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then invert the instrument, and move the central index, according to the order of the divisions of the limb, by a quantity equal to twice the arc passed over by the horizon index (or twice the distance of the sun and moon);* direct the plane of the instrument to the objects; look directly at the moon, and the sun will be seen in the field of the telescope; fasten the central index, and make the contact of their nearest limbs complete, in the middle between the two parallel wires of the telescope, by means of the tangent screw of the central index, and note the time of observation; then half the are shown by the central index will be the distance of the nearest limbs of the sun and moon, and half the sum of the times will be the mean time of observation.

Having finished these two observations, two others may be taken in the same manner, setting out from the points where the indices then are, and moving first the horizon index, then the central index: proceed thus till as many observations as are judged necessary be taken, always observing that the number of them be even; then the angle shown by the central index (or that angle increased by 720°, or 1440°, &c., if the index has been moved once or twice, &c., round the limb), being divided by the whole number of observations, will give the mean distance; and the sum of all the . times, divided in like manner, will be the mean time of observation.

To measure the distance between the moon and star by a circular instrument.

Fix the central index on 0, and, if the moon be bright, and the distance between 5° and 35°, place a large green glass before the central mirror at a, a, otherwise a small one at C; hold the instrument so that its plane may be directed to the objects with its face downwards when the moon is to the right of the star, otherwise with its face upwards; direct the sight through the telescope to the star; move the horizon index, according to the order of the divisions of the limb, till the reflected image of the moon appears in the telescope, and the enlightened limb of the moon be nearly in contact with the star; fasten the index, and make the coincidence perfect, in the middle between the parallel wires of the telescope, by means of the tangent screw belonging to that index, and note the time of observation; then invert the instrument, and move the central index, according to the order of the divisions of the limb, by a quantity equal to twice the arc passed over by the horizon index;* direct the plane of the instrument to the objects; look directly at the star, and the moon will be seen in the field of the telescope; fasten the central index, and make the contact of the enlightened limb of the moon and the star complete, in the middle between the two parallel wires of the telescope, by means of the tangent screw of that index, and note the time; then half the arc shown by the central index will be the distance of the star from the enlightened limb of the moon, and half the sum of the times will be the mean time of observation; these two observations being completed, others may be taken in the same manner, according to the directions above given for measuring the distance of the sun from the moon.

In continuing to take these cross observations by a circle furnished with the arc WSR, and slides U, X, it will be very easy to bring the reflected image into the field of the telescope; but if the instrument is not thus furnished, it will be often difficult to bring the image into the field of the telescope, and much time will be lost, and the observations rendered tedious by that means; to remedy this, a small table of the angles, at which each index should be placed, ought to be made before beginning the observation; this table is easily formed, as follows:-Find roughly, according to the directions heretofore given, the point at which the horizon glass must be placed to be parallel to the central glass, when the central index is on 0; then find what point of the arc the horizon index stands upon, after measuring the first distance, as directed above; the difference between these two points will be the angular distance of the objects; the double of this distance, being successively added to 0°, and to the angle pointed out by the horizon index after the first observation, will give the points of the arc where the indices must be placed at the 2d, 3d, 4th, &c. observations. Thus, if the point of parallelism is 471°, and the point where the horizon index is at the first observation is 525°, the difference, or 54°, will be the angular distance; the double of this, or 108°, being added to 525°, gives 633°, which is the point of the arc where that index must be placed at the third observation; 633° added to 108° gives 741° or 21° (because the divisions recommence at 720°), which is the point where the index must be placed at the fifth observation, &c., as in the adjoined table. The central index being at

Central Horizon
Index.

Inder.

525

108

633

216

21

324

129

432

237

540

&c.

&c.

first on 0°, after the second observation it will be on 108°, at the fourth on 108° +108° =216°, at the sixth on 216° +108°324°, &c. Thus, by constantly adding 108°, or twice the distance of the objects, the angles at which the indices must be placed will be obtained; and by fixing them at these angles, the reflected image will be brought into the field of view without any trouble.*

Having explained the methods of adjusting and using the circle of reflection, it remains to show how to calculate the error arising from not observing the contact of the objects in the middle between the parallel wires of the telescope, and also to estimate the errors arising from the want of parallelism of the mirrors and colored glasses. These verifications are much more necessary in a sextant than in a circle, and they may be in general neglected in a circle.

To estimate the error arising from not observing the contact of the objects in the middle between the parallel wires of the telescope.

To estimate this error, it is necessary to know the angular distance of the wires of the telescope, which may be thus determined:

:

Turn round the eye-piece of the telescope till the wires are perpendicular to the plane of the instrument, and put the central index on 0; direct the telescope to any well-defined object, at least 12 feet distant, and move the horizon index till the direct and reflected image of the object coincide; then make one of the wires coincide with the object, and turn the central index till the reflected image of the object coincides with the other wire-and the arc passed over by that index, will be the angular distance between the wires. This angle being obtained, the observer must, by means of it, estimate, at each observation, how much the place where the contact is observed is elevated above, or depressed below, the plane passing through the eye and the middle line between the two parallel wires of the telescope: the correction in Table XXXV., corresponding to this angle, is to be subtracted from the observed angular distance of the objects: thus, if the distance between the wires is 2°, one of them will be elevated above that plane 1°, and the other depressed below it, by the same quantity; if, in taking an observation, the point of contact is estimated to be one third part of the distance from the middle towards either wire, the angle of elevation or depression will be one third part of 1°, or 20; and if the observed distance is 120°, the correction in Table XXXV. will be 12", subtractive from the observed distance.

The correction for each observed distance being ascertained, in the above manner, the sum of them must be subtracted from the whole angle shown by the central index, and the remainder, divided by the whole number of observations, will be the mean distance.

Verification of the parallelism of the surfaces of the central mirror.

This verification is to be made ashore, by observing the angular distance of two well-defined objects, whose distance exceeds 90° or 100°, having previously well adjusted the instrument: after taking several cross observations, and finding the mean distance, take out the central mirror, and turn it so that the edge which was formerly uppermost may now be nearest the plane of the instrument; rectify its position, and take an equal number of cross observations of the angular distance of the same two objects; half the difference between the mean of these and that of the former, will be the error of the observed angle, arising from the defect of parallelism of the central mirror. If the first mean exceeds the second, the error is subtractive, otherwise additive, the mirror being in its first position; but the contrary when in its second position. Thus, if 10 observations are taken at each operation, and in the first the angle shown by the index is 1199° 53', and in the second 1200° 64', by dividing by 10 the mean angles are found to be 119° 59′ 21′′ and 120° 0′ 39′′, and their difference is 78"; the half of it, or 39", is the error of the mirror, additive when it is in its first position, subtractive in the second. The error for any other angle may be found by Col. 4, Table XXXIV., when the inclination of the plane of the horizon glass to the axis of the telescope is 80°, by saying, As the tabular error corresponding to 120°, that is, 1' 30", is to the error found in the glass 39', so is the tabular error for any

* If the distance of the object varies during the observation, these angles will require correction as you proceed with the observations. Thus, if the distance was increasing, and at the sixth observation it was found that the central index was on 326° instead of 324°, the increase being 2°, you must add 2° to the rest of the numbers in the table, and place the horizon index, at the seventh observation, on 129° +20=131°, and the central index, at the eighth observation, at 432° + 2o 434°, &c.

other angle 85°, which is 0' 28", to the error of the glass corresponding 12"; and in this manner a table of errors may be made, not only for the cross observations, but also for observations to the right or to the left.*

It may be remarked that the errors are much less in the cross observations than in the observations to the right, which are those made with a quadrant or sextant; so that the circle has, in this respect, greatly the advantage of those instruments.

The angle between the plane of the horizon glass and axis of the telescope produced being nearly the same in all observations and adjustments of the circle, no sensible error can arise from the want of parallelism in the surfaces of that glass.

Verification of the parallelism of the colored glasses.

Place one of the dark-colored glasses at C, and another at D; fix the central index at O, direct the telescope to the sun, and move the horizon index till the limbs of the direct and reflected image coincide; then turn the dark glass placed at C, so that the surface which was farthest from the horizon glass may now be nearest to it, and if the contact of the same two limbs be complete, the surfaces of the glass placed at C are parallel; but if the limbs lap over or separate, the central index must be moved to bring them again in contact; then half the arc passed over by that index will be the error arising from the want of parallelism of the glass C. If great accuracy is required, the operation may be repeated by setting out from the point where the indices then are, and taking 4 or 6, &c., observations; then the arc passed over by the central index, being divided by 4, 6, &c., will be the sought error. The other small glasses may be verified in the same manner; and, by placing one of the larger glasses before the central index at a, a, and one of the smaller ones at D, the former may be verified as above. The green glasses may be verified by observing the diameter of the full moon, or by some bright terrestrial object.

It may be remarked, as one of the greatest advantages of the circle, that, in measuring an angle by the cross observations, no error can arise from the want of parallelism in the surfaces of the smaller dark glasses; for if these glasses give too great an angle by an observation to the right, they will give too little by the same quantity by an observation to the left. It is not so with the large glasses placed at a, a, because the incidence of the rays on these glasses is more oblique in one observation than in the other, so that the errors do not wholly balance each other; however, as these glasses are used only in measuring angles less than 35°, where the errors are nearly the same as if the incidence of the rays were perpendicular, the errors of these glasses will also nearly compensate each other in the cross observations; and if such observations only are used, it will be unnecessary to verify the dark glasses. Even when taking observations to the right, or observations to the left, the error of the dark glasses will be destroyed, if the glass is turned at each observation, and the number of observations is even; but there are some cases in which an angle can only be measured by one observation; then it will be necessary to allow for the error of the dark glass, if the distance is required to be found within a few seconds.

*If the inclination of the plane of the horizon glass and the axis of the telescope differ from 80°, you may find the tabular numbers by the method given in the explanation of Table XXXIV. affixed to the tables.

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