instead of the last minute, accordingly as the eye can best judge by the magnifying-glass of the small quantity defective of perfect coincidence; in this the observer will be assisted by examining the places of the preceding and following strokes.

Many of the instruments now made by the best makers have the degree divided into 20, 15, and 10 minutes, by means of a vernier with divisions and subdivisions acting with divisions and subdivisions on the limb. On examining the limb of an instrument, as, for instance, a sextant, it will be observed that it is divided at every 5 degrees by a longer divisional line than the others; these spaces contain each 5 degrees, all of which are again subdivided by shorter lines representing minutes; if there be four spaces, each will be 15 minutes, and if six, then each will be 10 minutes. On the vernier, likewise, the divisions marked by longer lines are subdivided by shorter ones, the divisions being minutes and the subdivisions being seconds. If the spaces between each long division be divided into four, each will be 15", and if into six, then 10"; the number of subdivisions between the minutes of the vernier are generally, though not necessarily so, the same as the subdivisions between the degrees on the limb, so that if the limb is divided into 20', then the vernier is divided into 20".

With regard to reading off an angle on the limb of an instrument provided with a vernier; if the zero of the vernier coincide with a division on the limb, such division gives the angle measured; thus, if it coincide with 30°, then 30° is the measure of the angle; if with the first or second long division beyond the thirty, then the reading will be 31° or 32°, as the case may be; but supposing that the degree on the limb is subdivided into three parts, each, of course, indicating 20', and that the zero of the vernier coincides with the first of these subdivisions beyond 31°, then the reading will be 31° 20", and if with the second, the reading will be 31° 40'. In the next place let us consider that, on a limb divided as above, and with a vernier having also three short divisions between the long ones, and therefore reading to 20", the zero of the vernier has passed by 31° 20', but not reached the 40'; we must now look for a division on the vernier coinciding with one on the limb; let it be the first long division, then the reading will be 31° 21'; now let us suppose that it is the second short division beyond the first long one beyond zero, then the reading will be 31° 21' 40".

The Figures 52, 53, 54, 55, and 56, have been introduced to illustrate our observations on the vernier; in Fig. 52, the degrees from 35° to 40° are shown on the limb, each degree being divided into four divisions of 15' each; in the vernier each

READING THE VERNIERS.

67

division between the long strokes being also divided into four, gives 15" for each subdivision; in our example the zero has passed by 35° 15', and the second subdivision counting 30" beyond the two main divisions counting 2' on the vernier, coincides with a stroke on the limb; then this quantity of 2' and 30" is to be added to 35° 15', making together 35° 17′ 30′′. In example 53, each degree on the limb is divided into six parts of 10' each, and each minute on the vernier is divided into four spaces of 15' each, and the vernier reads 76° 52′ 45′′. In example 54 we read 131° 44′ 40′′. In example 55, each degree on the limb is divided into two spaces, and the vernier reads to minutes, and we have 22° 44' for the reading of the instrument. In Fig. 56, supposed to give a scale of a mountain barometer, every inch on the limb is divided into ten parts, and each tenth part is divided into two, each of which will be fivehundredths of an inch; the scale of the vernier is nine-tenths of an inch divided into ten; the long divisions give one-hundredth of an inch, the subdivisions reading thousandths; in our example 56 the instrument reads 30-420 inches. With any instrument provided with a vernier before him, the reader will have but little difficulty in understanding this with the help of the magnifying-glass, for it must be remarked that, for the sake of plainness, our scales in the examples are much exaggerated; the first thing to be looked at is, into how many divisions each degree on the limb is divided, each division being a certain number of minutes accordingly; and in the next place, into how many divisions, if any, the minutes of the vernier are divided; this ascertained and remembered, there will be no difficulty after a little practice in reading off the instrument.

It will no doubt be readily perceived that the accuracy of observations thus made, of necessity must depend on the correct division of the limb of the instrument, and still more so where it occupies the whole circumference of the circle; another source of error may be that the centre of the arm of the index may not be exactly coincident with that of the instrument itself. To correct this, two opposite verniers are generally applied to theodolites, and by taking the mean of the vernier readings this correction is made; the best proof of the wisdom of the introduction of this double vernier is, that in most cases very minute observation will show that they do not read exactly alike. Two opposite indices will correct such errors as those just mentioned, as will also three verniers placed equally distant from each other; these three are often placed on theodolites with a large limb.

If we have the limb of a theodolite slightly compressed on one side, and elongated on the other, but with the index re

volving round the point of intersection of the long and the short diameters; now, although the end and side divisions are altered by the compression, the readings of two verniers placed diametrically opposite to each other will not correct the error, inasmuch as the alternations correspond at the opposite ends of the diameter; but three verniers equidistant from each other, although they do not give an exactly mathematical correction, approximate sufficiently near for all practical purposes; and quite as well as four, with which some instruments are provided, though not for this purpose.

At the ends of the limbs of sextants are small additional portions of arc, which are called the arcs of excess, the nature of which will be explained under the head of "the sextant." When reading off on this arc, we first observe what quantity on the limb is passed over from the zero by the zero of the vernier; in the next place we proceed to read backwards on the vernier counting from the last division which we now consider as the zero.

THE SEXTANT AND BOX-SEXTANT.

The principle of construction of the sextant, as well as of the quadrant, is that the total deviation of a ray of light after successive reflection in the index and the horizon glasses is double the angle of inclination of those two glasses or mirrors.

That the angle of incidence is equal to the angle of reflection needs no demonstration; of this mention has been made in the article on the optical square; the fact being admitted, let A M, Fig. 57, be a mirror, and let S be an object from which the radiant SB falls on A M at B; the angle M BC is equal to the angle A BS, equal to 45°; let DB be perpendicular to A M; now let the mirror revolve on B into the position am, SB remaining fixed as before, and let A B a, DB d, and M Bm be equal to 10°; ABS being equal to 45°, a BS will be equal to 35°, and the angle of incidence being equal to the angle of reflection M B c, will also be equal to 35°, and it will at once be seen that C has moved to c in an arc equal to twice A a or D d.

In Fig. 58 let Ii and H h be two glasses, which are known as the Index glass and the Horizon glass, let them be placed with regard to each other at an angle IA H, and let SI be a radiant from S making with the mirror Ii the angle SIi; the reflected. ray I H will make with I i and with Hh the angles H IA and IHE, each equal to the angle of incidence SIi. The angle

=

AND BOX-SEXTANT.

=

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A HV the exterior angle I Hh, and the two interior angles HIA and HAI of the triangle IHA; and HVA = IVE, therefore VIE + IEV VHA + HA V, and therefore to HIA 2 HAI, but VIESI is also HIA; taking away these equals, the remainder SEH = the remainder twice the angle H A I.

=

The above is the principle of construction of quadrants, sextants, and reflecting circles; for I is the index mirror fixed on and revolving with the index bar, I A, by the vernier of which the angle is measured or read off along the limb LM; H is the horizon glass through the unsilvered portion of which any object is seen by direct vision from the point E, where the eye is applied, whilst the silvered part receives the reflection from the index mirror I of an object, as at S, as this mirror is made to revolve until the image of contact is effected at H. It will be seen that whilst the angle formed by two objects with the eye is thus obtained, the angle really measured is that formed by the bearings of the two glasses relatively to each other; but the limb L M is so figured as to give twice the angle between the glasses; thus, if the spaces on the limb measure 15', they are figured 30'; therefore the reading of the instrument when in adjustment, gives the angle between the bearing of the object from the centre of the index mirror and the bearing of the reflected image from the place of the eye.

Fig. 59 is a representation of one of Elliott's sextants; L L'is the limb, and V the vernier, graduated according to the size of the radius of the instrument; I is the index mirror fixed perpendicular to the plane of the instrument, and over the centre of motion of the index IV; M is a microscope connected with the index by an arm, and by the aid of which the whole of the vernier may be examined. The limb is also divided from zero towards L', forming the arc of excess by which the index error of the instrument is determined, as will be seen hereafter. In reading off on the arc of excess, the vernier must be read backwards; that is, the division on the limb next to the left of the zero of the vernier being read, the divisions of the vernier to be added must be read from the other end, marked 30, to a division coinciding with a division upon the limb. A clamp C is attached to the index to fasten it to the limb; T is the tangent screw by which the index may be moved any small quantity after it is clamped; this tangent screw having a most delicate motion, enables the observer to perfect the contact of the objects observed in a manner which could not be done by the hand acting on the index merely.

H is the horizon glass, the lower half of which only is silvered; this glass, also, must be perpendicular to the plane of the instrument, and in such a position that its plane must be parallel to the plane of the index mirror when the vernier is set at zero; a deviation from this constitutes the index error; for adjusting the horizon glass there is a small screw at the bottom of the frame in which it is set.

O is a telescope carried in a ring, attached to a stem called the up and down piece; this may be raised or lowered by means of a milled headed screw, till the objects seen in the telescope directly, and by reflection, appear of the same brightness, which will be the case when the field of view in the telescope is bisected by the line on the horizon glass that separates the silvered from the unsilvered part. A plain tube and two telescopes, one showing objects erect, and the other inverted, are supplied with the sextant; the inverting telescope having a higher magnifying power than the other, shows the contact of objects better; the adjustment for distinct vision is obtained by sliding the tube at the eye end of the telescope in the inside of the other.

Four dark glasses of different shades and colours are placed at G, any one of which may be turned down between the index mirror and the horizon glass, when the light seen from any object by reflection is too intense for the eye; similarly, there are three more at B, any one of which may be brought before the horizon glass when too bright a light proceeds from the object seen by direct vision. For the same purpose, to the eye end of the telescope a dark glass may be fixed, one or two of which are generally in the box containing the sextant; they are, however, principally used in observing the sun's altitude with an artificial horizon, or for ascertaining the index error, as the use, for this purpose, of the shades attached to the instrument would involve any error they might possess.

In the instrument case is a key for turning the adjusting

screws.

When in use, the instrument is held by a handle placed underneath.

By means of the up and down piece above mentioned, the axis of the telescope is brought to the same height above the plane of the instrument as the height of the line of separation between the silvered and unsilvered portions of the horizon glass, to prevent parallax of the reflected rays; by this adjustment, also, the image of one object may be made as luminous as the real body of the other, by increasing the light of one and decreasing that of the other; that is, by making more of the silvered or of the