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In it the degrees of longitude and those of latitude have always to each other their due proportion: the equator is conceived to be extended out into a straight line, and the meridians are straight lines at right angles to it, as in the figure. Altogether the general cha
racter of maps on this projection is not very dissimilar to what would be produced by referring every point in the globe
to a circumscribing cylinder, by lines drawn from the centre, and then unrolling the cylinder into a plane. Like the stereographic projection, it gives a true representation as to form of every particular small part, but varies greatly in point of scale in its different regions-the polar regions, in particular, being extravagantly enlarged; and the whole map, even of a single hemisphere, not being comprisible within any finite limits.
It would occupy too much space to pursue the subject further; but explanations of all the most useful projections will be found in the sixth chapter of Francœur's "Geodesie."
BEFORE proceeding to the solution of the few simple problems, by which the latitude, longitude, and time can be determined under different circumstances, it is considered advisable to commence with the definitions of such terms as are most constantly met with in practical astronomy, and an explanation of the corrections necessary to be applied to all observations.
The Sextant, Reflecting Circle, or Dolland's Repeating Circle, are the description of portable instruments used in taking astronomical observations for the above purposes. In an observatory, or on any extensive geodesical operations, instruments of firmer construction, and, admitting from their size, of more minute graduation, are required, but these are necessarily confined to permanent establishments.
In all reflecting instruments the angle formed by the planes of the two mirrors is only half the observed angle, but the arc or circle is graduated to meet this effect of the principle of their construction; thus an angle of 60° is marked on the limb of the sextant 120°, and the entire circle reads 720°.
Descriptions of the methods of using and adjusting the sextant and reflecting circle are given in Mr. J. Simms' "Treatise on Mathematical Instruments," which little work is, or should be, in the hands of every person practising to take obervations; but as no allusion is made in this work to the repeating circle, which is at all events in theory the most perfect of the class of reflecting instruments, a short description of the method of using it is given below.
Set the vernier which moves on the circumference of the inner
The repeating circle here spoken of, is a reflecting circle, having the power of repetition.
circle (as do also the horizon glass and telescope at the extremities of arms having one common centre), to zero, or 720° on the graduated outer circle, and clamp it there. Unclamp the vernier at the end of the arm carrying the index-glass, which, when the two glasses are parallel, should read zero. Take the required altitude or angular distance by moving the index forwards till a perfect contact is obtained, and clamp it to the outer circle, noting the time, if required, but merely reading approximately the angle.
Unclamp the arm to which the telescope is attached, and, reversing the instrument, make the contact again on the other side, by moving forward this arm concentric with that carrying the horizon glass, which can be done very rapidly by setting it to the approximate angle already read, but on the other side of the zero of the inner circle, which is graduated each way to 180°, and perfecting the observation by the tangent screw. The angle now read on the outer circle is evidently double that observed for the mean of the times, freed from any index error by the reversal of the instrument. This process may
be repeated over and over again all round the circle as often as required, and the last angle shown by the vernier of the horizon glass is the only one which requires to be read, and divided by the number of observations.
Instead of setting the vernier at first to 720°, it may be read off at any angle as with the theodolite; but the method described above is preferable.
The general outline of the phenomena of the heavens, with which it is absolutely necessary to be acquainted, is beautifully explained in Sir J. F. Herschel's "Treatise on Astronomy," originally published in the "Cabinet Cyclopædia," from which work some of the definitions below have been taken.
The terms answering to terrestrial longitude and latitude, when referred to the celestial sphere, are right ascension and declination; the former being measured on the equinoctial (or the plane of the equator produced to the heavens), commencing from the first point of Aries, which for many reasons has been taken as the conventional point of departure in the celestial sphere; and the latter being measured on great circles perpendicular to the equinoctial, and meeting at the poles, is reckoned north or south of this plane.
A confusion is caused, often puzzling to beginners, by the introduction of the terms longitude and latitude in the celestial nomenclature, but having a different meaning from the same expressions as applied to the situation of places on the earth; they have reference to the ecliptic instead of the equinoctial, celestial longitudes commencing also from the intersection of these two planès, called the "first point of Aries." This point having a constant gradual retrograde motion on the ecliptic, from causes which will be found clearly explained in the third chapter of Woodhouse's "Astronomy," under the head of "Precession of the Equinoxes," and at p. 282 of the work of Sir J. Herschel, already alluded to, it is evident that the longitudes, as well as the right ascensions and declinations, even of the fixed stars, are constantly undergoing a slight change, though imperceptible to measurement in short intervals of time. The correction for their places on this account, as well as on that of their "annual variations,' "aberration and nutation," are all allowed for in the "catalogue of the hundred principal stars," given in the "Nautical Almanac" for every tenth day.
Great circles perpendicular to the horizon, and meeting in the zenith and nadir, are called vertical circles; on these the altitudes of objects above the horizon are measured; the complements to these altitudes are termed zenith distances; and the arc of the horizon contained between a vertical circle, passing through any object, and the plane of the meridian, is termed the azimuth of that object. The altitude and azimuth of any object being known, its place in the visible heavens at that moment is determined, whereas the latitude. and longitude, or the right ascension and declination, fix its place in the celestial sphere.
The right ascension and declination of any celestial object can evidently be determined from its latitude and longitude, and vice versa; the obliquity of the ecliptic, or the angle it forms with the equinoctial being known.
The sensible horizon is an imaginary plane tangential to the earth, at the place of the observer; whereas the rational horizon, to which all altitudes must be reduced by the correction for parallax, is a plane parallel to the former, and passing through the centre of the globe: an altitude, however, is subjected to another correction for the effects
of refraction.* It has been already explained, in page 73, that from this cause the apparent place of any object is always elevated above its real place, and the correction is therefore subtractive.
The first correction alluded to, that for parallax,† is always additive. This term, as applied in its limited sense, to altitudes of celestial objects, is meant to express the angle subtended by the semidiameter of the earth at the distance of the object observed. Altitudes of the moon, from her proximity to the earth, are most affected by parallax: it is also always to be taken into account in observing altitudes of the sun, or any of the planets, but the fixed stars have no sensible parallax, owing to their immeasurable distance from our globe. In the figure below, HO is the sensible, and RL the rational horizon S the real place of the object, and S' its apparent place, elevated by refraction; S'OH is the angle observed; SOH the altitude corrected
for refraction, and SLR the same altitude corrected both for refraction and parallax, being equal to the angle SOH+OSL the parallax.
It is evident that the equatorial parallax of any object, which is that given in the "Nautical Almanac," being subtended by the semi-diameter of the earth at the equator, is always the greatest, and that at the poles the least. The diminution, according to the latitude of the place of observation, can be obtained from tables constructed for the purpose. The parallax in any latitude is also greatest at the horizon, and diminishes as the object approaches the zenith, where it vanishes. Another correction that must be applied to the observed altitudes of 'he sun or moon is for their semi-diameters, plus or minus, according
* See the tenth chapter of Woodhouse's "Astronomy," for the explanation of the method of obtaining the constant of refraction, and the different values of this quantity, generally estimated at 57".
For a further explanation of Parallax in a more general sense, see Sir J. F. Herschel's "Astronomy," p. 47.