the theodolite is used, can only be obtained by calculation, but is, in reality, seldom required. This must be carefully borne in mind in computations of heights and distances.

The last instrument to be described is the Quadrant or Sextant, commonly called Hadley's, after its inventor.* The sextant and quadrant are, in reality, the same instrument, differing only in the extent of the graduated arc; the quadrant measuring angles of 90°, and the sextant those of 120°. When the whole circle is complete, the instrument becomes what is called the Reflecting Circle; being then made in the most perfect manner, and accompanied by several appendages, which render it altogether a very refined and elegant instrument." The quadrant is never made with the same nicety of construction as the sextant, and the reflecting circle is used only when extreme accuracy is required: when, therefore, the instrument is alluded to, in the description and exercises which follow, I shall speak of it by the general name of

66

sextant.

It is an instrument beautifully adapted, by means of a peculiar ingenious contrivance, to nautical use; and, since it is the only instrument of observation which can be used at sea, its value to the navigator is beyond estimation. Such is the instability of every vessel, from the restless motion of the water, as absolutely to prevent the use of any apparatus placed on a fixed support. Whatever mechanism is employed, must, it is obvious, be held in the hand. But to take the angular distance between any two objects, with an instrument so held, is evidently impossible, unless both objects can be viewed at the same instant, and, consequently, in the same direction. This is precisely what Hadley's contrivance effects. By reflecting mirrors it deflects one of the objects apparently from the direction in which it would otherwise be seen, and brings it into seeming close contact with the other object, so that both can be seen at the same moment; and we have only to keep the telescope of the sextant steadily upon the one object (which can be done, notwithstanding the motion), to see the other also; while the adjustment of the mirrors necessary to effect the apparent coincidence of the two objects, serves, at the same time, to show the angular distance of the objects from each other.

Hadley first gave it to the world; but priority of invention is attributed to Sir Isaac Newton.

Thus, in the diagram, P and Q being the two objects,

and M and N two small mirrors attached to the instrument, half of the glass forming the mirror N is left transparent, and the object Q is seen directly through the telescope T, and through the transparent half of the glass N, while the other object, P, is reflected first from the mirror M to the mirror N, in which it is seen, through the telescope, apparently close to Q, provided that the

mirrors are suitably placed with regard to each other. The adjustment of the mirrors, with that view, is effected by turning the moveable radius MI, to which the mirror M is attached, until the apparent coincidence is effected. The degrees are then read on the arc AB, being pointed out by the index, I, on the moveable radius, MI.

The arc of the sextant is really of 60 degrees, but is divided into 120 equal parts, counted as degrees, and actually measuring degrees in the observed angles, since any movement of the index, I, deflects the object (P) from its apparent position, by twice as many degrees as those passed over by the index. In the same manner the arc of the quadrant is really of 45 degrees, but measures angles up to 90 degrees. The two instruments are not named analogically, the sextant taking its name from the actual arc of the instrument, the quadrant from the angle it is capable of observing.

While the theodolite is confined to vertical and horizontal angles, the sextant measures angles in any oblique plane. The latter instrument can be applied to the measurement of horizontal angles, as such, only when the objects observed are previously known to be both on the same level with the observer, as in the case of a survey of a harbour or of a sea shore, where signals can be placed at the same height, above the surface of the water, with the point of observation. For vertical angles, again, the sextant cannot be used without the aid either of the sealine or of an instrument called an artificial horizon.

When the surface of the ocean is seen unbroken by land and bounded only by the sky, if the observer is raised but little above its surface, its outline nearly coincides with the horizon, and may, therefore, with a slight correction,* be taken as the horizon, the angular elevation of any object above the sea-line being taken, with that correction, as its elevation above the horizon. When the sea immediately below the object is bounded by land, its outline is still taken as the horizon, when its distance is known; but, in this case, a different correction† must be made for the depression of the observed outline, since it always appears lower than the sea-line would have appeared, had the prospect been unbounded.

When the sextant is used on land for taking angles of elevation, since the sea-line cannot then be used as a horizon, a small basin of mercury is taken in which the object is seen reflected. The angle between the object and its reflection is then observed, and half of that is taken as the angle of elevation, since the angle of reflection is always equal to the angle of incidence. To prevent agitation by the wind, the mercury is frequently covered by a roof of plate glass. The basin or box thus fitted up is called an Artificial Horizon. When mercury is not at hand, any other liquid presenting a reflecting surface may be used. It happens very conveniently that the artificial horizon is available precisely in those circumstances in which the sea-line cannot be employed, that is, in observations on land; and that, where the instability of the water renders the artificial horizon useless, it is peculiarly there that the outline of the sea comes in to supply its place.

In surveying, the sextant may be used when the actual distances only are required, without reference to heights or to horizontal dimensions. But if the heights are required, or if the distances are to be reduced to a horizontal plane, then the theodolite is the appropriate instrument, and by far the most convenient.

That correction may be computed by Trigonometry from the Earth's diameter and the height of the observer (See Exercise 44 of this Chapter); but, in practice, it is always taken from a Table to be found in works on Navigation, and entitled "Dip of the Sea Horizon," being Table vi in this volume. When such a Table is not at hand, the dip may be found approximately by the following Rule :-Take the square root of the height in feet for the dip in minutes. This will be correct within half a minute, unless the height exceed 100 feet.

The correction in this case, dependent on the distance of the shore, may also be computed by Trigonometry. But it is more readily obtained from another table entitled "Dip of the Sea at different Distances," and forming Table VII of this Treatise.

EXERCISE 1. Suppose that a ladder cannot be placed with safety within 15 degrees of an upright position: what height can be reached by a ladder 50 feet in length? Ans. 48 ft. 3 in. 2. What is the angular inclination of a regular ascent, or gradient as it is called, of 1 in 85; that is, of 1 foot of vertical rise for 85 of horizontal distance?

Ans. 40′ nearly.

3. If the inclination of a gradient is 1 degree, how many feet of horizontal distance does that give to a rise of 1 vertically, and how many feet of rise to a mile of horizontal distance?

Ans. The rise is 1 in 57.29, or 92.16 feet in a mile. 4. The town of B lies 17 miles East of A, and A is 19 miles North of C: what is the bearing of B from C?

Ans. N. 41° 49' E.

5. My writing-desk, when open, is 5.4 inches high at the back, and 1.7 inch at the front, and 20.6 inches wide from front to back: what is the angle at which it slopes? Ans. 10° 11'.

6. A ship sails SE by S, 81 miles: what is her departure, and her difference of latitude?

Ans. Departure, 45.00 miles. Dif. of Lat., 67.35 miles.

NOTE. The preceding and the three following questions in Navigation are intended to be resolved by the method of Plane Sailing, as it is called, which proceeds on the assumption that the small part of the Earth's surface taken in is a plane. The Departure, means the distance that the vessel deviates from the meridian which she leaves,—that is, the number of miles she gets further East or West. Her Difference of Latitude, is the number of miles she gets further North or South. Thus, let AB be the distance sailed, and NS a meridian line through the ship's first position, A,—that is, a line drawn due North and South; and let BC be perpendicular to NS: then AC is called the "difference of latitude," and BC the "departure."

A

C

IN

B

The angle BAC is indicated, in this question, by the given direction, SE by S. For this the Mariner's Compass must be referred to, which will be found delineated in Table v. Its circumference is divided into 32 equal s parts, distinguishing the 32 Points of the Compass, as they

are called,-8 in each quadrant: each of the 32 parts is also divided into 4 quarters. The number of degrees between two adjacent points must, consequently, be the eighth part of 90°, or 11° 15'. The four points, N, S, E, and W, are I called the Cardinal Points. The direction in which a ship sails, called its Course, or the Bearing of one point from another, is indicated either by naming the point of the compass, as NE, or SSW, or WNW W, or by naming the number of degrees from the meridian, counted either from the North or from the South towards the East or West, as N 45° E, or S 221° W, or N 75° 56′ 15′′ W. Any given point of the compass is expressed in degrees, by counting the number of points from the North or South, and multiplying 11° 15′ by that number, or by taking the degrees at once from the middle column in Table v.

7. Suppose a ship to sail SSW, until its departure is 198 miles: what is the distance sailed, and the difference of latitude?

Ans. Distance 517.4 miles. Dif. of Lat. 478.0 miles. 8. A ship sailed between North and West 255 miles, and thereby made her difference of latitude 157 miles: on what course did she sail?

Ans. N 51° 51' W, or NW W nearly.

NOTE. The course is first found in degrees, and, when required, these are converted into points by dividing by 11° 15', or by referring to the middle column in Table V, taking the nearest number of degrees in that column and observing the point opposite in one of the other columns ; the particular column to be taken being known from the question, or from our previous knowledge of the two cardinal points between which the vessel sails.

10. At a point 169 feet from a wall, and on a level with its base, the angle of elevation of its summit is observed to be 24° 26': what is the height of the wall?

Ans. 76 78+ feet.

11. A surveyor, standing on the parapet of a bridge whose height he knew to be 105 feet, observed a boat at anchor at a short distance, directly opposite. Taking an observation, he found its angle of depression 26° 17'. What was the distance of the boat from the summit of the bridge, and its distance from the bridge measured horizontally,