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 1000, 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 were taken at each operation, and in the first the angle shown by the index, was 1199° 53', and in the second 12000 6'; by dividing by 10 the mean angles are found to be 119° 59′21′′ and 120° 0′ 39′′, the difference of which is 78", the half of which 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 is180, 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 other at 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 the observations to the right or 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 coloured glasses.

Place one of the dark coloured glasses at C and another at D, fix the cftral index at 0, 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 arch 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 arch 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 plac ing 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 abce. 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

*If the inclination of the plane of the horizon glass and the axis of the telescope differ from 80°, voi may find the tabular numbers by the method given in the explanation of Table XXXIV. prefixed 19

the Tabits.

want of parallelism in the surfaces of the smaller dark glasses; for if those glasses give too great an angle by an observation to the right, they give too little by the same quantity by an observation to the left. It is not so with the larger glasses placed at a a, because the incidence of the rays on those glasses is more oblique in one observation than in the other, so that the errors do not wholly balance each other; however, as those glasses are used only in measuring angles less than 35°, in which the errors are nearly the same as if the incidence of the rays was perpendicular, the errors of those glasses will also nearly compensate each other in the cross observations; and if those observations only were used, it would 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 would be destroyed, if the glass was turned at each observation, and the number of observations was even; but there are some cases in which an angle can only be measured by one observation, then it would be necessary to allow for the error of the dark glass, if the distance was required to be found within a few seconds.

ON PARALLAX, REFRACTION,

AND

DIP OF THE HORIZON.

PARALLAX (or diurnal parallax) is the difference between the true altitude of the sun, moon, or star, if it were observed at the centre of the earth, and the apparent altitude observed at the same instant by a spectator at any point on the surface of the earth.

Thus in Plate IX. fig. 3, let ABC be the earth, C its centre, A the place of a spectator, EDF part of the moon's orbit, e d G part of the orbit of a planet, and KZ part of the starry heavens. Then if at any time the moon be at D), she will be referred to the point H by a spectator supposed to be placed at the centre of the earth, and this is called the true place of the moon, but the spectator at A will refer the moon to the point b, and this is called the apparent place of the moon, the difference Hb (or the angle HDb=ADC) is called the moon's parallax in altitude, which is evidently greatest when the moon is in the horizon at E, being then equal to the arch K I, and decreases from the horizon to the zenith and is there nothing. The parallax is less as the objects are farther from the earth: thus the parallax of a planet at d is represented by a b, being less than that of the moon at D; and the horizontal parallax K f of the planet is less than the horizontal parallax KI As the parallax makes the objects appear lower than they really are, it is evident that the parallax must be added to the apparent altitude to obtain the true altitude. Having the horizontal parallax, the parallax in altitude is easily found by the following rule-As radius is to the co-sine of the apparent altitude, so is the horizontal parallax to the parallax in altitude. This rule may be easily proved: for in the triangle CAE we have CE: CA: radius: sine CEA; and in the triangle CDA we have CD (or CE): CA: :sine CAD: sine CDA; hence we have, radius: sine CEA :: sine CAD: sine CDA, but CEA=horizontal parallax, CDA=parallax in altitude, and sine CAD co-sine app. alt. Hence we have radius co-sine app. alt.:: sine hor. par. : sine par. in alt. but the parallaxes of the heavenly bodies being very small, the sines are nearly proportional to the parallaxes, hence we may say, as radius: co-sine app. alt. : : hor. par. : par. in alt.

of the moon.

The sun's mean parallax in altitude is given in Table XIV. for each 50 or 10 of altitude. The moon's horizontal parallax is given in the 7th page of the month of the Nautical Almanac, for every noon and midnight at the meridian of Greenwich.

REFRACTION OF THE HEAVENLY BODIES.

It is known by various experiments that the rays of light deviate from their rectilinear course in passing obliquely out of one medium into another of a different density, and if the density of the latter medium continually increase, the rays of light in passing through it will deviate more and more from the right lines in which they were projected towards the perpendicular to the surface of the medium: This may be illustrated by the following experiment: make a mark at the bottom of any bason or other vessel, and place yourself in such a situation that the hither edge of the bason may just hide the mark from your sight, then keep your eye steady, and let another person fill the bason gently with water: as the bason is filled, you will perceive the mark come into view, and appear to be elevated above its former situation. In a similar manner, the light in passing from the heavenly bodies through the atmosphere of the earth deviates from its rectilinear course, by which means those objects appear higher than they really are, except when in the zenith; this apparent elevation of the heavenly bodies above their true places, is called the refraction of those bodies. To illustrate this, let ABC (Fig. 2, Plate IX.) represent the atmosphere surrounding the earth DEF, and let an observer be at D, and a star at a, then if there were no refraction, the observer would see the star according to the direction of the right line D a, but as the light is refracted, it will, when entering the atmosphere near A, be bent from its rectilinear course, and will describe a curve line from A to D, and at entering the eye of the observer at D will appear in the line D b, which is a tangent to the curve at the point D, and the arch ab will be the refraction in altitude or simply the refraction, which must be subtracted from the observed altitude to obtain the true.

At the zenith the refraction is nothing, and the lesser the altitude the more obliquely the rays will enter the atmosphere, and the greater will be the refraction: at the horizon the refraction is greatest. In consequence of the refraction, any heavenly body may be actually below the horizon, when appearing above it. Thus when the sun is at T below the horizon, a ray of light TI proceeding from T comes straight to I, and is there, on entering the atmosphere, turned out of its rectilinear course, and is so bent down towards the eye of the observer at D, that the sun appears in the direction of the refracted ray above the horizon at S.

The mean quantity of the refraction of the heavenly bodies is given in Table XII. All observed altitudes of the sun, moon, planets, or other heavenly bodies, must be decreased by the numbers taken from that Table corresponding to the observed altitude of the object. The refraction varies with the temperature and density of the air, increasing by cold or greater density, and decreasing by heat and rarity of the atmosphere. The corrections to be applied to the numbers taken from Table XII. for different heights of Fahrenheit's Thermometer and the Barometer, are given in Table XXXVI.* Thus, if the refraction was required for the apparent altitude 50, when the thermometer was at 200 and the barometer at 30,67 inches, we should have the mean refraction by Table XII. equal to 9′ 53′′, and by Table XXXVI. the correction corresponding to the height of the thermometer 200 equal to +48", and for the barometer 30,67 equal to+22", hence the true refraction will be 9' 53"+48"+22′′=11′ 3′′.

There is sometimes an irregular refraction near the horizon caused by the vapours near the surface of the earth; the only method of avoiding the error arising from this source, which is sometimes very great, is to take the observations at a time when the object which is observed is more than 100 above the horizon.

The refraction makes any terrestrial object appear more elevated than it really is; the quantity of this elevation varies at different times from to

* This table is to be entered with the height of the Thermometer or Barometer at the top, and the apparent altitude at the side, under the former, and opposite the latter, will be the correction corresponding to the Thermometer or Barometer, which is to be applied to the mean refraction by addition or subtraction according to the signs at the top of the columns respectively.