SPHERE. 



SPHERE, DOCTRINE OF THE. 



tin u nearly u we plena equal to a great circle : the words greater 

 and tmaUrr would I* more correct. 



The centre of a circular Motion it found by drawing a perpendicular 

 from the centre of the cohere to the plane of the Motion. All Motion* 

 whoM pliarii are parallel bare their centre* on one straight hue, 

 namely, the perpendicular to the plane* which paam through the 

 centre of the inhere. The great circle in such a gyrtoui (cu AB) if 

 called the ;>riMilirr, the common perpendicular (roq) the asii, all the 

 email circles (Diro, KLMN, &c.) parallel*, the extremities of the axis 



(p and Q) pofe, and all great circles pausing through the axis and poles 

 (PCQB, PCQ, FAQ, 4c.) tecondariet. 



By the angle made by two great circles is always understood the 

 angle made by their plaues, which U also that made by their tangents 

 at the point of intersection, and that made by the intersections of the 

 two circles with the third circle to which both are secondary. It is 

 also the angle made by the axes of the two circles. Thus the spherical 

 angle EPF is the angle made by the planes PEQ and PKQ, or the atogle 

 made by tangents to the circles drawn through P, or the angle u o A. 



The angle made by two straight lines drawn from the centre (as OA 

 and OB) is often confounded with the arc (AB) which that angle marks 

 out on the sphere. When this causes any confusion, which at first it 

 will sometimes do, instead of each arc mentioned, read its augle : thus 

 for the arc AB read the "angle subtended by the arc AB" or A OB. 

 Thus when we say that the angle made by two great circles is the arc 

 intercepted between their poles, we mean not to equate the angle to 

 the length of an arc, but to the angle which that arc subtends at the 

 centre. 



The following propositions are essential to the doctrine of the sphere 

 in geography and astronomy; they may be easily proved, and will 

 serve as exercises in the meaning of the preceding terms : 



1. If the poles of a first circle lie upon a second, the poles of the 

 second will also lie upon the first. 



2. If a sphere be made by the revolution of a semicircle round its 

 diameter, the diameter will be an axis, the middle point of the semi- 

 circle will describe the primary, all other points will describe parallels, 

 and every position of the generating circle will be a secondary. 



3. If a point on a sphere be distant from each of two other points 

 (not opposite) by a quadrant of a great circle, the fin* point must be a 

 pole of the great circle which joins the second and third. 



J. The arc of a parallel (as E p) is found from the corresponding arc 

 of the primary (AU) by multiplying the latter by the cosine of the 

 angle (FOA) which is subtended by the intercepted arc (AF) of the 

 secondary. 



5. The surface of the zone intercepted between any two parallels is 

 the rectangle contained under the circumference of the primary and 

 the perpendicular distance between the parallels. 



6. The surface of a lune contained between two great circles is such 

 a proportion of the whole surface of the sphere as the angle contained 

 between the two great circles is of four right angles. 



7. The part of a lune contained within any zone made by two of its 

 parallels (as EF DA) is such a proportion of the whole zone as the angle 

 of the circleH forming the lune is of four right angles. 



We are now to show the method of CO-ORDINATES by which points 

 in the sphere are ascertained, and their relative positions described. 

 Take any great circle cu AB, and choose any point u as an origin, and 

 either direction to be that in which arcs ore measured. Say for in- 

 stance that UA, in preference to uc, shall be the direction in which 

 arcs are measured. The position of any point in this great circle is 

 then ascertained simply by determining its distance from U, since there 

 is a tacit understanding as to the direction in which that distance 

 shall be measured. If we give a name to that distance, be it longitude, 

 right ascension, or any other, the point whose right ascension (it it be 

 right ascension) is 80" means the point which is at 80 distance from r 

 in the direction r A. Again, if we wih to describe any other point, 

 not in the great circle chosen, as F : through r draw a secondary to 

 the great circle (PFAQ), then the point F will be known as soon as A is 



described, in the manner just laid down, and also as soon as the arc A r 

 is given, and the pole towards which it is measured* These tv. 

 ordinatec, VA and AK, when described in magnitude and direction, 

 form a complete description of the position of the point r mi the 

 sphere; and the angles subtended by UA and AF are generally used 

 instead of VA and AF. 



For the first steps of the application of spherical geometry to astro- 

 nomy, see the next article. 



SPHERE, DOCTRINE OF THE. This phrase is generally used 

 to signify the application of the simple geometrical notions iu the 

 article SPHERE to geography and astronomy. It comes between 

 spherical trigonometry and those two sciences, being merely the 

 explanation of the circumstances under which the former is to be 

 applied to the Utter, and the nomenclature which is employed to faci- 

 litate explanation. 



In geography the end is almost gained when a distinct notion is 

 acquired of the meaning of the terms terrestrial latitude and terrestial 

 longitude, generally abbreviated into latitude and longitude. These 

 are only names given to a pair of spherical co-ordinates as described in 

 SPUEBK, the axis of rotation of the earth furnishing the means of pre- 

 scribing the necessary data. The earth revolves round an axis, say p y 

 (see the diagram in SPHERE), and the great circle perpendicular to that 

 axis is the equator (OCA u). An arbitrary point u is chosen as an 

 origin ; and p being the pole which is called north, r A is the east 

 direction and u c the west. The English choose the point r in such a 

 way that the secondary p u passes through the Observatory at Green- 

 wich : the French pay the same compliment to their Observatory at 

 Paris, and so on. The co-ordinate u A (or its angle) is called longitude, 

 east or west according as it falls ; and the co-ordinate A F (or its an^le) 

 is called latitude, north or south according to the pole towards which 

 it is directed. Thus the place F (p u passing through Greri 

 would be described as in longitude u A east of Greenwich, and 

 north latitude ; but if the fundamental secondary, p u, be moved any 

 number of degrees to the east, every east longitude must be diminished 

 and every west longitude increased as much ; and all places which 

 the secondary posses over in the transfer, must have the names of 

 the directions of their longitudes changed, and take for their new 

 longitudes the excesses of the angle of transfer over their former 

 longitudes. Again, longitude might be measured all the way 

 round in one direction : thus o, instead of being described as in u c 

 of west longitude, might be considered as in 360 u c of east 

 longitude. 



There are few problems of much interest connected with geography 

 merely; and it must be remembered that the common terrestrial 

 globe, with its brazen secondary to the equator (called a meridian, very 

 incorrectly, except as meaning that it may be made a meridian to any 

 place), its ecliptic, and figured horizon, is almost as much a represen- 

 tative of the sphere of the heavens as of the earth ; an J the most 

 useful problems are those in which the sphere is used conjointly iu these 

 capacities. But, merely to show what we asserted at first, that the 

 description and nomenclature which are called the doctrine of the sphere 

 are nothing but the connecting link of geography, &c., and spherical 

 trigonometry, let us ask the following question : Given a table of 

 latitudes and longitudes, required the distance between two places 

 mentioned ? Let D and u be the places (see diagram in Sniuii:), then 

 r D is the co-latitude of D, or 90 lat. of D, and P u (on account of M'B 

 south latitude) is 90 + lat. of M ; while the spherical angle, D P M 

 (which is the angle of the arc A c), is, on account of the longitudes 

 being of different names, the sum of the longitudes of o and M. 

 Hence, if D and M be joined by the arc of a great circle, we have given 

 (from the tables) two sides and the angle included, in the spherical 

 triangle D p M. From these data the third side, D M, can be found, in 

 degrees, &c. : convert this into miles, at the rate of 69 miles to a 

 degree (which is accurate enough for the purpose), and the result will 

 be the distance required. 



We now make the passage from the terrestrial to the cr 

 sphere. The latter is a fiction, derived from the impossibility of dis- 

 tinguishing the distances of the heavenly bodies, on which account 

 they all seem at the same distances, on a sphere so great that the 

 earth, its centre, is but a point in comparison. But it must be 

 remembered that the appearances of the heavenly bodies conform 

 themselves to this fiction, so that the development of the consequences 

 of the latter amounts to an explanation of the phenomena of the 

 heavens. And first, the rotation of the earth from west to ea< 

 to the sphere of the heavens an apparent motion from east to west, 

 round an axis which is obtained, by lengthening the axis of the earth. 

 The point of the heavens which answers, for the moment, to tho 

 spectator's position on tho earth, is that point which is directly over 

 his head, or bis :cilh. And since the spectator is not exactly at tho 

 centre of the celestial sphere, we give the following diagram, illustra- 

 tive of the manner in which the effect of this misplacement is destroyed 

 by the largeness of the sphere. 



The eye of the spectator is at E, and his zenith-line is o'z. Tho 

 smaller circle is a section of the earth, and the larger of the sphere of 

 the heavens. The figure is drawn of dimensions so false, that thu 

 sphere of the heavens is represented about as well as a common 

 represents the solar system. Tho HORIZON is the small circle drawn 

 perpendicular to oz through xn ; the altitude of the pole of the 



