that is, unless the perpendicularity and parallelism of the two reflectors be palpably defective. Hence, instead of making the above adjustments, the judicious observer will, in all cases, and at all times, prefer finding the index error of his sextant: this may be conveniently done, as thus,

56. Screw the telescope into its socket, adjust it to distinct vision, and turn the tube or eye-part thereof, until two of the cross wires are parallel to the plane of the sextant. Raise the collar or socket for the telescope, by means of the milled screw at its back, till the field of view of the telescope is bisected by the line which separates the silvered and the transparent parts of the horizon-glass. Make the O on the vernier of the index correspond with the O or zero on the limb; then clamp the index sufficiently tight for observation. Hold the sextant in a vertical position, and direct the sight, through the telescope, to the horizon of the sea, when it is clear and well defined :-then, if the reflected horizon and the real horizon be exactly in the same plane, or form one continued straight line, there will not be any error in the index; but, should the two horizons not coincide, which very frequently will be the case with inferior instruments, make their coincidence perfect by means of the tangent-screw of the index; then, the angle indicated by the vernier on the arch or limb, will be the index error of the sextant; which will be subtractive when the angle is on the arch; that is, when the O on the vernier is to the left of zero on the limb; but additive when it is off the arch, viz., when the O on the vernier is to the right of zero on the limb.

57. The index error may be found with more exactness than as above, by adopting the following method, viz. :-Place one or two of the dark screens, according to the brightness of the sun, so as to intervene on each side of the horizon-glass; bring the index to zero on the limb, and tighten its clamp sufficient for observation: hold the sextant with its face up, and so that its plane may be parallel to the horizontal diameter of the sun; direct the sight to the sun, through the telescope, move the index forward, by means of the tangent-screw, till the right-hand limb of the reflected sun makes a perfect contact with the left-hand limb of the real sun, which is seen by direct vision through the transparent part of the horizon-glass. Read off the angle by means of the magnifying lens; note down its value, and it will express the measure of the sun's diameter to the left of zero, or of O on the arch. Ease the clamp and bring the index back to zero on the limb; then, tighten the clamp sufficient for observation, as directed above, and move the index backward, by means of the tangent-screw, till the left-hand limb of the reflected sun makes a perfect contact with the right-hand limb of the real sun. Read off the angle by means of

the microscope; note down its value, and it will express the measure of the sun's diameter to the right of zero, or of O off the arch. Now, if both angles, or horizontal diameters of the sun be of the same value, there is no index error in the sextant; but, if the diameters are of unequal value, half their difference will be the index error of the instrument; which will be subtractive when the diameter to the left of zero is greater than that measured to the right of zero; otherwise, it will be additive.

Note. In reading off the value of the diameter to the right of zero, or off the arch, it must be remembered that this is a retrograde operation; and, therefore, it is the complement of the minutes and seconds shown by the vernier that is to be added to the angle indicated by the first point of the index :-hence, should the vernier (one extended to 15 minutes) cut, or coincide at 12:40"; the complement of this, or 2:20 is the true arc of excess to be added to the angle, or divisions of a degree, pointed out by O on the index.

58. The above would be a very correct way of determining the index error of a sextant, provided the index-bar was inflexible or non-elastic, and that it had no play round its centre of motion: but, because the index is an elastic bar that must be made so as to move freely round its centre of motion beneath the moveable reflector; it is, therefore, invariably bent, or forced from the true line of the radius of the sextant, whenever it is moved backward or forward by means of the tangentscrew; its clamp being duly tightened for observation: and thus, as the O on the vernier part of the index is forced beyond the extremity of the direct line of the radius, the angle measured by the progressive motion of the tangent-screw will be something too much; whilst that measured by the retrograde motion of the screw will be too little. Of this singular fact the reader can easily satisfy himself in the following manner, viz.— Move the index to any degree on the limb, no matter which; but let us say, for the sake of perspicuity, 60; then, fasten the clamp, and move the index forward by means of the tangent-screw, to 60:30, keep the sight, by means of a good magnifying lens, fixed upon this point;-now, suddenly release the clamp, and the spring of the indexbar will become perceptible; for it will be seen to fall back something upon the arch or limb. The converse of this will take place if the index be moved backward to the same extent; for, the instant its clamp is released (this being done suddenly), it will be seen to spring forward a little upon the arch or limb. Hence, it is clearly manifest that the preceding method of finding the index error of a sextant is subject to a certain degree of incorrectness. And, as this is a subject of vast importance to the practical navigator, I shall therefore discourse of it more generally in the following article.

The true Method of finding the Index Error of a Sextant, so as to guard against the Errors arising from the Flexibility and the Friction of the Index-bar.

59. The customary method of finding the index error of á quadrant or sextant (as directed by writers on the use of these instruments,) is by measuring the vertical diameter of the sun to the right and left of O on the arch, with a motion of the index in contrary directions (that is, by bringing the reflected image to touch the lower and upper limbs of the direct object alternately), and then taking half the difference of those measures for the index error of the instrument. This method, it must be observed, is very far from being correct; because it is the horizontal diameter of the sun, and not its vertical diameter, that should be measured; for while the former remains invariably the same, the latter is subject to continual alterations, owing to the effects of atmospherical refraction, as will appear evident by an inspection of the last column of Table V.-Moreover, since the index is not an inflexible bar, and since it does not turn upon its centre without suffering some slight degree of friction; it is therefore evident that the measure of the sun's diameter taken by the progressive motion of the index will, in most cases, be more than the truth: whilst that taken by the contrary or retrogressive motion will, in general, be less than the truth :-hence, the index error established upon the above principles must frequently mislead the mariner by rendering inaccurate what, otherwise, might be a very correct observation. And this accounts for the result of the evening observations, taken on shore by means of an artificial horizon, so very seldom agreeing with the result of those taken in the morning; even though all imaginable care be used, and though the observer keeps the same plane and roof of the horizon directed to him during the time of both observations.

Now, to guard against the errors arising from the bending and the friction of the index-bar, as well as that proceeding from the contraction of the sun's vertical diameter; let the following observations be attended to, and the joint effects of the whole will be obviated.

First. To find the Error for a Progressive Motion of the Index :—

Screw the inverting telescope into its place. Arrange the shades for observation. Slack the clamp. Turn the tangent-screw backward to nearly as far as it will go. Put the vernier to about 1:15 to the right of O on the arch, and then fasten the clamp sufficiently tight for observation. Hold the sextant so that its plane may be parallel to the horizontal diameter of the sun: direct the sight to that object, and

turn the tangent-screw forward until the limbs of the sun seen by reflection and direct vision make a perfect contact.-Note down the angle and it will express the measure of the sun's diameter to the right of O on the arch.-Direct the sight again to the sun, and turn the tangent-screw still forward until the opposite limbs are in perfect contact: note down the angle, and it will be the measure of the sun's diameter to the left of O on the arch.-Now, if both measures of the diameter are the same, there is no error in the angles shown by the progressive motion of the index; but if those measures do not correspond, half their difference is to be taken as the index error of the instrument, which error will be additive when the diameter measured to the right of O exceeds that measured to the left; otherwise, subtractive. Then, this error is to be considered as a constant quantity (so long as the instrument does not meet with any accident), and to be applied to all increasing angles, either of altitude or distance, which may be taken by the progressive motion of the index.

Again. To find the Error for a Retrogressive Motion of the Index :—

Slack the clamp. Turn the tangent-screw forward to nearly as far as it will go. Put the vernier to about 1:15 to the left of O on the arch, and then fasten the clamp sufficiently tight for observation.Hold the sextant as before; direct the sight to the sun, and turn the tangent-screw backward until the limbs of the sun seen by reflection and direct vision make a perfect contact :-note down the angle, and it will express the measure of the sun's diameter to the left of O on the arch.-Direct the sight again to the sun, and turn the tangent-screw still backward until the opposite limbs are in perfect contact; read off the angle, and it will be the measure of the sun's diameter to the right of O on the arch.-Now, if both measures of the diameter are the same, there is no error in the angles shown by the retrogressive motion of the index: but if those measures do not correspond, half their difference is to be taken as the index error of the instrument; which error will be additive when the diameter measured to the right of O exceeds that measured to the left; otherwise, subtractive. Then, this error is to be considered as a constant quantity (so long as the instrument does not meet with any accident), and to be applied to all decreasing angles, either of altitude or distance, which may be taken by the backward or retrogressive motion of the index.

Hence it is very probable that two errors may be established for the same instrument; the one for increasing, and the other for decreasing angles. The true values of those errors should be noted down (for the

future guidance of the observer), with a black-lead pencil on the inside of his sextant-case in the following manner, viz. :—

Error for the forward or progressive motion of the index 0:10° subtractive.

Error for the backward or retrogressive motion of the index 1:40: additive.

Or whatever the errors may be.

And thus the correct values of the index error will be properly determined, whilst the errors arising from the spring and the friction of the bar, together with that proceeding from the contraction of the sun's vertical diameter will be all safely provided against.

60. Now the index error of the sextant being thus truly established, the instrument is properly adapted for making celestial observations: that is, for taking the altitudes, and measuring the angular distances of the heavenly bodies.

1. To take the Sun's Altitude at Sea.

61. Prepare the sextant as directed in Article 50, omitting the telescope, and let one of the darkest shades intervene between the indexglass and the fixed reflector. Slacken the clamp, and bring the index to zero on the limb. Hold the sextant in a vertical position. Look through the sight-vane, or socket, direct the sight to the sun, and his reflected image will be seen in the silvered part of the horizon-glass : move the index forward by hand, till the lower limb of the image is seen near to the horizon; then tighten the clamp sufficient for observation; screw the telescope into its place, and make the contact of the limb and the horizon perfect by means of the tangent screw: taking care that it be made at that point of the horizon which is exactly under the sun, and which would be touched by a plumb-line let fall from its centre. And to be certain that the contact is so made, give the sextant an immediate vibratory motion, and the reflected image will appear to describe an arch which will be above the horizon at all points except that to which the real sun is perpendicular. Now, the sun's lower limb being thus made to touch the horizon, the degrees, minutes, and seconds indicated by the vernier, and read-off by means of the magnifying lens, will be its observed altitude.

Note. In rough weather, or when the sea runs high and the ship pitches, the plain tube may be substituted for the telescope.