43 * ASTRONOMY. 
ti:e common Tables. As a-{-x, a'—j— a', are the true ze¬ 
nith diftances of one of the liars below and above the 
pole, the true zenith diftance of the pole will be one-half 
®f a-\-x-\-a'~\-x', which is the complement of the latitude 
o-f the place. 
Dr. Bradley determined the refraftion in the following 
manner: He obferved the pole liar, and other circumpo¬ 
lar liars, above and below the pole, and from thence de¬ 
duced the apparent zenith diftance of the pole. By the 
apparent and equal zenith diftances of the Sun at the two 
equinoxes, having at the fame time oppofite right afcen- 
fions, as found by comparing its obferved tranfits over the 
meridian with thofe of the fixed ftars, he found the appa¬ 
rent zenith diftance of the equator, which diminilhed by 
parallax, and added to the apparent zenith diftance of 
the pole, gave a fum lefs than ninety degrees by the fum 
of the two refractions belonging to the pole and meridian 
•altitude of the equator. For the fum of the two true ze¬ 
nith diftances equal ninety degrees; but the true diftance 
of each is diminilhed by refraction, and therefore the fum 
(after the correction for parallax) mult be lefs than ninety 
degrees by the fum of the two refraftions. Now, he ob¬ 
ferved, that the difference of the refraftions at thefe alti¬ 
tudes came out w ithin 2", or 3", from the belt tables then 
extant, whether deduced folely from obfervations, or part¬ 
ly from obfervation and partly from theory. Hence, know¬ 
ing the fum and difference of the refractions, he knew the 
retraftion at each altitude. He afterwards more accurate¬ 
ly divided the fum of the two refraftions, by taking the 
parts in proportion to the tangents of the zenith diftances. 
The apparent zenith diftance of the equator, by the mean 
of twenty obfervations in 1746-47 he found to be 51 0 27' 
28''; and the mean apparent zenith diftance of the pole, by 
obfervations made between 1730 and 1752,was 30' 35" 
the fum of which being 89° 58' 3'', the fum of the two re¬ 
fractions is 1' 5•/"; confequently the polar refraftion is 
451", and the equatorial 1' 11^"; therefore, the latitude 
of Greenwich Obfervatory is 51 0 28' 394''. Dr. Bradley 
here fuppofed the Sun’s horizontal parallax to to be iof"; 
but Dr. Malkelyne obferves, that had he taken it 8J", as 
determined from the two laft tranfits of Venus over the 
Sun; the refraftion at forty-five degrees, which lie fixed at 
57”, would have come out 56^", and the latitude of the 
Obfervatory 5i°28'4o". Dr. Bradley having thus fet¬ 
tled the refraftion at the altitude of the equator and pole, 
could calculate the refraftion at all higher altitudes, or for 
all ftars between the equator and pole, by taking it as the 
tangent of the zenith diftances, which would be very ac¬ 
curate for all filth altitudes. Hence, by taking the alti¬ 
tudes of the circumpolar ftars above and below the pole, 
and knowing the refraftion above, he immediately got the 
refraftion at the lower altitudes; for, knowing the retrac¬ 
tion at the altitude above the pole, he knew the true alti¬ 
tude above, and, knowing the altitude of the pole, lie got 
the true diftance of the ftar from the pole; which, fub- 
trafted from the altitude of the pole, gave the true altitude 
beloiv, the difference between which and the apparent al¬ 
titude was the refraftion. When the weight and tempera¬ 
ture of the air remains the fame, he found that the refrac¬ 
tion varied as the tangent of the zenith diftance diminilhed 
by three times the refraftion found by the common rule ; 
and, having fixed the refraftion at forty-five degrees (whofe 
tangent, if radius —1, is unity) to be 57", if r— the re¬ 
fraftion in the tables, z— the apparent zenith diftance, he 
got this proportion, r : 57" :: tan. z —3 r : x. The appli¬ 
cation of this rule to find the refraftion at all altitudes is 
thus: Let the apparent zenith diftance be z, then the re¬ 
fraftion will be nearly 57"X tan. z, which put —r; and 
the correft mean refraftion will be 57" X tan. z —3r. If 
at very low altitudes it fhould be required to have the re- 
•Vaftion more correftiy, put 57"Xtan.z — 2 r — r '> and the 
sefraftion becomes 57" X tan. 2—3 r'. Let the refraftion 
at rhe apparent zenith diftance 70° be required; The tan- 
1 
gent of 70 0 is 2-747; hence 57"x2-747=2'36-6", which 
multiplied by 3 and fubtrafted from 70° gives 69° 52' 10", 
the tangent of which is 2-728; therefore 57"X2-728— 
2 ' 35'5"> the mean refraction at the apparent zenith dif¬ 
tance 70 degrees. 
The inftrument invented by Mr. Ramfden, Called a cir~ 
cular injlrument, and explained under the article Qua¬ 
drant, is admirably calculated to determine the quantity 
of refraftion at all altitudes ; for, by taking the altitude 
and azimuth of a body whofe declination is known, the 
true altitude may be immediately computed from the lac 
titude of the place, declination of the body, and obferved 
azimuth; hence the difference between the obferved and 
computed altitudes gives the refraftion at that apparent 
altitude. 
The refraftion being found to vary in different dates of 
the air, the next enquiry is, what allowance muft be made 
for any variation of the temperature and weight of the air, 
from any ftandard which we make the mean. Dr. Bradley 
made 29-6 incites the mean ftandard for the barometer; 
and, as Mr. Haukfbee had determined from experiment 
that the refraftion was in proportion to the denfity of the 
air, it muft alfo be as the altitude of the mercury in the 
barometer. Now, in the mean (tate of the air, that is, 
when the barometer is at 29-6 inches, and Fahrenheit’s 
thermometer at fifty degrees, the refraftion : 57" :: 
tan. z—3r : 1 ; hence, at any altitude (a) of the mercury. 
the refraftion : 57" :: ax tan. z—31- : 29-6. The re¬ 
fraftion, thus correfted for the variation of the weight of 
the air, agrees very well with obfervations. The next 
tiling to be done is, to find bow the refraftion varies in 
different temperatures. M. de la Caille found, that the 
refraftion was diminilhed -Jj part from an increafe of ten 
degrees in the altitude of the mercury in the thermome¬ 
ter of Reaumur. Mayer obferved that the refraftion va¬ 
ried about 2*2- part for ten degrees of variation. M. Bon¬ 
ner made fome experiments in order to determine the va¬ 
riation of refraftion arifing from that of the temperature; 
calling the refraftion unity for the altitude ten degrees of 
the thermometer, he found the refraftion to be 0-92 at 
the altitude thirty degrees, or diminilhed for a varia¬ 
tion of ten degrees; and, at eight degrees below nothing, 
he found the refraftion to be 1-085 or — 1 —, for a varia¬ 
tion of ten degrees. The mean of thefe differ but very 
little from the determination of Mayer. The obferva¬ 
tions, upon which Dr. Bradley formed his rate of variation, 
have never been publilhed. He ufed Fahrenheit’s ther¬ 
mometer, and fixed the mean temperature at fifty degrees; 
and, if h° be any other altitude, he found that the refrac- 
Jl Q - i--? ro° 
tion varied in the ratio of 400° : ^°4-35°°> or 1 :-—— 
Hence, allowing for the variation of temperature and 
weight, he found, the true refraftion : 57" :: ° - X tan. 
29-6 
- 1 - ^^ 0 ° 
z—3 r '• -5—• And this agrees very accurately with 
the rule deduced by Mayer. 
When the Sun is in the horizon, the rays in palling ve¬ 
ry obliquely through the atmofphere are fo far feparated, 
that M. Bouguer, in a work entitled Traite d’Optique fur 
la Gradation de la Lumiere, lias concluded from experi¬ 
ment, that the intenfity of light is 1354 times Lefs than 
when the Sun is in the zenith. M. de Mairan thinks, that 
the weaknefs of the Sun’s rays in the former cafe is prin¬ 
cipally to*-be attributed to the quantity of vapours with 
which the lower parts of the atmofphere are always filled. 
To determine the Place of the Moon’s Apogee. 
Compare the obferved place of the Moon at any time 
with the place obferved at any time afterwards ; take the 
mean motion eorrefponding to the interval of time, and 
add 
