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Horizontal parallax of the planet=23 Seconds, proportional log.= 2.6717 Apparent altitude of ditto 30 Degrees, log. secant . 10.0625
Parallax in altitude
20 Seconds, proportional log. 2. 7342
Parallax of the Sun in Altitude.
The difference between the places of the sun, as seen from the surface and centre of the earth at the same instant, is called his parallax in altitude, which may be computed in the following manner.
To the log. cosine of the sun's apparent altitude, add the constant log. 0.945124, (the log. of the sun's mean horizontal parallax estimated at 8”.813,) and the sum, rejecting 10 from the index, will be the log. of the parallax in altitude; as thus,
Given the sun's apparent altitude 20 degrees; required the corresponding parallax in altitude ?
Sun's apparent altitude 20 degrees, log. cosine : 9.972986
0.945124 Parall. corresponding to the given altitude 8". 282 Log. 0.918110 This Table, which contains the correction for parallax, is to be entered with the sun's apparent altitude in the left-hand column; opposite to which, in the adjoining column, stands the corresponding parallax in altitude ;thus, the parallax answering to 10: apparent altitude is 9 seconds; that answering to 40: apparent altitude is 7 seconds, &c. &c.-And since the parallax of a celestial object causes it to appear something lower in the heavens, than it really is ; this correction for parallax, therefore, becomes always additive to the sun's apparent altitude.
Mean Astronomical Refraction.
Since the density of the atmosphere increases in proportion to its proximity to the earth's surface, it therefore causes the ray of light issuing from a celestial object to describe a curve, in its passage to the horizon; the convex side of which is directed to that part of the heavens to which a tangent to that curve at the extremity of it which meets the earth, would
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.
Let the apparent altitude of a celestial object be 45., required the corresponding refraction?
9.999279 Log. cosine 9.849485
Log. cosine 9.818764
0.5':42". ; 6 = 0.57" ; which, therefore, is the mean astronomical refraction answering to the given apparent altitude.
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? Apparent altitude 5 degrees,-mean refraction in Table VIII = 9.54" Opposite to 5 degrees, and over 29. 15, in Table IX, stands . . - 0.9 Opposite to 5 degrees, and under 48 degrees, in ditto. + 0. 3
True atmospheric refraction, as required
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 degreest, 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.
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?
As mean height of barometer
29. 60. Log. ar.
co. Is to observed height of ditto. . 28. 56. Log. So is mean refraction 24:29!= . 1469" Log.
To corrected refraction.
Correction for barometer
- 52", which is negative, because the corrected refraction is the least.
And As 350: + 32:=.
382: Log. ar. co.
7.417937 Is to
2. 602060 So is mean refraction 24:29"= . 1469" Log.
To corrected refraction
Correction for thermometer + 69"= 1.9", which is affirmative,
because the corrected refraction is the greatest.
Example 2. 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?
Correction for barometer .
+ 2", which is affirmative.
And As 350 + 72:=.
422: Log. ar. co. . 7.374688 Is to ..
2. 602060 So is mean refraction 7'20. = 440 Log.
To find the Latitude by an Allitude 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 1824 ; 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 1824, and 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.
Example 1. Required the correction of the polar star's altitude in January 1834, the right ascension of the meridian being 6 hours and 22 minutes ? Correction of altitude answering to 6:20:, is
0:16: 9% Proportional part to 2 minutes of right ascension
Correction of polar star's altitude in January 1824= .
+ 2".90 Number of years after 1824
Correction of the polar star’s altitude in Jan. 1834, as required 0:15:48".