STRONOMY. 
437 
To DETERMINE the RefRaCTIONT 
Take the altitude of the Sun, or a (tar whole right af- 
ceniion and declination are known, and note the time by 
the clock ; obferve alfo the times of their tranfits over the 
meridian; then find the hour angle ; hence, in the triangle 
P 7 .x, we know PZ and Pw the complements of latitude 
and declination, and the angle x P Z, to find the fide Z x, 
the complement of which is the altitude, the difference 
between which and the obferved altitude is the refraction 
at that altitude. 
Ex. On May i, 1738, at 5h. 20' in the morning, Caffini 
obferved the altitude of the Sun’s centre at Paris to be 
5 0 o' 14", and the Sun paffed the meridian at iah. o', o". 
to find the refraCtion, the latitude being 48° 5©' 19", and 
the declination was 15 0 o' 25". The Sun's diftance front 
the meridian was 6h. 40', which gives ioo° for the hour 
angle w PZ ; alfo PZ=4i° 9' 50, and P*=:74 0 59' 35''; 
hence Zx— 85° 10' S", confequently the true altitude was 
4 0 49' 52”. Now to 5 0 o' 14", the apparent altitude, add 
9" for tire parallax, and we have 3 0 o' 23" for the appa¬ 
rent altitude corrected for parallax ; hence 5 0 o' 23" — 
4° 49' 52" 10' 31'', the refraCtion at the apparent alti¬ 
tude 5 0 o' 14". 
O t r, Take the greateft and 1 -eaft altitude of a circumpo¬ 
lar (tar which pafies through, or very near, the zenith, 
when it pafies the meridian above the pole ; then the re- 
fraCtion being nothing in the zenith, we (hall have the true 
diftance of the ftar from the pole at that obfervation, the 
altitude of the pole above the horizon being previoufly 
determined; but, when the ftar pafies the meridian under 
tiie pole, we (ball have its difiance affeCted by refraCtion, 
and the difference of the two obferved diftances above and 
below the pole gives the refraCtion at the apparent alti¬ 
tude below the pole. 
Ex. M. de la Caille obferved at Paris a fiar to pafs the 
meridian within 6' of the zenith, and confequently at the 
diftance of 41 0 4'from the pole ; hence it muft pafs the 
meridian under fhe pole at the fame diftance, or at the 
altitude 7 0 46'; but the obferved altitude at that time was 
7 0 52' 25"; hence the refraCtion was 6' 25" at that appa¬ 
rent altitude. 
M. de ia Caille alfo employed obfervations made at Pa¬ 
ris, and the Cape of Good Hope, in 01 der toafeertain the re¬ 
fraCtion. The method he made ufe of was this : The 
diftance of the parallels of Paris and the Cape was found 
to be about 82° 46', the half of which is 4i°23'; there¬ 
fore a ftar vertical to a parallel in the middle between Paris 
and the Cape, muft be at the zenith diftance of 41° 23' 
from each. Now, the fum of the apparent zenith diftan- 
cesof fuchaftarwas found to be 82° 44'46", which there¬ 
fore is the diftance of the two parallels, diminiftied by^tlie 
fum of the two refraCtions at the zenith diftance 41 0 23', 
for refraCtion elevating a ftar, muft make tire apparent ze¬ 
nith diftance of each ftar lefs than the true diftance. Next, 
the apparent altitude of the pole at the Cape was obferved 
Vol. If. No. 81. 
to be 33 0 56' 49 , i", and the altitude at Pans'to be 48° 52' 
27-5", the fum of tlvefe two apparent altitudes is X2 0 49' 
16’6", the diftance of the parallels increafed by the fum 
of the two refraCtions correfponding to the altitude of 
eacli pole. The difference of thefe two determinations is 
4' 30'6", for the (uni of the four refraCtions. Now, ta¬ 
king the-refraCtion to be as the tangent of the zenith dif- 
tance, he found the tangents of 41° 23', and of the com¬ 
plement of the altitudes of the two poles, and divided 
4' 30-6" into four parts in the ratio of thefe tangents, nu¬ 
king the refraCtion a fortieth part lefs at the Cape than at 
Paris, as he had obferved it; hence he got 1'36 5" for 
the refraCtion at the altitude 33 0 56' 49-1'-’ at the Cape, 
and 58-2" at the altitude 48° 52' 27-5" at Paris ; alfo 57-2" 
for the refraCtion at the zenith diftance 41 0 23'at the Cape, 
and 38-7" for the refraction at tiie zenith diftance 41° 23, 
at Paris. The altitudes and zenith diftances corrected by 
thefe refraCtions give 82° 46'42''-for the true diftance of 
the parallels of Paris and the Cape. 
Having determined the refraCtion at the altitude 4S 0 52' 
27-3" at Paris, he calculated the refraCtions from that al¬ 
titude up to the zenith, upon fuppofttion that they were 
as the tangents of the zenith diftances, and hence he knew 
the refraCtions at thefe altitudes at the Cape. Therefore, 
by taking the meridian altitudes of ftars from (even to 
forty-eight degrees at Paris, and tlie correfponding meri¬ 
dian altitudes at the Cape, and correcting thele latter foi 
refraCtion; he got the refraCtion from feven to forty-eight 
degrees at Paris; for the fum of the two true zenith dif¬ 
tances was 82° 46' 42"; therefore, knowing the true zenith 
diftance at the Cape, the true zenith diftance at Paris v\us 
known, the difference between which and the apparent 
zenith diftance was the refraCtion. Thus M. de la Caille 
formed his Table of refraCtions. His method was very 
ingenious ; but, from more accurate obfervations lince his 
time, it appears, that his refraCtions are a little too great. 
This Dr. Mafkelyne lias clearly (hewn in the Phil. Tranf. 
for 1787.. By comparing the fum of the two apparent 
zenith diftances of ftars obferved at a low altitude at Pa¬ 
ris, and confequently at an high altitude at the Cape, and 
at an high altitude at Paris, and therefore at a low altitude 
at the Cape*-he found the refraCtion at the Cape to be a 
fortieth part lefs than at Paris. 
Bofcovich proposes toftind the refraCtion by the circum¬ 
polar ftars, only by knowing its variation at different alti¬ 
tudes. Let a and a' be the apparent meridian zenith dif¬ 
tances of a ftar above and below the pole ; x and x' the 
refpeftive refraCtions ; b and b‘ the apparent meridian ze¬ 
nith diftances of another ftar below and above the pJe ; 
z and z' the correfponding refraCtions ; then the true dif¬ 
tance will be a-f-.v, a'-f-.r', and b-\-z, i'-j-z'; and, as the 
diftance of the pole from the zenith is equal to half the 
fum of the greateft and leaft true zenith diftances, a+.v 
-j- a -j- x b —|— 1 z *-j— b' —z ; hence (dj r-j— x — z — z b b' 
—a—a'. Now taking, at firft, the refractions to be as the 
tangent of the zenith diftances, we have tan. a : tan. a ':: 
, xtan.a' - _ _ .vtan .5 
x:x— -; for the lame realon z ==-, z' 
tan. a tan. a 
vx fan. b : 
tan. a 
; fubftitute thefe in the equation ( A), and we 
get A; 
b-\-b'—a —-a'x tan. a 
— ; hence the other re- 
’tan. a-j- tan. a'— tan. b — tan. b' 
fractions are known. But, as the refraCtions vary mere 
accurately as the tangent of the zenith diftance diminiftied 
by three times the refraCtion, put a —3.x -=m, a' —3-v'~ 
m\ b — 3z — n, o' — 32' — it', and we have x — 
b-\-b‘ —a—a' X fan. m 
the correCt refraCtion 
at the apparent altitude a; hence we know x'zzz 
x tan. m' tX lan.a , , jcftui. 1/ 
-, —-, and z — -The 01 c- 
tan. m tan. m. tan. m 1 
ration may be ftiortened, by taking 3.V, 3.x', 32, 3 z‘, from 
5 T ta’e 
