be directed. Hence it is, that the celestial objects are apparently more elevated in the heavens than they are in reality; and this apparent increase of elevation or altitude is called the refraction of the heavenly bodies; the effects of which are greatest at the horizon, but gradually diminish as the altitude increases, so as to entirely vanish at the zenith.

In this Table the refraction is computed to every minute in the first 8 degrees of apparent altitude; consequently this part of the Table is to be entered with the degrees of apparent altitude at the top or bottom, and the minutes in the left-hand column: in the angle of meeting, stands the refraction.

In the rest of the Table the apparent altitude is given in the vertical columns, opposite to which in the adjoining columns will be found the corresponding refraction. Thus, the refraction answering to 3:27. apparent altitude, is 13:14"; that corresponding to 9:46. is 5.52"; that corresponding to 17:55 is 2.54%, and so on. The refraction is always to be applied by subtraction to the apparent altitude of a celestial object, on account of its causing such object to appear under too great an angle of altitude. The refractions in this Table are adapted to a medium state of the atmosphere; that is, when the Barometer stands at 29. 6 inches, and the Thermometer at 50 degrees; and were computed by the following general rule, the horizontal refraction being assumed at 33 minutes of a degree.

To the constant log. 9.999279 (the log. cosine of 6 times the horizontal refraction) add the log. cosine of the apparent altitude; and the sum, abating 10 in the index, will be the log. cosine of an arch. Now, onesixth the difference between this arch and the given apparent altitude will be the mean astronomical refraction answering to that altitude.

Example.

Let the apparent altitude of a celestial object be 45%, required the corresponding refraction?

Difference.

0:5′42′′ 6 = 0.57"; which, therefore,

is the mean astronomical refraction answering to the given apparent alti

tude.

TABLE IX.

Correction of the Mean Astronomical Refraction.

Since the refraction of the heavenly bodies depends on the density and temperature of the atmosphere, which are ever subject to numberless variations; and since the corrections contained in the foregoing Table are adapted to a medium state of the atmosphere, or when the barometer stands at 29. 6 inches, and the thermometer at 50 degrees: it hence follows, that when the density and temperature of the atmosphere differ from those quantities, the amount of refraction will also differ, in some measure, from that contained in the said foregoing Table. To reduce, therefore, the corrections in that Table to other states of the atmosphere, the present Table has been computed; the arguments of which are, the apparent altitude in the left or right hand margin, the height of the thermometer at the top, and that of the barometer at the bottom of the Table; the corresponding corrections will be found in the angle of meeting of those arguments respectively, and are to be applied, agreeably to their signs, to the mean refraction taken from Table VIII, in the following manner :

Let the apparent altitude of a celestial object be 5 degrees; the height of the barometer 29. 15 inches, and that of the thermometer 48 degrees; required the true atmospheric refraction?

0. 9

Apparent altitude 5 degrees,-mean refraction in Table VIII = .. 9:54"
Opposite to 5 degrees, and over 29. 15, in Table IX, stands..
Opposite to 5 degrees, and under 48 degrees, in ditto . . + 0.3

True atmospheric refraction, as required

9.48%

The correction of the mean astronomical refraction, may be computed by the following rule, viz.

As the mean height of the barometer, 29. 6 inches, is to its observed height, so is the mean refraction to the corrected refraction; now, the difference between this and the mean refraction will be the correction for barometer, which will be affirmative or negative, according as it is greater or less than the latter.-And,

As 350 degrees increased by the observed height of Fahrenheit's thermometer, are to 400 degrees †, so is the mean refraction to the corrected refraction; the difference between which, and the mean refraction, will be the correction for thermometer; which will be affirmative or negative, according as it is greater or less than the latter.

* Seven times 50 degrees, the mean temperature of the atmosphere.
+ Eight times 50 degrees, the mean temperature of the atmosphere.

Example 1.

Let the apparent altitude be 1 degree, the mean refraction 24.29%, the height of the barometer 28.56 inches, and that of the thermometer 32 degrees; required the respective corrections for barometer and thermometer ?

Let the apparent altitude be 7 degrees, the mean refraction 7:20", the height of the barometer 29.75 inches, and that of the thermometer 72 degrees; required the respective corrections for barometer and thermometer?

As mean height of barometer. . 29.60. Log. ar. co.

Is to observed height of ditto.

So is mean refraction 7:20% =

29.75. Log.

440" Log.

2.643453

TABLE X.

To find the Latitude by an Altitude of the North Polar Star.

The correction of altitude, contained in the third column of this Table, expresses the difference of altitude between the north polar star, and the north celestial pole, in its apparent revolution round its orbit, as seen from the equator: the correction of altitude is particularly adapted to the beginning of the year 1836; but by means of its annual variation, which is determined for the sake of accuracy to the hundredth part of a second, it may be readily reduced to any subsequent period, (with a sufficient degree of exactness for all nautical purposes,) for upwards of half a century, as will be seen presently.

The Table consists of five compartments; the left and right hand ones of which are each divided into two columns, containing the right ascension of the meridian: the second compartment, which forms the third column in the Table, contains the correction of the polar star's altitude: the third compartment consists of five small columns, in which are contained the proportional parts corresponding to the intermediate minutes of right ascension of the meridian; by means of which the correction of altitude, at any given time, may be accurately taken out at the first sight: the fourth compartment contains the annual variation of the polar star's correction, which enables the mariner to reduce the tabular correction of altitude to any future period: for, the product of the annual variation, by the number of years and parts of a year elapsed between the beginning of 184, and 36 any given subsequent time, being applied to the correction of the polar star's altitude by addition or subtraction, according to the prefixed sign, will give the true correction at such subsequent given time.

Note. In taking out the proportional parts for the intermediate minutes of right ascension from the upper part of the Table, or between the double horizontal line and the top, whenever the odd minutes of R. A. exceed 5, let the double of any one, or the sum of any two proportional parts, be taken (answering to the two minutes that will make up the odd minutes of R. A.) in the line opposite to the nearest tabular R. A., and the result will be the required proportional part.

Example.

Required the correction of the polar star's altitude on the first day of January in 1848, the R. A. of the meridian being 6: 22′′ ?

C

Correction of polar star's altitude January 1st, 1848=

The corrections of altitude contained in the present Table were computed in conformity with the following principles :

Since to an observer placed at the equator, the poles of the world will appear to be posited in the horizon, the polar star will, to such observer, apparently revolve round the north celestial pole in its diurnal motion round its orbit. In this apparent revolution round the celestial pole, the star's meridional or greatest altitude above the horizon will be always equal to its distance from that pole; which will ever take place, when the right ascension of the meridian is equal to the right ascension of the star. In six hours after this, the star will be seen in the horizon, west of the pole; in six hours more it will be depressed beneath the horizon (on the meridian below the pole), the angle of depression being equal to its polar distance; in six hours after, it will be seen in the horizon east of the pole; and in six hours more, it will be seen again on the meridian above the pole allowance being made, in each case, for its daily acceleration.

Now, since the north celestial pole represents a fixed point in the heavens, and that the star apparently moves round it in an uniform manner, making determinable angles with the meridian; it is, therefore, easy to compute what altitude the star will have, as seen from the equator, in every part of its orbit; for, in this computation, we have a spherical triangle to work in, whose three sides are expressed by the complement of the latitude, the complement of the polar star's altitude, and the complement of its declination; in which there are given two sides and the included angle to find the third side; viz. the star's co-declination or polar distance and the complement of the latitude, with the comprehended angle, equal to the star's distance from the meridian, to find the star's co-altitude; the difference between which and 90 degrees will be the correction of altitude, or the difference of altitude between the polar star and the north celestial pole, as seen from the equator.

For the sake of conciseness, the polar star's altitude, as seen from the equator, may be determined at once by the following formula :—

To the log. rising of the star's horary distance from the meridian, add