ni 



SPHERE, DOCTRINE OF THE. 



SPHERICAL TRIGONOMETRY. 



are perpetually * I ""C < "C their values for any one star. The follow in,- 

 isritii ni will serve to try the reader 1 ! comprehension of these terma : 

 points on the north aide of the meridian are in azimuth 0, on the 

 south aide in azimuth 180 ; the zenith has all azimuths, and every 

 other point of the eatt aide of the prime vertical 90* of azimuth ; the 

 altitude of a star which set* U greateat when it in on the meri.li.in ; 

 the meridian altitude* of a circumpolar (tar are the greatest and leiut 

 of all iU altitudes, and their half turn is always the latitude of the 

 place of observation. [ALTITUDE; AZIMUTH.] 



1 Equatorial Sytttm. KgJtt Atct**iu and Inclination. The pri- 

 mitive circle bore b the equator ; the point of the equator called the 

 vernal equinox (presently described) is the origin, and the direction of 

 the sun's motion from west to east is the direction in which riyht 

 atcenavn is measured. In the diagrams T is not the vernal, but the 

 autumnal equinox, the point opposite to the vernal equinox, conse- 

 quently T has 180* of right ascension, and so have w, v, and all points 

 on the same half of the secondary PWT. The other co-ordinate, 

 dediitatum, is measured on the secondary to the equator north or south 

 according to its direction : thus s has for its right ascension 180* + T o., 

 and q 8 of south declination ; while R has the same right ascension, 

 and q B of north declination. The secondaries to the equator are called 

 kour-cirrlti, and the difference of the right ascensions of two stars is 

 the angle made by their hour-circles: thus the angle q r 5, measured 

 by the arc q 5, is obviously the difference of the right ascensions of the 

 point B and 1. The equator moves with the sphere, so that the right 

 ascension and decimation of a star remain the same, as long as it moves 

 only with the diurnal motion. The right ascension is generally ex- 

 pressed in time, as before described ; and the following assertions will 

 serve for exercise in the meaning of these terms : the sidereal day 

 beginning when the vernal equinox is on the meridian, the right 

 ascension of any star, turned into time, expresses the moment of the 

 sidereal day at which that star will be on the meridian ; when the 

 vernal equinox is on the meridian of Greenwich, the longitude of any 

 place, measured eastwards, is the same as the right ascension of a star 

 which is on the meridian of that place ; the meridian altitude of any 

 star, diminished by its declination (if north), or increased by its 

 declination (if south), U the co-latitude of tho place of observation ; 

 every star which has the same declination as the place of observation 

 has latitude, passes directly over the head of the spectator at that 

 place ; the time of rising of a star, and the time during which it 

 remains above the horizon, depend solely upon the declination, and not 

 at all upon the right ascension. [RIGHT ASCENSION ; DECLINATION.] 



3. Ecliptic System. Celatial Longitude and Latitude. The ecliptic 

 (B T s 6) is the circle which the sun appears to describe in the course of 

 a year, the direction of this orbital motion being from west to east. 

 One half of it is north, the other half south, of the equator ; and the 

 point of the equator in which the ecliptic cuts it, and through which 

 the sun passes when it leaves the southern and enters the northern 

 part of the ecliptic, is the rental equinox, the opposite point being the 

 autumnal equinox. Consequently, T, as drawn, is the autumnal 

 equinox, for motion from west to east, or in the direction B T s, makes 

 the sun pass from the northern to the southern side of the equator. 

 In this system of co-ordinates the ecliptic is the primitive circle, the 

 vernal equinox is the origin, longitude is measured from west to east 

 on the ecliptic, and latitude north or south, as the case may be, is 

 measured on a secondary to the ecliptic drawn through the star. In 

 fact, celestial longitude and latitude are to the ecliptic precisely what 

 right ascension and declination are to the equator. The obliquity of 

 the ecliptic is the angle made by the equator and the ecliptic ; and the 

 secondaries to the ecliptic, drawn through the vernal and autumnal 

 equinoxes, are the equinoctial and solstitial colures. [LONGITUDE AND 

 LATITUDE.] 



A complete understanding of all these terms makes the com]>ivln n- 

 sion of the globe easy, and also the application of spherical trigonometry 

 to those who know the latter science. We now describe the diagrams, 

 in order to point out how such applications are made. The point s is 

 the sun, of course in the ecliptic ; its right ascension is 180 + T Q, its 

 declination q a south, its longitude 180 + T 8, it* latitude 0, its 

 azimuth N I., it* altitude L s, its hour-angle ( a name given to the angle 

 made by the hour-circle of a star with the meridian) s r M, measured 

 by u q. The parallel to the equator CSKC would be the diurnal jath 

 of the sun, if it continued at the point s of the ecliptic ; but as the sun 

 has a slow motion of its own towards K, it is not strictly (though very 

 nearly) correct to say that, for the day in question, the sun continues 

 in the parallel. Hence we may say, without sensible error, that the 

 sun moves over c K during half the night, and through K c during half 

 the day. It rises when at the point K, and the angle K p s, turned into 

 time, shows the sidereal time elapsed since the rising, while the angle 

 a i P M shows the time which is yet to elapse before noon. As to the 

 time of the year, observe that the sun was at the autumnal equinox T 

 on the 21st of September, since which time it has moved over T s, 

 independently of the diurnal rotation of the sphere. We see then 

 what is meant by saying that the diagram represents some morning in 

 October. The use of the globe is thus explained, as far as setting it for 

 any hour and day is concerned. The pole p must first be elevated 



until tl Irvation is equal to the latitude of the place, the sun must 



then bo put in its proper place in the ecliptic for the time of the year, 

 and its hour-angle must then be made to represent the time which in 



wanted of noon, or has elapsed since noon. All this on the globe is 

 done without attending to the distinction of sidereal and solar time, 

 which need hardly be attended to when no greater degree of accuracy 

 is wanted than can be obtained on a globe. We now refer the reader 

 to works on the use of the globes, and shall conclude this article by a 

 few indications of the mode of applying spherical trigonometry. 



To find the time of sunrise, observe that in the spherical triangle 

 r K N, right-angled at N, we have p K given, being 90 + the sun's 

 declination, and also r N, the latitude of the place of observation. 

 Hence the angle K PN can be found, which lining turned into sidereal 

 time, gives a good approximation to the time of sunrise, refraction and 

 the sun's proper motion being neglected. 



Given 8 L the sun's altitude, and the latitude of the place ; required 

 the time of day. In the triangle s z r, we now know z s the sun's co- 

 altitude, 8 P which is 90* + declination, and 7. r the co-latitude of the 

 place. Hence the angle s p z can be found, and thence the time from 

 noon. If s, instead of the sun, were a known star, the question would 

 be solved in the same way, except that the sun's hour-angle is no longer 

 s p z, but that angle increased or diminished by the difference of 

 the right ascensions of the sun and star. 



Two known stars, w and s, are observed to be in the same circle of 

 altitude s w L at a given place ; required the time of day. Here p w 

 and P s, the co-declinations of the stars, are known, and also the angle 

 w p s, which is the difference of their right ascensions ; hence in tin- 

 triangle 8 w p the angle 8 w p can be found, and thence its supplement, 

 the angle z w p. Then, in the triangle w z r, we know the angle 7. w r, 

 PWthe co-declination of the star w, and / p the co- latitude of the 

 place : whence the angle w r z can be found ; and thence, by com- 

 parison of w with the sun, the time of day. 



For the actual applications we must refer to mathematical works on 

 astronomy. 



SPHERICAL ABERRATION'. If a lens or mirror could accom- 

 plish all that we should desire in it, it would refract or reflect rays 

 diverging from or converging towards a point so th.it the directions of 

 the refracted or reflected rays should accurately pass through a point. 

 This however can in general be only approximately effected; and tin- 

 failure of the rays to pass accurately through a point is i 

 aberration. It depends partly on the form of the surfaces, partly on 

 the compound nature of light itself. The former is termed spherical, 

 the latter chromatic aberration. Spherical aberration is of course the 

 only kind which exists in the case of a mirror. For the formula) 

 relating to the spherical aberration of lenses and mirrors, see LENS and 

 SPECULUM. 



SPHERICAL ANGLE. [SPHERICAL TRIGONOMETRY, &c.] 



SPHERICAL EXCKSS. [Si-in:i!ir.n. THI.;..XOMETRY, &c.] 



SPHERICAL TRIANGLE. [SPHERICAL TRIGONOMETRY, *c.] 



SPHERICAL TRIGONOMETRY, SPHERICAL TRIANGLE, 

 SPHERICS. We shall confine ourselves in the present article to such 

 a collection of the properties of a spherical triangle as may be useful 

 for reference, referring for demonstration to the treatise on the subject 

 in the ' Library of Useful Knowledge,' and to that on Geometry; 

 adding to the former nothing but a shorter mode of obtaining Napier's 

 Analogies. 



By a spherical triangle is meant that portion of the sphere which is 

 cut off by three arcs of great circles, each of which cuts the other two 



as A B c. It in now usual, however, to consider the spherical triangle as 

 a sort of repri'.--fiitativ<- of the solid angle formed at the centre of the 

 sphere by the planes AOB, BOC, COA.OS follows : The arcs A B, B c, 

 c A, are the measures of the angles A OB, B o c, c o A, and are used for 

 them : the spherical angles B A c, A c B, c B A are by definition the 

 angles made by the planes BOA and \ o c, A o c and COB, COB and 

 BOA. The sperical triangle then has six parts corresponding in 

 name to the six parts of a plane triangle ; but a tide of it means the 

 angle made by two straight llatt of a solid angle, while an anyle of it 

 refers to the angle made by two planes of the solid angle. 



Throughout this article we shall designate the angles by A, B, c, the 

 sides opposite to them by a, 6, c ; the half sum of the sides by *. 

 And by A', B', c", a', b', c', we mean the supplements of A, B, 4c., so 

 that A + A' = 180, a + a' = 180, &c. No triangle is considered 

 which has either a side or an angle greater than 180*. 



Three circles divide the sphere into eight spherical triangles. Of 

 these four are equal and opposite to the other four, with which they 

 agree in every respect but one [SYMMETRICAL] with which we have 

 nothing hero to do. Of the four which are distinct, if A B c be one, 

 there are three others thus related to it : the first has for its sides 

 a, V, c', and for its angles A, B', c*, ; the second has o', 6, c 1 , for sides, 



