442 
ASTRO 
the Sun to us at its mean diftance. Hence the Sun always 
appears twenty feconds backwarder than its true place. 
To find, the Aberration in Declination. Ettabcdbe. the 
ellipfe of aberration, and P the pole of the ecliptic, as 
defcribed in the laft figure ; on the major axis ac defcribe 
the circle apcq, which we will now fuppofe to lie in the 
plane of the ellipfe, and then every point of this circle 
will be projefted into the fame point of the ellipfe as be¬ 
fore ; let R be the pole of the equator, and perpendicular 
to RS draw the diameter MN of the ellipfe ; alfo draw 
BMC, YNW, perpendicular toac, and YB will be the 
correfponding diameter of the circle ; draw FS perpendi¬ 
cular to BY, FQJ} perpendicular to ac, and Qjtf per¬ 
pendicular to M S ; from any point X let fall XsE per¬ 
pendicular to ac; draw Xz, st, perpendicular to BY and 
M N refpeftively, and s v an ordinate to the diameter M N. 
As the point F of the circle lies at the diftance of ninety 
degrees from the diameter B Y, the diameter F sy will be 
projected into a diameter QSr which will be conjugate to 
MSN, and therefore a tangent at Q^is parallel to MN ; 
hence Qj;I is the greateft perpendicular on MN, and con- 
fequently it is the greateft aberration in declination ; for 
as MN is the projection of BY, which is perpendicular 
to the circle of declination RS, there can be no aberra¬ 
tion in MN ; alfo st is the aberration in declination at any 
point s. Now, when the apparent place of the ftar is at a, 
the ftar is then in conjunction ; and as the motion of the 
Sun is equal to the motion of the ftar in the circle apcq, 
whilft the ftar moves from s to CLjn the ellipfe, its mo- 
. tion in the circle would be XF, which therefore reprefents 
the Sun’s motion in the fame time, or the motion from 
the time when the ftar is at s to the time when the aber¬ 
ration in declination is the greateft. Alfo the arc F a thews 
the elongation of the ftar from the Sun when the ftar ap¬ 
pears at and Xa the elongation when at s. 
When the ftar is at s, st is the aberration in declination ; 
and, as the polition of st to sv is conftant, st varies as si/; 
but sv is the projection of Xz, and therefore in a given 
ratio to it; hence st varies as Xz the fine of XY, or co¬ 
tine of XF ; that is, the aberration in declination at any 
time is as the cofine of the Sun’s diftance from the point 
where it was when the aberration in declination was the 
greateft. 
To find QH, we have, by the property of the ellipfe, 
&HxSM=SdxS cl hence, = (be- 
r CM Sd Sd\ CMxBS CM CB 
“" 6 .£S=r=SjJ s3JxCfl = SM dmd ' d ^ 
N O M Y. 
fin. MSa 
confequently, QH : cS :: fin. MSa : fin. BSa 
fin. BSa 
fin. PSR : cof. FSa, hence QH 
ao"xfin. PSR 
the 
' cof. FSA 
greateft aberration in deelination. 
Let P, in the annexed figure, be the pole of the equator 
QJiW, O the pole of the ecliptic CEV, S the place of 
the ftar, P S A M a circle 
of declination, O S L a cir¬ 
cle of latitude; then L has 
the fame longitude as the 
ftar, and therefore it marks 
the place of the Sun when 
the aberration in latitude is 
nothing. Draw the circle 
STR perpendicular to 
PSA, and T will be the 
place of the Sun when the 
aberration in declination is 
thegreateff; for, by Co¬ 
nics, in the preceding figure, WN : WY :: Si : S* :: 
fin. ftar’s lat. : rad. alfo WN : WY :: tan. NSW or 
PSR: tan. YSW, or cot. FSa; hence fin. ftar’s lat. : 
rad. :: tan. PSR : cot. FSa. But in this figure fin. SL 
(the ftar’s lat.) : rad. :: cot. TSLortan.LSM : cot. TL; 
hence, the three firft terms in thefe two laft proportions be¬ 
ing refpeflively equal, the arc a F, in the preceding fi¬ 
gure, = TL, in the prefent figure ; and as a F reprefents 
the motion of the Sun from the time when the aberration 
in declination is the greateft to conjunftion, and L repre¬ 
fents the place of the Sun at conjunftion, T muft be the 
place at the greateft aberration. Hence the greateft aber- 
. , . 20"Xfin.MSL . 
ration in declination =- . ■ T —-. But, in the trian- 
cof. LT 
gle STL, cof. TSL or fin. MSL = fin. LTS x cof. LT; 
hence the greateft aberration in declination becomes 20" 
X fin. LTS. Alfo, in the triangle ETR, fin. ETR or 
- TfT ,„ fin.ER X fin.ERT 
fin. LTS =- - -——-= (becaufe the meafure of 
fin. ET 
ERT is AS, and AE 
cof. AE x fin.SA 
- ^ -. Hence we get the greateft aberration 
in declination=2o"xcof. right afcenfion xfin. dec. divid¬ 
ed by the fin. of the Sun’s longitude at the time when the 
aberration is greateft fubtraftive. 
Therefore, as the aberration at any other time is as the 
cofine of the Sun’s diftance from that place where it was 
when the aberration was the greateft, if L be the Sun’fe' 
longitude at the time of the greateft aberration in declina¬ 
tion fubtraftive, 5 its longitude at any other time, A the 
Sun’s right afcenfion, D its declination, 0 tiie ob¬ 
liquity of the ecliptic, the aberration at that time — 
— 20" X cof. A x fin. D X cof. L —S 
lin. L 
is the complement of ER) 
20" X cof. A x fin. D x cof. L x cof. S -f- fin. L x fin. S 
(becaufe 
cof. I. 
fin. L 
: cot. L) — 
20" X cof. A x fin. D x 
fin. 
cot. L X cof. S — 20" x cof. A X fin- D X fin. S. But, 
by Mauduit’s Trigonometry, or Crackelt’s Tranfiation, 
. cot. ERTyt, fin. E __ 
cot. ET, or cot. L~ -—-j- cof. E X cot. ER 
lin. ER 
—• (becaufe cot. ER — — tan. A) 
cot Dx fin. 0 
cof. 0 
BS 
cof. A 
X tan. A ; hence the aberration in declination becomes — 
20" X fin. D X cof. S x cot. D X fin. 0 -f- 20" x cof. ^ 
X fin. D x cof. S x cof. 0 X tan. A — 20" X cof. A X 
lin. D X fin. 5 — (becaufe fin. D X cot. D — cof. D, and 
cof. A X tan. A— fin. A) — 20'' X fin. 0 X cof. D X cof. 
S+2o"x cof. 0 x fin. A x fin. D X cof. S —2 o"X cof. A 
X fin. D x fm. S =—20"X fin. 0 X cof. D x cof. 5 — 
20" 
