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of aberration; and beginning from the center, divide this diameter each way to the circumference into as many parts as shall be found by this analogy. As cosine star's latitude : radius : : 20′′.25 : the number sought, which in this case will be 25".02. Because any arc of a small circle is to its corresponding arc of a great circle, as the cosine is to radius.

At right angles to this diameter, draw another to represent the star's circle of latitude, whose radius divide into 20".25 equal parts, because it is part of a great circle, and in this diameter take CE: CF :: radius: sine of the star's latitude, and CF will be the lesser axis of the star's aberration. Upon the axes CA and CF describe the ellipsis CFBT, which shall represent the apparent path of the star through the year; its true place being at C the center of the ellipsis. From S the place of the sun in the ecliptic let fall the perpendicular SG on the diameter CA, and cutting the ellipsis in L the place of the star in its ellipsis, and LG will be its aberration in latitude. Draw LN perpendicular to the lesser axis of the ellipsis, from the place of the star, and it will be the aberration in longitude. These may be measured on their respective scales, or computed as follows.

As EC: FC:: radius: S, star's latitude :: 20".25; 11".9 the maximum of aberration in latitude=FC. Hence as EC: FC:: SG: LG:: S, of the sun's distance from his opposition to the star; aberration in latitude at the time, that is as radius: maximum=11′′.9 =FC:: S, SCB=76`: 11".55=the aberration in latitude at that time.

Or as radius: S, star's latitudex S, sun's distance from opposition to the star:: 20".25: the aberration in latitude. Or, lastly, as cosine star's latitude: S,

sun's distance from point opposite the star :: 20".25: the aberration in latitude at the time.

Let O represent the sun's place in the ecliptic, when the star's apparent longitude or latitude is the same with the true, and tends to excess. For the aberration in latitude, is the point opposite to the star.

TO FIND THE ABERRATION IN LONGITUDE,

FOR WHICH PURPOSE IS 3 SIGNS AFTER THE STAR'S PLACE.

As cosine star's latitude : radius :: 20′′.25: maximum of aberration=BC=25".02=M.

As radius: S, sun's elongation from :: maximum : aberration in longitude at the time=A=6".21. Or, without the maximum. As cosine star's latitude : S, sun's elongation from O=14° : : 20′′.25: A=6".21.

TO FIND THE ABERRATION IN RIGHT ASCENSION AND DECLINATION.

We must determine the angle made by the star's circle of latitude and its circle of declination previous to any construction or computation of the aberration in declination or right ascension, which is easily done by spherical trigonometry; by saying, as cosine latitude of the star, is to the sine of its distance from the solstitial colure; so is the sine of the obliquity of the equator and ecliptic, to the sine of the angle between these circles, commonly called the star's angle of po sition; which in this case is 26° 50'.

As the north part of the star's circle of declination is so much removed to the east of its circle of latitude, make the angle ECP-26' 50′ to the eastward of EC, and draw the diameter PCQ, to represent the star's circle of declination, and divide its radii into 20′′.25

each. At right angles to PCQ, draw the diameter RCT, to represent a portion of a small circle parallel to the equator passing through the star; and divide its radii into as many parts as are found by this analogy; as cosine star's declination: radius: : 20.25: the number sought, which in this case is 30".5. Now draw LM, LN, and these lines being perpendicular to PC, and CT, respectively, will be the respective aberrations in right ascension and declination; and may be measured on their respective scales.

As there is frequently occasion, in practical astronomy, for computing the aberrations in right ascension and declination, we shall deliver these practical rules following, without their demonstrations, referring you, for this purpose, to the writers on this subject.

FOR THE ABERRATION IN RIGHT ASCENSION.

In the astronomical tables, find that point in the ecliptic, which answers to the right ascension of the star, which call N. This point is that, wherein the sun. being found, the aberration makes the right ascension the least. Take, in the same tables, the angle made by the ecliptic, and a meridian passing through the point N; and then say, as radiusx cosine star's declination: cosine sun's distance from NxS, angle between the ecliptic and meridian of N:: 20".25: aberration in A.R. which is additive or subtractive, according to the sun's situation with respect to the point N.

Otherwise thus; as S, star's latitude : radius:: cosine P-the star's angle of position: tangent of an angle Z.

Then if the star be, (with respect to that pole which is of the same name as the star's latitude) in a sign ascending or descending, P being acute, Z added to

the star's true place, or taken from that opposite to its true place; but P being obtuse, if the star be in an ascending or descending sign, Z taken from the star's true place, or added to that opposite to its true place, will give O. N. B. v, ≈, x, y, 8, п, are ascending signs with respect to the north pole, and descending signs with respect to the south pole, the rest are the contrary.

As cosine star's declinationx SZ: cosine Pxradius :: 20" 25: maximum of aberration in A.R.

Radius: S, sun's distance from :: M: aberration in A.R.

Or, cosine star's declinationxS,Z:: S, sun's distance from Oxcosine P:: 20".25: aberration in A.R.

N. B. is the sun's place in the ecliptic when the star's apparent longitude, latitude, declination, or A.R. being the same as the true tends to excess. And the angle Z is the sun's distance from the nearest syzygy with the star, at the time of O.

FOR THE ABERRATION IN DECLINATION.

As S, star's latitude: radius: : tangent P-the star's angle of position: tangent of an angle, which is Z. Then if P be acute, and the star, (in respect of that pole of the equator which is of the same name with the star's latitude,) be in a sign ascending or descending, Z taken from or added to the point opposite to its true place gives ; but if P be obtuse, and the star be in an ascending, or descending sign, Z added to or taken from its true place gives O; provided always, that the star's declination and latitude be both north or both south, but if one be north and the other south, then for the star's true place, read point opposite to its true place, and vice versa.

As S,Z: S,P:: 20".25: maximum of aberration in declination.

As radius: S, sun's elongation from :: maximum : aberration in latitude at the time proposed.

Or, as radiusx S,Z: S,PxS, of sun's elongation from :: 20".25: A the aberration in declination at the time.

Note. These rules give the value of A and M, in seconds of a degree; and if the sun's place be in that semi-circle of the ecliptic which precedes O, the aberration A must be taken from the star's true longitude, latitude, declination, or A.R. to give the apparent; but if in that semi-circle which follows O, A must be added.

The libratory variation of the equator to the ecliptic, which arises from the action of the moon and sun upon the redundant matter of the earth about the equator, is termed the nutation of the poles, which occasions another small apparent motion of the fixed stars, amounting to about 9".3 of a degree when greatest. This principally depends upon the position of the moon's nodes. The following method will serve to determine it with sufficient precision.

FOR THE NUTATION OR DEVIATION OF A STAR IN A.R.

With a radius equal to 9".3 describe a circle,* which graduate into signs and degrees. At the points 3o and 9 draw a diameter, on which as a transverse axis describe an ellipsis, whose axes are as 9.3: 7.1. Graduate the same into signs and degrees, from the right hand to the left, making 0', of the ellipsis to coincide with 3s in the circle. Graduate the ellipsis by drawing lines from every degree in the circle perpendicular to the * See Plate 20.

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