These are all the adjustments necessary in measuring au angular distance by cross observations; but if one observation only be taken to the right, or to the left, it will be necessary to find the division on which the horizon index must be placed, to make the horizon glass parallel to the central glass, when the central index stands on 0 These adjustinents are similar to those of a sextant; but a particular explanation of each will here be given.

To set the central glass perpendicular to the plane of the instrument.

This adjustment may be made by placing the eye in front of the central glass at L, a little above the plane of the instrument, and observing if the reflected image of that part of the limb nearest the eye appears to make one continued circular line with the parts of the limb towards T, seen to the right and left of the central glass; for, in this case, the glass is perpendicular to the plane of the instrument; otherwise it must be adjusted by means of the screws till the two images coincide.*

By examining this adjustment in different parts of the limb, it will be known if the limb be in the same plane. If any difference should be found, the central glass must be so fixed that the reflected image of the limb may appear as much above the direct image in some places as below it in others.

To set the horizon glass perpendicular to the plane of the instrument.

The central glass being previously adjusted, and the telescope directed to the line separating the silvered from the transparent part of the horizon glass, hold the instrument nearly vertical, and move either index till the direct and reflected image of the horizon, seen through the telescope, coincide; then incline the instrument till it is nearly horizontal, and, if the images do not separate, the horizon glass is perpendicular to the plane of the instrument; but if they do separate, the position of the glass must be rectified by means of the screws in its pedestal.

This adjustment may be also made by directing the sight through the telescope to any well-defined object; then if, by moving the central index, the reflected image passes exactly over the object seen directly, the glass is perpendicular; otherwise its position must be adjusted by means of the screws attached to the pedestal of the glass.

A planet, or star of the first magnitude, will be a good object for this purpose. If the sun is used, one of the colored glasses must be placed at C, and another at D.

To make the axis of the telescope parallel to the plane of the instrument.

The telescope may be raised or depressed by means of two screws attached to the standards i, k (fig. 2), and passing through two pieces of brass connected with the tube of the telescope. On each of these pieces is a mark or index, by which the telescope is to be adjusted; for, by bringing the indices to the same mark on each standard, the telescope will be parallel to the plane of the instrument. †

To find that division to which the horizon index must be placed to render the mirrors parallel when the central index is on 0.

Place the central index on 0; direct the telescope to the horizon glass, so that the line joining the silvered and transparent parts of that glass may appear in the middle of the telescope; hold the instrument vertically, and move the horizon index till the direct and reflected horizons agree, and the division shown by the horizon index will be that required.

This adjustment may also be made by measuring the diameter of the sun in

*When the instrument is furnished with adjusting tools, this adjustment may be made in the following manner Set the two tools on opposite parts of the limb at T and L.; place the eye at e, at nearly the same height as the upper edge of the tools, so that part of the tool at T may be hid by the central glass; move the central index till the reflected image of the tool nearest the eye appears in the central glass at the side of the other tool seen directly; then, if the upper edges of the tools are apparently in the same straight line, the central glass is perpendicular to the plane of the instrument; otherwise its position must be adjusted by the screws at the back of the frame.

If you suspect that the marks on the standards are inaccurate, you may examine them in the following manner-Lay the circle horizontally on a table; place the two adjusting tools on opposite parts of the limb, at T and L; and at about 12 or 15 feet distance let a well-defined mark be placed, so as to be in the same straight line with the tops of the tools; then raise or lower the telescope till the mark is apparently in the middle between the two wires; then the axis of the telescope will be parallel to the plane of the instrument, and the difference (if any) between the divisions pointed out by the indices on the graduation of the standards i, k (fig. 2), will be the error of the indices, and, this being known it will be easy, in future adjustments, to make allowance for it.

contrary directions; thus, the central index being fixed on 0, place a dark glass at C, and another at D; direct the telescope (through the transparent part of the horizon glass) to the sun, and move the horizon index till his reflected image appear in the telescope; bring the upper edge of the direct image to coincide with the lower of the other, and note the angle shown by the index; then, by moving the horizon index, bring the lower edge of the direct image to coincide with the upper edge of the reflected one, and note also the angle pointed out by the index; half the sum of these two angles will be the point of the limb where the horizon index must be placed to render the mirrors parallel. Thus, if the index, in the first observation, is on 473° 30′, and, in the second, on 474° 34', the half sum of the two, 474° 2, will be the point where the horizon index must be placed to make the mirrors parallel.

These are all the adjustments necessary to be made preparatory to measuring any angular distance. When the angle is measured by cross observations, the error arising from the want of parallelism of the surfaces of the mirrors and screens, will in general be very small; however, the method of verifying those glasses, and making allowance for any error in them, will be given hereafter.

To observe the meridian altitude of any celestial object, either by an observation to the right or to the left.

The method of observing the meridian altitude of an object with a circle, is exactly similar to that with a quadrant or sextant. The central index must be fixed on 0, and the horizon index on the point which renders the two mirrors parallel; then the altitude may be taken either by an observation to the right or to the left; but the former method, in which the large colored glasses are not necessary, is in general to be preferred, because these large glasses are more liable to cause an error in the observation than the small ones.

If an observation to the right is to be taken, a small dark glass must be placed at C, if the object be bright; then hold the instrument in the right hand, in a vertical position, move the central index, according to the order of the divisions of the limb, till the reflected image of the object, seen in the telescope, nearly touches the direct image of the horizon; tighten the index by the screw at the back of the instrument; make the contact complete in the middle between the parallel wires of the telescope, by the tangent screw, and by sweeping, exactly in the same manner as when observing with a quadrant, and the central index will point out the altitude of the object.

If an observation to the left is taken, and the object be bright, a large dark glass must be placed at a, a, if the altitude be between 5° and 35°, otherwise a small glass at C hold the instrument in the left hand, in a vertical position; move the central index contrary to the order of the divisions, and bring the reflected image in contact with the horizon as above; the angle shown by the central index, being subtracted from 720°, will be the sought altitude.

In both these methods of observing the meridian altitude of an object, the circle, the radius of which is only five inches, will hardly be so accurate as a good sextant of a larger radius; but, by the help of a well-regulated watch, the meridian altitude may be obtained, by the circle, to a much greater degree of accuracy than by a sextant, by observing in the following manner:-A few minutes before the object passes the meridian, begin to observe the altitude by cross observations (in the manner to be described in the next article), and note the time of each observation by the watch; continue to observe till a few minutes after the object has passed the meridian; then the angles shown by the central index, being divided by the whole number of observations, will give the approximate meridian altitude; the correction to be applied to it to obtain the true meridian altitude, may be found by means of Tables XXXII. and XXXIII., by a method which will be explained hereafter, when treating of finding the latitude by a single altitude of the sun.

In this article, the meridian altitude only has been spoken of, though it is evident

in some instruments, there is an adjustment of the horizon glass, to place it at its proper angle with the axis of the telescope; if an adjustment of this kind is necessary, it ought to be made before the other adjustments, in such manner that if a colored glass be fixed at C, none of the rays from the central glass can be reflected to the telescope from the horizon glass, without passing the colored glass. To effect this, the rentelle must be placed at D, and lowered so as to intercept the direct light entirely; then place the colored glass at C. and direct the telescope to the silvered part of the horizon glass; move the central index, and if no uncolo.ed images appear (reflected from the central glass), but all have the same tinge as that of the colored glass used, the horizon glass is in its proper position; otherwise it must

that the method is applicable to an object not on the meridian; but, in this case, the cross observations, which give to the circle all its advantages, may be used, and the mean of the altitudes taken ir stead of a single altitude. This method is peculiarly adapted to the taking of altitudes for regulating a watch; for this reason it will be particularly explained in the following article:

To take altitudes of the sun, or any celestial object, by cross observations, for regulating a watch.

Times of obs.

4h. 20m. Os.

4

21 10

4

92 15

4

23 0

4

24 45

4

25 30

16 40

Angle. 6)60° 24'

10 4

Fix the central index on 0, and if the object be bright, and the altitude between 5° and 35°, place a large colored glass before the central glass at a, a, otherwise a smail one at C; hold the instrument m the left hand, in a vertical position; nove the horizon index till the image of the reflected object be brought in complete contact with the horizon, in the middle between the two parallel wires of the telescope, as directed in the preceding article, and note the time of observation by the watch; then fasten the horizon index; hold the instrument in the right hand, in a vertical position; move the central index according to the order of the divisions, till the reflected image be again brought into complete contact with the horizon as above, and note the time of observation. Then half the sum of the times, and half the angle shown by the index, will be a mean time, and a mean altitude corresponding thereto.

6)26

4 22 47

If greater accuracy be required, the observation must be repeated, setting out from the points where the indices then are, and observing in the same manner by moving first the horizon index, then the central one; continue taking as many of these cross observations as are judged necessary, and note the times of each observation; then the sum of the times, divided by the whole number of observations, will be a mean time; and the angle shown by the central index, divided by the number of observations, will be a mean altitude corresponding thereto. Thus, if six† observations were taken, and the times noted as in the adjoined table, the angle shown by the index being 60° 24′ the mean time would be obtained by dividing the sum of the times, 26h. 16m. 40s., by 6, and the mean altitude by dividing 60° 24' by 6; therefore the mean time would be 41. 22m. 478., and the mean altitude corresponding 10° 4'.

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

The instrument being well adjusted, fix the central index on 0, and, if the object be bright, place a small dark glass at C; hold the instrument so that its plane may be directed to the objects with its face downwards when the sun is to the right of the moon; otherwise, with its face upwards; direct the sight through the telescope to the moon; move the horizon index, according to the order of the divisions of the limb, till the reflected image of the sun appears in the telescope, and the nearest limbs of the sun and moon are almost in contact; fasten the index, and make the coincidence of the limbs perfect, in the middle between the two parallel wires of the telescope, by means of the tangent screw of the horizon glass, and note the time of observation;

The are described on the limb by the central index, will be equal to twice the altitude of the object, or twice the angle passed over by the other index: if more cross observations be taken, each of the indices, when moved, will describe an arc equal to double the altitude of the object; the same is to be observed in measuring any other angular distance. If the instrument is furnished with the are WSR, and sliding pieces U, X, you must bring the slide X to the central index, after taking the first observation to the left, and place the slide U at the same degree, on the are SW, that X is on the are PR; then. in the next observation, the central index is to be brought to touch the slide U; in the next observation to the left, the slide X is to be brought to the central index, and so on for the other observations. Thus, by means of the slides, the indices may be placed at nearly their proper angles with each other at the beginning of the observation, which will save considerable time. After being thus fixed, the contact must be completed by means of the tangent screw of the index, which is to be moved.

The number 6 is a convenient number to use, because the remainder of the division of the hours by 6 gives the first figure of the minutes; and the remainder of the division of the minutes by 6 gives the first figure of the seconds. Thus, in the above example, in dividing 26h. by 6, we get 4h., and the remainder 2 is set down immediately for the first figure of the minutes; the second figure of the minutes is the quotient 2, found by dividing 16m. by 6, and the remainder 4 of this last division is the first figure of the seconds. We may remark that, as the term 4h. 20m. is common to all the 6 observations, it may be neglected; then adding the minutes in the column of units, and the seconds, the sum becomes 16m. 40s dividing this by 6 gives 2m. 47s., to be connected with 4h. 20m., making, as above, 4h. 22m 474

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 observatiou.

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 are 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 cnlightened 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 are 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 he adjoined table. The central index being at

Central Horizon
Index. Index.

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 parallelisin 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 inake 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° 534', 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 29° +20= = 131°, and the central index, a the eighth observation, at 432° + 2° = 434°, &c.