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required number of degrees of latitude (4); through one of these points of division (say 46°) draw CD intersecting the meridian at right angles, and likewise draw lines through the other points parallel to C D.

Take the breadth in minutes of a degree of longitude in lat. 46° = 41.63; from M towards C and D, set off each way one-half of this, 20.84, (ME. MG). Again, from N lay off on each side one-half of the length of a degree in lat. 47° 40·92 - N F, N H. Measure the diagonals G H, E F, and putting one point of the compasses successively on F, G, H, and E, describe the arcs, x x x x.

Take 41.68, the whole measurement of a longitudinal degree in lat. 46°, and lay off the distance, GO, EO, intersecting the arcs xxx at 00. Again, take the value of a degree in latitude 47° 40.92, and lay off the distances EP, HP.

This process continued until the parallels of 46° and 47° are completed, the whole projection may be carried on in the same manner, the two parallels first drawn furnishing the respective points of each meridian.

It would occupy too much space to pursue the subject further; explanations of all the most useful projections will be found in the sixth chapter of Francœur's "Géodesie," and in other works of the same character.




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 explain the meaning of such terms as are most constantly met with in practical astronomy, and the corrections necessary to be applied to all observations.

The Sextant; Reflecting Circle, or Dollond's Repeating Circle; with the Artificial Horizon and Chronometer; are the description of portable instruments generally used in taking astronomical observations. In an observatory, or for any extensive geodesical operation, instruments are required of firmer construction, and admitting from their size of more minute graduation; but these are mostly 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 observer; but as no allusion is there made to the repeating circle*, which is, at all events in theory, the most

* The repeating circle here spoken of, is a reflecting circle, having the power of repetition. For the determination of latitudes and longitudes on surveys of the magnitude of the Ordnance Survey of Great Britain, or for very important and delicate geodesical operations, the Zenith Sector, Altitude and Azimuth Instrument, and Portable Transit are employed. This latter, though properly an observatory instrument, can be used upon a stand formed by the stump of a large tree, or by three or four strong posts driven into the ground, supporting a top, on which the transit is placed. A rough pedestal of masonry or brick-work of course answers the same purpose, great care being taken to secure its steadiness, and prevent its

perfect of the class of reflecting instruments, a short description of the method of using it, is here given.

Set the vernier, which moves on the circumference of the inner 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. 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 nearly 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 perfect 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 aften 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, for the mean angular measurement answering to the mean of the times.

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 terms answering to terrestrial longitude and latitude, when referred to the celestial sphere, are right ascension and declination;

being affected by the movement of those about it, to ensure which, a sort of detached platform upon posts will be found efficient. Solid rock is considered not so suited for the foundation of this sort of pedestal as sand, or other species of earth, on account of its more readily conveying tremulous vibrations to the instrument. Transits of from 20 to 30 inches focal length were thus used upon the survey (in 1845) of the North American Boundary, a tent made of fine canvas being contrived to protect the lights from the wind.

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 on great circles perpendicular to the equinoctial and meeting at the poles, being 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, 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 commence also from the intersection of these two planes, 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 corrections 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 versâ; 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, passing through the centre of the globe an altitude requires also another correction for the effects of refraction *, which it has been already explained, in page 71, causes the apparent place of any object to be always elevated above its real place; 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 semi-diameter of the earth at the distance of the object observed. Altitudes of the moon, from her proximity to the earth, are most effected 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 appreciable parallax, owing to their immeasurable distance from our globe.

In the figure below, HO is the sensible, and R L 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

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altitude corrected for refraction, and SLR the same altitude corrected both for refraction and parallax, being equal to the angle SOH + OSL, the parallax.

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.

At least 5000 million times the diameter of the globe.

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