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rence, viz., the angle A SE is, therefore, the augmentation of the moon's semidiameter, which may be easily computed; thus

In the oblique angled triangle ASE, there are given the side A E =3958.75 miles, the earth’s semidiameter; the side AS,=AM - AE= 232733.6 miles, the moon's distance when in the zenith from the observer at A; and the angle A ES=15'30", the moon's mean semidiameter; to find the angle ASE=the greatest augmentation corresponding to the given horizontal parallax and horizontal semidiameter : therefore, As moon's zenith distance = AZ=232733.6 miles, Log. ar. co. 4.633141 Is to moon's semidiameter AES= 15:30 Log. sine

7.654056 So is earth's semidiameter EA= 3958.75 miles, Log.. 3.597558 To augment. of semidiam. ASE=0:16"

Log. sine

5. S84755 Now, having thus found the augmentation of the Moon's semidiameter, when in the zenith, answering to the assumed horizontal parallax and horizontal semidiameter; the increase of semidiameter at any given altitude, from the horizon to the zenith, may be computed in the following

manner.

Let SA be produced to F. and draw E F parallel to ZS; then will EF represent the greatest augmentation to the radius EZ. Let the moon be in any other part of her orbit, as at ) with an altitude of 45 degrees ; join ) E, and ) F, and make ) G=)E; then will EG (the measure of the angle EDG to the radius ED,) be the augmentation corresponding to the given altitude. Then, in the right angled triangle EGF, right angled at G, there are given the angle E F G=45 degrees, the moon's apparent altitude, and the side E F=16 seconds, the augmentation of semidiameter when in the zenith; to find the side EG, which expresses the augmentation of semidiameter at the given altitude. And, since the angles expressing the augmentations are so very small, the measure of each may be substituted for its sine, which will simplify the calculation; thus, As radius

90:00Log. sine ar. comp. 0.000000 Is to moon's greatest augment. of semidiam.=E F 16", Log. = 1. 204120 So is moon's given apparent alt.= _ EFG, 45. Log. sine = 9.849485 To the augmentation, or side EG = ll".31. Log. = 1.053605 which, therefore, is the augmentation of the moon's semidiameter corresponding to the given apparent altitude of 45 degrees; horizontal semidiameter 15:30". and horizontal parallax 57:30.

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Explanation of the Table.

This Table contains the augmentation of the moon's semidiameter (determined after the above manner,) to every third degree of altitude: the

augmentation is expressed in seconds, and is to be taken out by entering the Table with the moon's horizontal semidiameter at the top, as given in the Nautical Almanac, and the apparent altitude in the left-hand column ; in the angle of meeting will be found a correction, which being applied by addition to the moon's horizontal semidiameter will give the true semidiameter, corresponding to the given altitude. Thus the auginentation answering to moon's apparent altitude 30 degrees, and horizontal semidiameter 16:30is 9 seconds; and that corresponding to altitude 60: and semidiameter 16! is 14 seconds.

TABLE V.

Contraction of the semidiameters of the Sun and Moon.

Since all parts of the horizontal semidiameter of the sun or moon are equally elevated above the horizon, all those parts must be equally affected by refraction, and thereby cavse the horizontal semidiameter to remain invariable. But when the semidiameter is inclined to the plane of the horizon, the lower extremity will be so much more affected by refraction than the upper, as to suffer a sensible contraction, and thus cause the semidiameter, so inclined, to be something less than the horizontal semidiameter given in the Nautical Almanac, Hence it is manifest that the semidiameter of a celestial object, measured in any other manner than that parallel to the plane of the horizon will be always less than the true semidiameter by a certain quantity :--this quantity, called the contraction of semidiameter is contained in the present Table; the arguments of which are, the apparent altitude of the object in the left-hand column, and at the top the angle comprehended between the measured diameter and that parallel to the plane of the horizon ; in the angle of meeting will be found a correction, which being subtracted from the horizontal semidiameter in the Nautical Almanac, will leave the true semi-diameter. Thus, let the sun's or moon's apparent altitude be 5 degrees, and the inclination of its semidiameter 72 degrees; now, in the angle of meeting, of these arguments, stands 23 seconds ; which, therefore, is the contraction of semidiameter, and which is to be applied by subtraction to the semidiameter given in the Nautical Almanac.

To compute the contraction of Semidiameter. Rule.- Find by Table VIII. the refraction corresponding to the object's apparent central altitude, and also the refraction answering to that altitude augmented by the semidiameter; (which, for this purpose, may be estimated

at 16 minutes,) and their difference will be the contraction of the vertical semidiameter. Now, having thus found the contraction corresponding to the vertical semidiameter, that answering to a semidiameter which forms any given angle with the plane of the horizon, will be found by multiplying the vertical contraction by the square of the angle of inclination.

Example. Let the sun's or moon's apparent central altitude be 3. and the inclination of its semidiameter to the plane of the horizon 72.; required the contraction of the semidiameter?

Apparent central altitude 3: 0 Refraction=14:36!
Do. augmented by semidiam, = 3:16! Ditto . = 13.46.

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Contraction of the vertical semidiameter . 0:50” Log.=1.698970 Inclination of semidiameter. =72: twice the log. sine .=19.956412 Required contraction of semidiameter . . 45". 22 Log.=1.655382

And so on of the rest. It is to be remarked, however, that the correction arising from the contraction of the semidiameter of a celestial object is very seldom attended to in practice at sea.

Table VI.

Parallax of the Planets in Altitude.

The arguments of this Table are the apparent altitude of a planet in the left or right-hand margin, and its horizontal parallax at the top ; under the latter, and opposite the former, stands the corresponding parallax in altitude ; which is always to be applied by addition to the planets apparent altitude. Hence, if the apparent altitude of a planet be 30 degrees, and its horizontal parallax 27 seconds, the corresponding parallax in altitude will be 23 seconds; additive to the apparent altitude.

The parallaxes of Altitude in this Table were computed by the following,

Rule.-To the proportional logarithm of the planet's horizontal parallax add the log. secant of its apparent altitude, and the sum, abating 10 in the index, will be the proportional logarithm of the parallax in altitude.

Example.

If the horizontal parallax of a planet be 23 seconds, and its apparent altitude 30 degrees; required the parallax in altitude ?

Horizontal parallax of the planet=23 Seconds, proportional log.= 2.6717
Apparent altitude of ditto 30 Degrees, log. secant 10.0625

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ertical ang to forms

Parallax in altitude

20. Seconds, proportional log. 2.7342

.

TABLE VII.

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 86.813,) 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?

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Sun's

apparent altitude 20 degrees, log. cosine :

Constant log:

9.972956
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.

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TABLE VIII,

Mean Astronomical Refraction.

llas

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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

at 16 minutes,) and their difference will be the contraction of the vertical semidiameter. Now, having thus found the contraction corresponding to the vertical semidiameter, that answering to a semidiameter which forms any given angle with the plane of the horizon, will be found by multiplying the vertical contraction by the square of the angle of inclination.

Example. Let the sun's or moon's apparent central altitude be 3. and the inclination of its semidiameter to the plane of the horizon 72.; required the contraction of the semidiameter ?

Apparent central altitude 3: 0 Refraction=1436
Do. augmented by semidiam. = 3:16! Ditto . = 13.46.

Contraction of the vertical semidiameter . 0:50. Log.=1.698970 Inclination of semidiameter · =72: twice the log. sine =19.956412

Required contraction of semidiameter. ..

45". 22 Log.=1.655382

And so on of the rest. — It is to be remarked, however, that the correction arising from the contraction of the semidiameter of a celestial object is very seldom attended to in practice at sea.

Table VI.

Parallax of the Planets in Altitude.

The arguments of this Table are the apparent altitude of a planet in the left or right-hand margin, and its horizontal parallax at the top ; under the latter, and opposite the former, stands the corresponding parallax in altitude ; which is always to be applied by addition to the planets apparent altitude. Hence, if the apparent altitude of a planet be 30 degrees, and its horizontal parallax 27 seconds, the corresponding parallax in altitude will be 23 seconds; additive to the apparent altitude.

The parallaxes of Altitude in this Table were computed by the following,

Rule. To the proportional logarithm of the planet's horizontal parallax add the log. secant of its apparent altitude, and the sum, abating 10 in the index, will be the proportional logarithm of the parallax in altitude,

Example. If the horizontal parallax of a planet be 23 seconds, and its apparent altitude 30 degrees; required the parallax in altitude ? .

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