The _celestial sphere_ is that imaginary surface upon which all
celestial objects are projected. Its radius is infinite.
The _earth's axis_ is the imaginary line about which it revolves.
The _poles_ are the points in which the axis pierces the surface of the
earth, or of the celestial sphere.
A _meridian_ is a great circle of the earth cut out by a plane passing
through the axis. All meridians are therefore north and south lines
passing through the poles.
From these definitions it follows that if there were a star exactly at
the pole it would only be necessary to set up an instrument and take a
bearing to it for the meridian. Such not being the case, however, we are
obliged to take some one of the near circumpolar stars as our object,
and correct the observation according to its angular distance from the
meridian at the time of observation.
For convenience, the bright star known as Ursæ Minoris or Polaris, is
generally selected. This star apparently revolves about the north pole,
in an orbit whose mean radius is 1° 19' 13",[1] making the revolution in
23 hours 56 minutes.
[Footnote 1: This is the codeclination as given in the Nautical Almanac.
The mean value decreases by about 20 seconds each year.]
During this time it must therefore cross the meridian twice, once above
the pole and once below; the former is called the _upper_, and the
latter the _lower meridian transit or culmination_. It must also pass
through the points farthest east and west from the meridian. The former
is called the _eastern elongation_, the latter the _western_.
An observation may he made upon Polaris at any of these four points,
or at any other point of its orbit, but this latter case becomes too
complicated for ordinary practice, and is therefore not considered.
If the observation were made upon the star at the time of its upper or
lower culmination, it would give the true meridian at once, but this
involves a knowledge of the true local time of transit, or the longitude
of the place of observation, which is generally an unknown quantity; and
moreover, as the star is then moving east or west, or at right angles to
the place of the meridian, at the rate of 15° of arc in about one hour,
an error of so slight a quantity as only four seconds of time would
introduce an error of one minute of arc. If the observation be made,
however, upon either elongation, when the star is moving up or down,
that is, in the direction of the vertical wire of the instrument, the
error of observation in the angle between it and the pole will be
inappreciable. This is, therefore, the best position upon which to make
the observation, as the precise time of the elongation need not be
given. It can be determined with sufficient accuracy by a glance at the
relative positions of the star Alioth, in the handle of the Dipper,
and Polaris (see Fig. 1). When the line joining these two stars is
horizontal or nearly so, and Alioth is to the _west_ of Polaris, the
latter is at its _eastern_ elongation, and _vice versa_, thus:
[Illustration]
But since the star at either elongation is off the meridian, it will
be necessary to determine the angle at the place of observation to be
turned off on the instrument to bring it into the meridian. This angle,
called the azimuth of the pole star, varies with the latitude of the
observer, as will appear from Fig 2, and hence its value must be
computed for different latitudes, and the surveyor must know his
_latitude_ before he can apply it. Let N be the north pole of the
celestial sphere; S, the position of Polaris at its eastern elongation;
then N S=1° 19' 13", a constant quantity. The azimuth of Polaris at the
latitude 40° north is represented by the angle N O S, and that at 60°
north, by the angle N O' S, which is greater, being an exterior angle
of the triangle, O S O. From this we see that the azimuth varies at the
latitude.
We have first, then, to _find the latitude of the place of observation_.
Of the several methods for doing this, we shall select the simplest,
preceding it by a few definitions.
A _normal_ line is the one joining the point directly overhead, called
the _zenith_, with the one under foot called the _nadir_.
The _celestial horizon_ is the intersection of the celestial sphere by a
plane passing through the center of the earth and perpendicular to the
normal.
A _vertical circle_ is one whose plane is perpendicular to the horizon,
hence all such circles must pass through the normal and have the zenith
and nadir points for their poles. The _altitude_ of a celestial object
is its distance above the horizon measured on the arc of a vertical
circle. As the distance from the horizon to the zenith is 90°, the
difference, or _complement_ of the altitude, is called the _zenith
distance_, or _co-altitude_.
The _azimuth_ of an object is the angle between the vertical plane
through the object and the plane of the meridian, measured on the
horizon, and usually read from the south point, as 0°, through west, at
90, north 180°, etc., closing on south at 0° or 360°.
These two co-ordinates, the altitude and azimuth, will determine the
position of any object with reference to the observer's place. The
latter's position is usually given by his latitude and longitude
referred to the equator and some standard meridian as co-ordinates.
The _latitude_ being the angular distance north or south of the equator,
and the _longitude_ east or west of the assumed meridian.
We are now prepared to prove that _the altitude of the pole is equal to
the latitude of the place of observation_.
Let H P Z Q¹, etc., Fig. 2, represent a meridian section of the sphere,
in which P is the north pole and Z the place of observation, then H H¹
will be the horizon, Q Q¹ the equator, H P will be the altitude of P,
and Q¹ Z the latitude of Z. These two arcs are equal, for H C Z = P C
Q¹ = 90°, and if from these equal quadrants the common angle P C Z be
subtracted, the remainders H C P and Z C Q¹, will be equal.
To _determine the altitude of the pole_, or, in other words, _the
latitude of the place_.
Observe the altitude of the pole star _when on the meridian_, either
above or below the pole, and from this observed altitude corrected for
refraction, subtract the distance of the star from the pole, or its
_polar distance_, if it was an upper transit, or add it if a lower.
The result will be the required latitude with sufficient accuracy for
ordinary purposes.
The time of the star's being on the meridian can be determined with
sufficient accuracy by a mere inspection of the heavens. The refraction
is _always negative_, and may be taken from the table appended by
looking up the amount set opposite the observed altitude. Thus, if the
observer's altitude should be 40° 39' the nearest refraction 01' 07",
should be subtracted from 40° 37' 00", leaving 40° 37' 53" for the
latitude.