which it forms a part, may be carried without inconvenience in a breast-pocket; and also that it may be used as an ordinary ruler.

The reason that this instrument has been called a Quintant, as distinguished from a Sextant, lies in this; viz.—that whereas, in Hadley's Sextant, as well as in the Pocket-Sextant and the Reflecting Semicircle of Sir Howard Douglas, the greatest angle that can be taken is 120°; in this instrument angles can be realized up to 160o. None however less than 10o can be relied upon. For terrestrial observations this is a matter of no consequence, since the student is warned, in all surveying books, against protracting very small angles; nothing under 20° being admissible. As twice 60o, giving 120o, is called a Sextant, so may 2 x 720 144° be called a Quintant.

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The SIGHT-VANE is in the short leg of the instrument; on which may be perceived a quadrangular cut about two inches long, and half an inch broad. That mark is the boundary of the bottom of a MAGNETIC NEEDLE-BOX. This box revolves longitudinally on two pivots, and must be turned right over by pressing against the outer upper edge of it with the left thumb. When it can go no further, the lid of the box (which has a cut in it, and is hinged at the end farthest from the sector's joint,) must be raised to perpendicularity. This is the SightVane of the instrument; for use when observation is made by Reflection.

When the Polymeter is to be used as an angle-finder, the Mirror must be turned upwards on its Side Pivots. When it stands perpendicular to the plane of the instrument and the Sight-Vane is set up as above noted, the instrument, when opened to 90o, is in its proper form for use as a QUINTANT.

The angle required is to be found as follows:

On one of the faces of the mirror will be observed a Vertical

Line, which stands just over the true centre of the joint of the sector. The observer must raise the Sector to the level of the eye, holding the instrument as nearly as possible in the plane of the objects to be observed; then looking over the line on the mirror at one object, he will move the left leg of the instrument by opening the sector, if the object to be reflected lie to the left hand, or closing it, if it lie to the right hand, until he finds the object sought, reflected on the line on the mirror; and so, exactly under the other object. When this is the case, the OBSERVATION is completed, i. e. the indication of the angle required is found.

The next process is to make it available; that is, to READ IT OFF, either to record it, or to protract it.

The annular disc of the sector's joint is graduated into the ordinary number of divisions, namely 360, forming four distinct quadrants of 90° each; but it is not requisite to graduate more than three of the Quadrants, or 270°, nor to number more than two of them: i. e. 180°; the third, without numbers, being merely for use in measuring the minutes of the degree in connection with the right-hand numbered Quadrant.

On the two double-numbered quadrants only, will the result, of observations by reflection be read off. A half-arrow will be seen at the inner end of the sector's leg. The graduation upon the quadrant, with which that half-arrow coincides, is, when multiplied by 2, the required angle. Why this doubling of the angle read off is necessary is thus explained: (see Plate). If, first, at the vertex c of an isosceles (or equal-legged) triangle acb, a mirror be erected perpendicularly, and it be so placed as that the base of its plane, as mcr, shall lie at right angles to one of the sides of the triangle as ac:

If, secondly, (see Annexed Figure) on placing the eye at the opposite end b of its other leg cbhg, so that the visual ray

may run along the edge bc of this leg; and any object, as the tree at t, be seen in the mirror at the vertex c of the triangle acb:

If, thirdly, a line ct be drawn from that vertex c in the direction of that reflected object, viz. the tree at t:

Then will the angle tca, which is contained between such line tc and the side of the triangle or the sector's inner edge ac to which the mirror was placed at right angles, be exactly equal to the original angle acb of the isosceles triangle abc, which angle is the angle of opening of the sector's legs. Here the angle formed by and lying between the legs of the sector is, what is called in optics, the Angle of Reflection; the equal angle at which the ray from the observed object "falls upon" the mirror is called the Angle of Incidence. This is the explanation and illustration of what is meant by the phrase in optical science; "the angle of incidence is equal to the angle of reflection, and vice versa." The definition of which angles is thus given in all treatises on optics:

"The Angle of Incidence is the angle which is formed by the incident ray, with a perpendicular to the reflecting surface at the point of incidence; the Angle of Reflection is the angle formed by the same perpendicular and the reflected ray."

A reference to the diagram at the end of the book will shew, that whereas we have been speaking of and wishing to obtain the angle between the points b and t, we have only got the measure of the angle between a and b; but as we know that the greater angle is double of the former, all that will be necessary is, to multiply the half angle by 2, and then we shall have the angle sought. Hence, if by the graduation on the inner quadrant on the disc at the point marked west, we find the legs of the sector are expanded to an angle of 25o, we know then that the angle tcb is 50o.

The angle, however, that is wanted is, in reality, not the angle bct but the angle tco, co being the prolongation of the line cb, which is the inner edge of the sector in the direction o, an obelisk, or any object at a distance convenient for observation with the tree at t.

N.B. It is to be observed that the point c in the diagram is always supposed to be the observer's position.

It is proper here to note, that since the first publication of the account of the instrument, a considerable improvement has been suggested with regard to the position of the mirror, which will be in future at right angles to that which it occupies in the plate. This change, while it makes no difference in the principle of the instrument, is attended with these benefits: First, that the angle may be taken either from the right-hand object to the left or from the left to the right, whichever may happen to be the more convenient.

Secondly, that any Index Error in the instrument is readily discoverable, by taking the angle in both these ways, and noting the difference of the two observations. Half of this difference will be the index-error to be added to the side which takes the least angle, or subtracted from the side which takes the greater.

When the required angle has been realized, and the graduated disc has enabled the observer to read it off in terms of degrees of the circle, he proceeds either to note this acquired angle in his memorandum-book, or sketching-tablets, or fieldbook, in order to lay it off on paper when at home, or to do so on his sketch in the field. For the manner of doing this see page 5.

N.B. The observer will be careful not to set off his angle on the wrong side of his Line of Observation.

IV*.

THE ANNEXE OR DOUBLE SEXTANT.

It has been said that an advantage which the Quintant has over the Sextant, is that it takes an angle up to 150° or 160°; and a disadvantage, in that it will not take a very small angle. But by means of an Annexe placed at a fixed angle, any angle, however small, or however great, up to 240°, can be taken by the Polymeter. This Annex is so arranged as that it may fall into a recess cut for it on the face of the instrument, like the semi-circular mirror, already described.

This Annex is a piece of SPECULUM placed at an angle of 50° with the Sight-Line of the instrument, the consequence of which is, that an additional angle of 100° is obtained to whatever angle is realized by the Quintant.

When the angle to be taken lies between 40° and 140o the angle will be taken by the Quintant, i. e. with the single Reflector but in the event of a smaller or a larger angle being required it becomes necessary to use the Annex, i. e. to use two reflections.

When the observation is made by means of the Annex, it will be found that the Sight-Vane need not be used, as the angle to be realized will equal twice the angle at which the mirrors are inclined to each other, and that is independent of the position of the Line of Sight.

When the Annex is wanted, it must be set up perpendicularly, and the observer must place himself in such a position as to bring the right-hand object required anywhere upon the face of the Annex. If now the observer find his angle to be greater than 140°, he opens the instrument beyond the right angle; and, keeping the image of the right-hand object on the Annex, moves the other leg until he gets that of the left-hand object