3 56 
PAR 
PAR 
Thus let AR (Plate Mised. fig. 1 Si.) be a 
quadrant of a great circle on the earth’s sur- 
face, A the place of . the spectator, and the 
pomt \ in the heavens the vertex and zenith. 
Vrv tl represent the starry firmament, 
AL) the sensible horizon, in which suppose 
the star C to be seen, whose distance from 
tne centre of the earth is TC. If this star 
was observed from the centre T, it would 
appear in the tirmament in E, and elevated 
above the horizon by the arch DM: this point 
J> is called the true place of the phenomenon 
or star. Rut an observer viewing it from the 
surface of the earth at A, will see it at T), 
which is called its visible or apparent place ; 
and the arch DM, the distance between the 
true and visible place, is what astronomers 
call the parallax of the star, or other pheno- 
mena. 
If the star rises higher above the horizon to 
M, its true place visible from the centre is P, 
and its apparent place N ; whence its paral- 
lax will be the arch PN, which is less than 
the arch DM. The horizontal parallax, 
therefore, is the greatest; and the higher a 
star rises, the less is its parallax ; and if it 
snouid come to the vertex or zenith, it would 
have no parallax at all : for when it is in Q 
^ seen both from T and A in the same line 
IA\, and there is no difference between its 
true and apparent or visible place. Amfin, 
t.ic faitbei a star is distant from the earth, so 
much the less is its parallax: thus the paral- 
, of the star I* is only GI), which is less 
l oan Dc, the parallax of C. Hence it is plain 
that the parallax is the difference of the dis- 
tances of a star from the zenith when seen 
from the centre and from the surface of the 
earth : for the true distance of the star M 
from the zenith is the arch VP, and its appa- 
rent distance VN, the difference between 
which PN is the parallax. 
1 hese distances are measured by the angles 
VTM and VAM, but VAM — VTM = 
TMA. For the external angle VAM — z. 
A I M -j- A AMT, the two inward and oppo- 
site angles ; so that AM 1' measures the pa- 
rallax, and upon that account is itself fre- 
quently called the parallax: and this is al- 
ways the angle under which the semidiame- 
tei of the earth, A I , appears to an eye placed 
in the star ; and therefore where this semi- 
diameter is seen directly, there the parallax 
is greatest, viz. in the horizon. When the 
star rises higher, the sine of the parallax is 
always to the sine of the star’s distance from 
the zenith, as the semidiarneter of the earth 
to the distance of the star from the earth’s 
centre: hence if the parallax of a star is 
known at any one distance from the zenith, 
we can find its parallax at any other dis- 
tance. 
If we have the distance of a star from the 
earth, we can easily find its parallax: for on 
the triangle TAG ‘(tig. 181.) rectangular at 
A, having the s ipidiumeter of the earth, and 
TC the distance of the star, the angle AC M, 
which is the horizontal parallax, is found by 
trigonometry; and, on tne other hand, if we 
have this parallax, we can find the distance 
of the star; since in the same triangle, hav- 
ing AT, and the z. ACT, the distance TC 
may be easily found. 
Astronomers, therefore, have invented se- 
veral methods for finding the parallaxes of 
stars, in order thereby to discover their dis- 
tances from the earth. However, the fixed 
PAR 
stars are so remote as to have no sensible pa- 
rallax; and even the sun, and all the pri- 
mary planets, except Mars and Venus when 
in perigee, are at so great distances from the 
earth, that their parallax is too small to be 
observed. In the moon, indeed, the parallax 
is found to be very considerable, which in 
the horizon amounts to a degree or more, 
and may lie found thus : In an eclipse of the 
moon, observe when both its horns are in the 
same vertical circle, and at that instant take 
the altitudes of both horns: the difference of 
these two altitudes being ha ved and added 
to the least, or subtracted from the greatest, 
gives nearly the visible or apparent altitude 
of the moon’s centre; and the true altitude 
is nearly equal to the altitude of the centre 
of the shadow at that time. Now we know 
the altitude of the shadow, because we know 
the place of the sun in the ecliptic, and its 
depression under the horizon, which is equal 
to the altitude of the opposite point of the 
ecliptic in which is the centre of the shadow. 
And therefore having both the true altitude 
of the moon and the apparent altitude the 
difference of these is the parallax required. 
Rut as the parallax ot the moon increases as 
she approaches towards the earth, or the pe- 
liga'iim of her orbit, therefore astronomers 
have made tables, which shew the horizontal 
paiallax for every degree of its anomaly. 
I he parallax always diminishes the altitude 
of a phenomenon, or makes it appear lower 
than it would do if viewed from the centre 
ot the earth ; and this change of the altitude 
may, according to the different situation of 
the ecliptic and equator in respect of the ho- 
rizon of the spectator, cause a change of the 
latitude, longitude, declination, and right 
ascension of any phenomenon, which is call- 
ed their parallax. The parallax, therefore 
increases the right and oblique ascension; 
diminishes the descension ; diminishes the 
northern declination and latitude in the east- 
ern part, and increases them in the western; 
but increases the southern both in the eastern 
and western part ; diminishes the longitude 
in the western part, and increases it m the 
eastern. Hence it appears, that the parallax 
has just opposite effects to refraction. 
Parallax, annual, the change of the ap- 
parent place of a heavenly body, which is 
caused by being viewed from the earth in 
different parts of its orbit round the sun. The 
annual parallax ot all the planets is found 
very considerable, but that of the fixed stars 
is imperceptible. 
PARALLEL, in geometry, an appellation 
given to lines, surfaces, and bodies, every 
where equidistant from each other ; and 
winch, though' infinitely produced, would 
never meet. 
Parallel planes, are such planes as 
have all the perpendiculars drawn betwixt 
them equal to each other. 
I arallel rays, in optic-, are those which 
keep at an equal distance from the visible 
object to the eye, which is supposed to be 
infinitely remote from the object. 
Parallel ruler, an instrument consist- 
ing of two wooden, brass, &c. rulers, equally 
broad everywhere; and so joined together 
by cross blades as to open to different inter- 
vals, accede and recede, and yet still retain 
tneir parallelism. See Instruments, ma- 
thematical. 
1 he use of this instrument is obvious; for 
| one of the rulers being applied lo a given 
line, and the other withdrawn to a given point, 
a right line drawn by its edge through that 
, point, is a parallel to the given line, 
j Parallels, or Parallel circles, in 
i geogi apuy, called also parallels or circles of 
i latitude, are lesser circles of the sphere con- 
; ceived to be drawn from west lo east, through 
| ail the points of the meridian, commencing 
from tlie equator to which they are parallel, 
and terminating with the poles. They are 
called parallels of latitude, because all places 
lying under the same parallel, have the ‘same 
latitude. 
Parallels of latitude, in astronomy, are 
lesser circles of the sphere parallel to the 
ecliptic, imagined to pass through every de- 
giee and minute of the colures. T hey are 
represented on the globe by the divisions on 
the quadrant of altitude, in its motion round 
the globe, when screwed over the pole of the 
ecliptic. See Globe. 
Parallels of altitude, or Almucantars, 
are circles parallel to the horizon, imagined 
to pass through every degree and minute of 
the meridian between the horizon and zenith, 
having their poles in the zenith. They are 
represented on the globe by the divisions on 
the quadrant of altitude, in its motion about 
the body of the globe, when screwed to the 
zenith. 
Parallels of declination, in astronomy, 
aie the same with parallels of latitude in geo- 
graphy. 
Parallel sphere, that situation of the 
spheie, wherein the equator coincides with 
j the horizon, and the poles with the zenith and 
nadir. In this sphere all the parallels of the 
equator become parallels of the horizon, con- 
sequently no stars ever rise or set, but all 
turn round in circles parallel to the horizon ; 
and the sun, when in the equinoctial, wheels 
round the horizon the whole day. After his 
rising to the elevated pole, he never sets for 
six months; and after his entering again 
on the other side of the line, never rises lor 
six months longer. 
I his is the position of the sphere to such 
as live under the poles, and to whom the 
sun is never higher than 23° 30'. 
1 arallel sailing, in navigation, is the 
sailing under a parallel of latitude. See Na- 
vigation. 
1 ARALLELEPIPED, or Parallelopi- 
pel>, hi geometry, a regular solid compre- 
hended under six parallelograms, the oppo- 
site ones whereof are similar, parallel, and 
exjuai. See Geometry. 
All parallelepipeds, prisms, Cylinders, &c. 
who-e bases and heights are equal, are them- 
selves equal. 
A diagonal plane divides a parallelepiped 
mto two equal prisms; so that a triangular 
pi ism is hall a parallelep.ped upon th^same 
base and ot the same altitude. 
All parallelepipeds, prisms, cylinders, foe. 
are m a ratio compounded of their bases and 
-altitudes: wherefore, if their bases are equal 
tney are in proportion to their altitudes; and 
conversely. 
All parallelepipeds, cylinders, cones Sec. 
are in a triplicate ratio of their homologous 
sides, and also ot their altitudes. 
Equal parallelepipeds, prisms, cones, cy- 
linders, See. reciprocate their bases and aki- 
tildes. 
