E

B

F

Example. Let the three objects on shore, A, B, C, be fixed in position; and let the angle subtended at D by A B be equal to 50°, and the angle subtended by BC be equal to 40°; to find the point D by construction. Subtract double the

angle A B D from 180°, and take half the remainder, equal 40°. Lay off this angle at A and B, the two lines

forming the angles with A B will meet in E, the centre of a circle passing through A, B, D (Euc. III. 20). Again, subtract double the angle B D C from 180°, and take half the remainder, equal to 50°. Lay off this angle at B and C; the two lines forming the angles with B C will meet in F, the centre of a circle passing through B, C, and D. The point D, where the two circles intersect, marks the station required.

But this process, although much simpler in point of construction than that previously explained, would yet be exceedingly tedious where a great number of stations had to be determined. To simplify the construction, an instrument, called the station-pointer, has been invented: it affords means of laying down the work with great rapidity, and with sufficient accuracy for all practical purposes. The following is a description of the instrument.

The Station Pointer

Is formed by three limbs or rulers, A, B, C, which revolve round a common centre, in such a manner that B and C may be set to form any angles with A. "The middle ruler is double, and has a fine wire stretched along its opening; the other rulers have likewise a fine wire stretched from end to end, and so adjusted by the little projecting pieces which carry them, that all the three wires tend to the centre of the instrument, where they would meet if produced.

Through the centre is an opening sufficiently large to admit a steel pricker." The middle limb carries at the