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44. On a planisphere of this kind, the stars are placed according to their right ascensions and declinations. Then the ecliptic is traced, as well as its poles, the circles of latitude all intersecting mutually at those two poles, and then the parallels to the ecliptic.
Thus, on the radius marked 270°, at a distance cr from the centre = tan 11° 44', we mark the north pole as of the ecliptic. From that point, with an opening of the compasses = cosec 23° 28′ we mark the point E, or, which amounts to the same, we make cF = cot 23° 28′. Through the point E we draw the indefinite perpendicular vex, which is evidently the locus of the centres of all the circles of latitude intersecting mutually in II and w: E will be the centre of that circle of latitude which passes through 0 sand 180°, or of the equinoctial colure, which will be the circle vaxII. resi will be the solstitial colure. *
In general, make Ex G' = longitude: from the centre G' with the radius G’z we describe a circle, it will be the circle of latitude which answers to the longitude supposed. ... Repeating the same operation on the other side of the line z F, we shall have the circles of latitude of the other hemisphere. For the circles parallel to the ecliptic we employ the formulae tan: (p + A) + tan? (D - A).
2 - * tan #(p + A) – tan #(D - A). 2.
p being = 23° 27'49", or nearly 23°28', the distance between the poles of the ecliptic and the equator; and A the polar distance of the parallel. When A > D, the sign of the second term will be changed.
45. The principal defect of this projection is the little resemblance and proportion between the arcs of the sphere and their projections. Thus, the arcs wa and All represent arcs of 90°; a D represents an arc of 113° 28, and D+, though greater, only represents 66° 32'; the arcs zu, a 2, a H, &c, are of 90°; DE represents 23°
-28’ = DAH = inclination of the ecliptic. Ahq to the equator ADQ. It is true, however, that the greatest inequalities are out of the circle ADQB which is properly the projection. If we regard the circle avs]xr as a map of half the terrestrial globe, then rv, wy, we, we, will represent arcs of 90°, though rv will be the only one of othose four which is actually a quadrant. , , , 46. Another inconvenience of this projection, is the - * difficulty of finding the true distance of two points of which we have the projections. Yet, let M and N in the diagram to art. 42, be two such points: produce cM, CN, to m and n respectively; the arc mm will give us McN which is the same as on the sphere. CM and CN are the tangents of the half distances from the pole of the projection. The spherical triangle will give (see chap. vi. equa. 2), cos MN = cos CM cos CN + sin CM sin CN cos C _ (1 – tan” #cM) (l – tan" #cN) + 4tan #cM tanocN cose - (1 + tan” AcM) (1 + tan” #CN) (1 – CM”) (1 – c.N”) + 4cM. CN cos c (1 + CM”) (1 + CN”) Take cM' = cM and draw om'M”, AM” will be equal to the arc represented by C.M. Proceed similarly for cN. Then cos MN may be computed from the above. The third member of the equation is obtained from the second by substituting for cos CM, sin CM, their values in tan #CM, &c. deduced by means of equa. R, chap. iv. 47. The projections here treated serving for the usual purposes of astronomy, we need not enter upon the explication of the other kinds of projection devised by geometers for different purposes. The principal of these is the gnomonic projection, in which the eye is supposed at the centre of the sphere, and the plane of projection a tangent plane to the sphere at any assumed point. All the points within adequate limits have their projections at the extremities of the tangents of their distances from the p. of contact; and those tangents form respectively the same angles as the arcs that measure the several distances from the principal point. In this projection too, a less circle will evidently be projected into an ellipse, a parabola, or a hyperbola, according as the distance of its most remote point is less, equal to, or greater than 90°, from the centre of the plane of projection.
But for a farther developement of these properties, and for the geometrical constructions derived from them, such as want to enter more minutely into this subject may consult Emerson’s Projection of the Sphere, the treatise in Bishop Horsley's Elementary Treatises on Practical Mathematics, or that in the Traité de Topographie, par Puissant.
1. DIALLING, or gnomonics, is the art of drawing on the surface of any given body, whether plane, anular, or curved, a sun-dial, that is, a figure, the dif#. lines of which, when the sun shines, indicate by the shadow of a style or gnomon the time of the day. 2. The general principles which serve as a basis to the theory of dialling, cannot be more aptly illustrated than they have been by Ozanam and Ferguson, in the following contrivance. Suppose a hollow transparent sphere DPRp, of glass, to represent the earth as transparent, and its equator divided into 24 equal parts by so many meridian semicircles a, b, c, d, e, &c, one of which is the geographical meridian of any given place, as London, which it is supposed is at the point a; then if the hour of 12 were marked at the equator, both upon that meridian and the opposite one, and all the rest of the hours in order on the other meridians, those meridians would be the hour circles of
(Zoo London. Because, as o the sun appears to move
round the earth, which is in the centre of the visible heavens, in 24 hours, he will pass from one meridian to another in an hour. Then, if the sphere had an opaque axis, as PEp, terminating in the poles P and p, the shadow of the axis, which is in the same plane with the sun and with each meridian, would fall upon every particular meridian and hour when the sun came to the plane of the opposite meridian, and would consequently show the time at London, and at all other places on the same meridian. If this sphere were cut through the middle by a solid plane ABCD in the rational horizon of London, one half of the axis EP would be above the plane, and the other half below it; and if straight lines were drawn from the centre of the plane to those points where its circumference is cut by the hour circles of the sphere, such lines would be the hour lines of an horizontal dial for London: for the shadow of the axis would fall upon each particular hour line of the dial, when it fell upon the like hour circle of the sphere. If the plane which intersects the sphere be imagined upright, and at the same time to face the meridian of the assumed place, the intersections of the several hour circles with the plane in this position, would give the hour lines of an erect, direct, south, or north dial. And proceeding, in like manner, to contemplate
planes, or other surfaces, any way posited in this hollow terrestrial sphere, we should by the several intersections of these surfaces with the hour cirches, obtain the hour lines for every variety of dial. And since the earth itself compared with its distance from the sun, may, in reference to this branch of inquiry, be regarded as a point, if a small sphere of glass, or a small sphere constituted of a wire axis and wire hour circles, be placed upon any part of the earth's surface, so that its axis be H. to the axis of the earth, and the sphere have ines upon it and planes within it, such as those above described, it will indicate the time of the day as accurately as if it were placed at the earth’s centre, and the earth itself were as transparent as glass. These general notions being premised, it will be proper to annex a few definitions. 3. The plane erected perpendicularly to the face of the dial, and the upper edge of which marks and bounds the shadow, is called the gnomon; the superior edge of the said plane is called the style of the dial, and it is always parallel to the earth’s axis.” 4. The line in which the plane of the gnomon intersects the plane of the dial is denominated the substyle. 5. The angle included between the style and the substyle is called the elevation of the style; in the following formulae it will be denoted by E. In a horizontal dial this is, evidently, equal to the latitude of the place, or = L. 6. While those dials whose planes are parallel to the plane of the horizon are called horizontal dials; such as have their planes perpendicular to the horizon are called erect dials. 7. Those erect dials whose planes are either parallel or perpendicular to the plane of the meridian, are called direct erect dials; they face one or other of the four cardinal points. o 8. All other erect dials are called declining dials. 9. Those dials whose planes are neither parallel nor