PRACTICAL ASTRONOMY- AND GEODESY: INCLUDING THE PROJECTIONS OF THE SPHERE SPHERICAL TRIGONOMETRY. jpor t&e se of t&e l&opl JMteg GDoIIqp. BY JOHN NARRIEN, F.R.S. & R.A.S. i% PROFESSOR OF MATHEMATICS, ETC. IN THE INSTITUTION LONDON: PRINTED FOR LONGMAN, BROWN, GREEN, AND LONGMANS, PATERNOSTER-ROW. 1845. NOTICE. THE Publishers of this work beg to state that it is private property, protected by the late Copy- right Act, the 5 & 6 Victoria, c. 45. They beg also to state that any person having In his pos- session, within the United Kingdom, for sale or hire, one or more copies printed abroad of any English work protected by the Act referred to, is liable to a penalty, which, in cases affecting their interests, they intend to enforce. The Public are farther informed that the Act 5 & 6 Victoria, c. 47. 8. 24. prohibits the im- portation of all works printed in foreign countries, of which the copyright is not expired. Even single copies, though for the especial use of the importers and marked with their names, are excluded ; and the Customs officers in the different ports are strictly enjoined to carry this regulation into effect. N.B. The above regulations are in force in all the British colonies and dependencies, as well as in the United Kingdom. LONDON : Printed by A. SPOTTISWOODE, New-Street- Square. ADVERTISEMENT. THE following Treatise on Practical Astronomy and Geodesy, including Spherical Trigonometry, is the fifth of a series which is to constitute a General Course of Mathematics for the use of the gentlemen cadets and the officers in the senior department of this Institution. The course, when completed, will comprehend the subjects whose titles are sub- joined: I. Arithmetic and Algebra. * II. Geome- try. * III. Plane Trigonometry with Mensuration, f IV. Analytical Geometry with the Differential and Integral Calculus and the Properties of Conic Sections. V. Practical Astronomy and Geodesy, including Spherical Trigonometry. VI. The Prin- ciples of Mechanics, and VII. Physical Astronomy. Royal Military College, 1844. * Published. f Ready for the press* PREFACE. THE courses of study pursued at the military and naval seminaries of this country have, within a few years, been greatly extended, in order that they might be on a level with the improved state of the sciences relating to those branches of the public service, and also that they might meet the necessity of qualifying officers to conduct the scientific opera- tions which have been undertaken, both at home and abroad, under the authority of the government. An education com- prehending the higher departments of mathematics and na- tural philosophy has also been found necessary for the quali- fication of such as have been, or may be, appointed to superintend Institutions established in the remoter parts of the British empire for the purpose of promoting the advance- ment of physical science, and of preserving or extending its benefits among the European, and, in certain cases, the native inhabitants of the countries. An attempt to supply the want of a treatise on the ele- ments of Practical Astronomy and Geodesy is what is proposed in the present work ; which, it is hoped, will be found useful to the scientific traveller, and to persons employed in the naval or military service of the country who may accompany expeditions to distant regions, where, with the aid of port- able instruments, they may be able to make observations valuable in themselves, and possessing additional importance from their local character. The extent to which the subjects are here carried will probably be found sufficient for the proposed end, and may be useful in preparing the student for the cultivation of the highest branches of astronomical science. A brief notice of the phenomena of the heavens forms the commencement of the work ; but no more of the merely descriptive part of astronomy has been given than is necessary for a right understanding of the subjects to which the pro- cesses employed in determining the elements are applied. A tract on spherical trigonometry constitutes the third chapter ; yi PREFACE. and that tract has been introduced in this 'work because the general theorems are few in number, and have their chief applications in propositions relating to astronomy : the trans- formations which the theorems are made to undergo are, almost always, investigated for the purpose of rendering them convenient in computations ; and, as both the investigations and the examples chosen for illustrating the formulae ge- nerally refer to astronomical subjects, it is evident that faci- lities are afforded and repetitions avoided by comprehending the tract in a work for which it is immediately required. There is given a description of the principal instruments employed in making observations; and, after investigations of the formulae for refraction and parallax, there follow out- lines of the methods by which the elements of the solar, lunar, and planetary orbits are determined : these are succeeded by formulae for computing the apparent displacements of celestial bodies, produced by the actions of the sun, moon, and planets on the earth, and by the motions of the latter. In a chapter on Nautical Astronomy there is given a series of propositions relating to the geographical positions of places on the earth, the determination of local time and the declination of the needle ; and it may be right to state, here, that the examples which illustrate the several propositions are taken from the book of sextant-observations made by the students at the Observatory belonging to the Institution. Each computation is made from a single observation, and the instruments used are graduated so as to give thirds or, at best, quarters of minutes: these circumstances will account for the discre- pancies, generally amounting however to a few seconds only, in the results. After an outline of the methods of computing eclipses of the moon and sun, and the occultations of stars by the moon, there are given formulae and examples for determining ter- restrial longitudes by those phenomena ; and, in the chapter on geodesy, there are given the methods of executing trigo- nometrical surveys for the purpose of determining the figure of the earth, with propositions relating to spheroidal arcs and angles, also the manner of making pendulum experiments for a ike purpose, together with notices of the principal formulae relating to terrestrial magnetism. CONTENTS. CHAPTEE I. THE EARTH. PHENOMENA OF THE CELESTIAL BODIES. FORM OF THE EARTH AND ITS ROTATION ON ITS AXIS. APPA- RENT MOVEMENT OF THE STARS. - REVOLUTION OF THE MOON ABOUT THE EARTH. HYPOTHESIS OF THE EARTH'S ANNUAL MOTION. PHASES OF THE MOON. APPARENT MOVEMENTS OF THE PLANETS. THE CHICLES OF THE SPHERE. Art. Page 1 . Proofs that the surface of the earth is accurately or nearly that of a sphere - - - - - 1 2. Definition of the plane of the horizon, the plane of the meridian,, a meridian line, its north and south points, a vertical line, the zenith and nadir, a vertical plane, the prime vertical, the east and west points - 1 3. Diurnal movements of the stars, as they would appear to a spectator when proceeding from north to south on the earth's surface - - - 2 4. Hypothesis that the earth revolves daily on an axis : posi- tion of that axis - - - - 3 5. The celestial sphere : its assumed rotation - 3 6. Apparent movements of the sun and moon eastward from the stars : inference that the moon revolves monthly about the earth ; and hypothesis that the earth revolves annually about the sun - 3 7. The orbits of the earth and moon are differently inclined to a plane which is perpendicular to the axis of the earth's rotation - 4 8. The axis of the earth's rotation continues, during the annual movement, to be nearly parallel to itself - 4 9. Apparent deviation of the moon from the sun eastward, and her subsequent approach to him - - 5 10. Proof of the globular figure of the moon. Eclipses - 5 11. The apparently independent movements of the planets: the elongations of Mercury and Venus from the sun ; and proof that these planets revolve about the sun within the earth's orbit - - 6 12. The other planets, and the comets, revolve about the sun : the sun and planets revolve on their axes. The Earth, Jupiter, Saturn, and Uranus have satellites - 6* A 4 viii CONTENTS. Art. Page IS. The zodiac and its constellations - 14. Ancient method of finding the magnitude of the earth - 7 15. The sun, the earth, and the planets constitute a particular group of bodies in the universe. The celestial sphere considered as infinitely great with respect to the group 16. Trace of the ecliptic ; its poles : circles of celestial longitude - 8 17. Division of the ecliptic into signs. Direct and retrograde movements defined 18. The equator and circles of declination ; the terrestrial me- ridians : geographical longitude and latitude : celestial longitude and latitude ; right ascension and declination - 9 19. Ecliptical and equatorial systems of co-ordinates - - 10 20. Horizontal systems of co-ordinates ; azimuth and altitude defined 1 21. Amplitude defined - H CHAP. II. PROJECTIONS OF THE SPHERE. NATURE OF THE DIFFERENT PROJECTIONS EMPLOYED IN PRACTICAL ASTRONOMY AND GEOGRAPHY. PROPOSITIONS RELATING TO THE STEREOGRAPHICAL PROJECTION IN PARTICULAR. EXAMPLES OF THE ORTHOGRAPHICAL, GNOMONICAL. GLOBULAR, AND CONICAL PROJECTIONS. MERCATOR'S DEVELOPMENT. 22. The use of diagrams for astronomical investigations - - 12 23. Nature of the stereographical, globular, orthographical, gnomonical, and conical projections of the sphere - 12 24. The primitive circle defined. A projected great circle intersects it in one of its diameters - - - 14 25. (Prop. I.) Circles of the sphere, passing through the eye, are projected in straight lines - - - - 14 26. (Prop. II.) If a circle of the sphere be projected on a plane passing through the centre, the projecting point being on the exterior of the sphere, and the circle not passing through it ; the projected figure is an ellipse, or a circle - 15 Cor. 1. When the projecting point is on the surface, the projected figure is a circle - 16 Cor. 2. When the projecting point is infinitely remote, the projection of an oblique circle is an ellipse - - 17 27. Schol. The orthographical projection of an ellipse, on any plane, is an ellipse - - - 1 7 28. (Prop. III.) If a plane touch a sphere, and from the centre, lines be drawn through the circumference of a small circle perpendicular to the plane, the projection on the plane is an hyperbola - - _ - 18 NOTE. The fifteen following propositions relate to the stereographical projection only. 29- (Prop. IV.) The angle contained between the tangents to two circles of the sphere is equal to the angle contained between the projections of those tangents ' J - . .'. _ 19 CONTENTS. IX Art. Page 30. (Prop. V.) If a circle of the sphere he described about each of the intersections of two circles of the sphere as a pole, at equal distances, the intercepted arcs are equal - - 20 31. (Prop. VJ.) The projected poles of any circle of the sphere, and the centre of the projection of the same circle, are in the representation of a great circle passing through the poles of the primitive and given circle - - 21 32. (Prop. VII.) The radius of the circle constituting the pro- jection of a great circle which does not pass through the projecting point is equal to the secant of its inclination to the primitive - - - - - -21 33. (Prop. VI II.) The distances of the extremities of the diameter of any projected circle from the centre of the primitive are equal to the tangents of half the arcs which measure the least and greatest distances of the circle from that pole of the primitive which is opposite the projecting point - 22 Cor. 1. The projection of a circle parallel to the primitive - 23 Cor 2. The projection of a circle perpendicular to the primitive - - - - - 23 34. (Prop. IX.) The distances of the projected poles of any circle of the sphere from the centre of the primitive circle are equal to the tangent and cotangent of half the inclination of the plane of the circle to the primitive - 24 35. (Prop. X.) To find the poles of a given projected great circle, and the converse - - - 25 36. (Prop. XI.) To find the poles of a given projected small circle, and the converse - 26 37* (Prop. XII.) To describe the projection of a great circle, through two given points - - 27 38. (Prop. XIII.) Through a given point in the circumference of the primitive circle, to describe the projection of a great circle, making with the plane of projection a given angle - 28 39. (Prop. XIV.) Through any given point to describe the pro- jection of a great circle making a given angle with the projection of a given great circle - - 28 40. (Prop. XV.) To describe the projection of a great circle making, with two projected great circles, angles which are given - - 29 41. (Prop. XVI.) To measure an arc of a projected great circle - 30 42. (Prop. XVII.) To measure the angle contained by the pro- jections of two great circles of the sphere - 31 43. (Prop. XVIII.) A line making any constant angles with the meridian, in a stereographical projection on the equator, is a logarithmic spiral - - - 31 44. Projection of a hemisphere orthographically on the plane of the equator - - - - - - 32 45. Projection of a hemisphere orthographically on the plane of a meridian ... - - 33 46. Projection of a portion of a sphere gnomonically on a tangent plane parallel to the equator - ~ - - 33 x CONTENTS. Art. Page 47. Projection of a portion of a sphere gnomonically on a tangent plane perpendicular to the equator - 34 48. Globular projection of the sphere - - 34 49. Conical projection of a spherical zone 50. Cylindrical projection of a zone about the equator - 51. Development of a spherical zone, on the surface of a cone cutting the sphere 52. Conical development of a spherical quadrant 53. Nature of Mercator's development - - 39 54. Investigation of the lengths of the meridional arcs, with an example - - - - 39 55. Application of Mercator's principle to a problem in navigation 41 CHAP. III. SPHERICAL TRIGONOMETRY. DEFINITIONS AND THEOKEMS. 56. The nature of spherical triangles - - 43 57. Manner of expressing their sides and angles - - 43 58. Spheres to which, in Practical Astronomy, the triangles are conceived to belong - - 44 59. Correspondence between the propositions of spherical and plane trigonometry - - 44 60. (Prop. I.) To express one of the angles of a spherical triangle in terms of the three sides - - - - 45 Cors. 1, 2. Deductions from Prop. I., for right-angled tri- angles - - - - - 47 61. (Prop. II.) The sines of the sides of any spherical triangle are to one another as the sines of the opposite angles - 47 62. (Prop. III.) To express one of the sides of a spherical tri- angle in terms of the three angles - 48 Cor. 1. Three deductions from the proposition, for right-angled triangles - - - - - 48 Cor. 2. Formula for the relation between two angles of a spherical triangle, a side adjacent to the angles and a side opposite to one of them - - - 49 63. Explanation of the middle part, and the adjacent and opposite parts, in right-angled spherical triangles. Napier's two rules for the solution of such triangles - - 49 64. Application of Napier's rules in the solution of oblique angled spherical triangles - 50 65. Reason for transforming the first and second of the preceding propositions and the second corollary of the third into formulae in which all the terms are factors or divisors - 52 66. (Prop. IV.) Investigations of formulae for finding an angle of a spherical triangle in terms of its sides - - 52 67. (Prop. V.) Investigations of formulae for determining two of the angles of a spherical triangle when there are given the other angle and the two sides which contain it - 53 CONTENTS. XI Art. Page 68. (Prop. VI.) Investigations of formulae for determining two sides of a spherical triangle when there are given the other side and two adjacent angles - - 56 69. Investigations of formulae for expressing the relations between an angle in any plane and its orthographical projection on a plane inclined to that of the angle - 56 70. Reduction of an arc on any small circle to the corresponding arc on a great circle parallel to it ; and the converse - 58 71. Two great circles making with each other a small angle at their line of section, and from a point in one of these an arc of a great circle being let fall perpendicularly on the other, also from the same point an arc of a small circle perpendicular to both being described ; there are inves- tigated approximative formulae for the distance between the two arcs and the difference of their lengths - - 58 CHAP. IV. DESCRIPTION OF THE INSTRUMENTS EMPLOYED IN PRACTICAL ASTRONOMY. THE SIDEREAL CLOCK. MICROMETER. TRANSIT INSTRUMENT. MURAL CIRCLE. AZIMUTH AND ALTITUDE CHICLE. ZENITH SECTOR. EQUATORIAL INSTRUMENT. COLLIMATOR. REPEATING CIRCLE. REFLECTING INSTRUMENTS. 72. Nature of the observations made for determining the places of celestial bodies - - 60 73. The sidereal clock. Its adjustments for the purpose of in- dicating the commencement of the sidereal day, the right ascension of the mid-heaven, and of a star - - 60 74. Description of the micrometer - - - - 6l 75. Determination of the value of its scale by a terrestrial object - 62 76. The same, by the diameter of the sun and by a transit of the pole-star - - _ - 62 77- The position micrometer - - - . - 63 78. The micrometer microscope - - - 63 79- Manner of finding the value of a run of the micrometer screw 64 80. Manner of obtaining the subdivisions on a circular instrument when a micrometer microscope is used - - - 64 81. Description of a transit telescope - - - - 64 82. The fixed wires at the focus of the object glass, and the man- ner of enlightening them at night - - 65 83. The attached circle for obtaining altitudes approximative^ - 66 84. Description of the spirit level, and the manner of using it in placing the axis of motion horizontal - - - 66 85. Formula for computing the deviation of the axis of motion from a horizontal position ; and the corresponding correc- tion to be applied to the observed time of a star's transit - 68 86. Manner of determining the like deviation of the axis, from transits observed by direct view and by reflexion - 69 xii CONTENTS. Art. Page 87. Manner of bringing the meridional wire to the optical axis by trials - - 69 88. Process for determining the error of collimation by a micro- meter, and the manner of correcting the observed time of a transit on account of that error - - 69 89. Method of ascertaining that the meridional wire and those which are parallel to it are in vertical positions, and that the wire at right angles to them is horizontal - 70 90. Use of two, or a greater number of wires parallel to the me- ridional wire - - - - - -71 91. Manner of obtaining the correct time of a star's transit at the mean wire, when the transits have not been observed at all the wires - - - - 71 92. Formula for determining the distance of any wire from the mean wire, when the transit of a star near the pole is used 72 93. Correction to be applied in finding the time of transit at the mean wire, when a planet is observed - - 72 94-. Method of placing a transit telescope very near the plane of the meridian - - 73 95. Investigation of a formula for finding the azimuthal deviation of a transit telescope from the meridian, by stars differing considerably in altitude - - - - 73 96. The same, by transits of circumpolar stars - - 75 97. Description of a mural circle - - - 76 98. Methods of verifying the horizon tality of the axis, the line of collimation, and the position of the circle with respect to the meridian - - - - - - 76 99- Process for finding the polar point on the circle - - 77 100. Processes for finding the horizontal point - - - 77 101. Formula for obtaining the correction of a star's observed declination, when, at the time of the observation, the star is not on the meridian wire - - 78 102. Manner of correcting the observed declination of the moon when her disk is not wholly enlightened - - 78 103. The mural circles at Greenwich and at Gottingen - - 79 104. Description of the azimuth and altitude circle - - 79 105. Manner of verifying the positions of the wires - - 81 106. Methods of finding the error of collimation in azimuth and in altitude - - - _ _ -81 107. Description of the zenith sector - - _ - 83 108. Improved zenith sector by the Astronomer Royal - - 83 109. Manner of making the observations - - -84 110. Description of a fixed equatorial instrument - - 85 111. A portable equatorial - - . _ - 86 J 12. Method of finding the error of the polar axis in altitude and in azimuth - - _ _ _ - 87 1 13. Method of determining the error of collimation - - 88 114. Method of making the axis of the declination circle perpen- dicular to the polar axis _ - 88 115. Adjustment of the index of the equatorial circle - - 89 CONTENTS. Xlll Art. Page 116. Employment of the instrument in finding differences of right ascension and declination - 89 117- Adjustment for observing a star during day -light - - 90 118. Smeaton's equatorial - - 90 119. Employment of the micrometer with an equatorial,, in mea- suring angles of position, and small angular distances - 91 120. Manner of deducing from thence the differences of declina- tion and right ascension between two stars, and between the moon and a star - - - - - 92 121. Bessel's heliometer - - 93 122. Explanation of the terms north following, south following, north preceding and south preceding - 94* 123. Description of the floating collimator - - 9^ 124. Manner of using the horizontal and vertical collimators - 95 125. The spirit level collimator, and manner of using it for find- ing the error of collimation in a transit telescope or circle 95 126. Description of the repeating circle - 96 127. Manner of using the instrument in observing altitudes or zenith distances - 96 128. Manner of observing horizontal and oblique angles - -97 129. Hadley's reflecting octant, &c. - - 98 130. Manner in which, after two reflexions of light from an object, the image of that object may be made to coincide with that of an object seen by direct view - - - 98 131. The parallax of a reflecting instrument - -99 132. Proof that the angle between the first and last directions of a pencil of light reflected from two mirrors is equal to twice the inclination of the mirrors - - 99 133. Employment of the visible horizon, and the image of a celestial body reflected from quicksilver, as objects in observing altitudes at sea and on land, respectively - 99 134. Advantages of a reflecting circle - - 100 135. Adjustments of the index, and horizon glasses on a reflecting instrument - - - -- -101 136. Manner of finding the index error of such instrument - 101 137. Manner of placing the line of collimation parallel to the plane of the instrument - - - - 101 138. Captain Fitzroy's sextant - 101 139. Captain Beechey's sextant - 102 140. Reflecting circles having the property of repetition - 103 CHAP. V. REFRACTION: LATITUDE OF A STATION: PARALLAX. 141. Cause of the refraction of light : it takes place in a vertical plane 1 05 142. Refraction varies with the tangent of the zenith distance, as- suming the earth to be a plane and the upper surface of the atmosphere parallel to it - - - 106 143. Coefficient of refraction from experiments on the refractive power of air - - - - - - 106 xiv CONTENTS. Art. Pa*e 144. Formula of Bradley: values of the constant, according to Biot, Groombridge, and Atkinson 145. Manner of determining the refraction by circumpolar stars 107 Table of refractions 108,109 146. Refraction in high northern latitudes - 110 147. Formulae for the variations of polar distance and right ascen- sion in consequence of refraction - 110 148. Cause of the elliptical forms assumed by the disks of the sun and moon when near the horizon - 111 149. Proof that the decrements of the oblique diameters vary with the squares of the sines of their inclinations to the horizon 11 1 Table of the decrements - 112 150. The latitude of a station found by circumpolar stars - 1 12 151. Nautical and geocentric latitude explained - - 113 152. Formula of reduction from one kind of latitude to the other 113 153. Parallax defined - 114 154. Investigation 'of a formula for the parallax in altitude in terms of the apparent altitude - - 114 155. Formula for the parallax in altitude in terms of the true al- titude - 115 156. The decrements of the earth's semidiameters vary with the square of the sine of the latitude - 116 1 57. The difference of the sines of the horizontal parallaxes vary in like manner. Formula for the sine of the horizontal paral- lax at any station ; and a table of the decrements of the moon's horizontal parallaxes from the equator to the pole 117 158. Formula for the geocentric parallax in altitude at any station 118 159- The relative parallax of the moon or a planet - - 118 160. Method of finding the horizontal parallax of the moon or a planet by observation. The tangents of the horizontal parallaxes for different celestial bodies vary inversely with the distances of the bodies from the earth - 119 161. Investigation of the parallax of a celestial body, in right as- cension - 120 162. Investigation of the parallax in declination - 122 163. Formula for the apparent augmentation of the angle sub- tended by the moon's semidiameter when above the ho- rizon ; and a table of the augmentations - 1 24 164. The dip, or angular depression of the horizon - 126 165. Terrestrial refraction and its effects on that depression - 126 166. Determination of the dip when affected by refraction - 127 CHAP. VI. DETERMINATION OF THE EQUINOCTIAL POINTS AND THE OBLIQUITY OF THE ECLIPTIC BY OBSERVATION. 167. Manner of finding the declinations of celestial bodies by ob- servation - _ _ _ - 129 168. The declinations of celestial bodies are subject to variations 129 CONTENTS. XV Art. Page l69 The sun's declination goes through its changes during a year 129 170. The greatest observed declination of the sun is an ap- proximate value of the obliquity of the ecliptic - - 130 171. Manner of finding the sun's daily motion in right ascension from observation - - - 130 172. Manner of finding approximative^, by simple proportion, the instant that the sun is in the equinoctial point ; and of adjusting the sidereal clock - - 131 173. Determination of the instant that the sun is in the equi- noctial point, by trigonometry '''-' - - 131 174-. Precession of the equinoxes : lengths of the tropical and si- dereal years - - - 132 175. Manner of finding the obliquity of the ecliptic by computa- tion and by observation : its mean value and annual di- minution - - - - - 133 CHAP. VII, TRANSFORMATION OF THE CO-ORDINATES OF CELESTIAL BODIES FROM ONE SYSTEM TO ANOTHER. 176. Manner of indicating the position of a point in space by rectangular co-ordinates - - - 135 177. Designations of the rectangular co-ordinates of any point - 135 178. Transformation, when two of the co-ordinate planes turn about their line of section, the latter remaining fixed - 136 179' Application of spherical trigonometry in the case just men- tioned - 136 180. Transformation, when one of the planes only, turns on the line of its intersection with another - - 137 181. Application of spherical trigonometry in the last case - 138 182. Transformation, when two of the planes turn together, one of them on its intersection with the third, and both of them by a movement of this line of section about the centre of the co-ordinates in that third plane - - 140 183. Manner of finding an equation containing the rectangular co-ordinates of any point in a plane oblique to three rec- tangular co-ordinate planes - - 140 184. Transformation, when the centre is moved in one of the planes, the two other planes continuing parallel to their original positions _ - - 142 185. Investigation, from the co-ordinates of three points in a given plane, of the inclination of that plane to one of the co- ordinate planes - 143 186. Investigation, from the same data, of the position of the line in which the given plane cuts one of the co-ordinate planes - - 144 187. Processes for finding, from the observed azimuth and altitude of a celestial body at a given time, its right ascension and declination, and the converse - - - - 144 XVI CONTENTS. Art. Page 188. Processes for finding, from the like data, the latitude and longitude of a celestial body ; and the converse - - 146 189- Manner of computing the right ascension and declination of the sun, having his longitude and the obliquity of the ecliptic ; and the converse - .*;: - 147 190. Investigation of a formula for computing the difference between the sun's longitude and right ascension - 148 1.91. Manner of obtaining the longitude and latitude of the moon, a planet, or a star, when the right ascension and declination are given ; and the converse - - - - 151 CHAP. VIII. THE ORBIT OF THE EARTH. ITS FiaURE SHOWN TO BE ELLIPTICAL. SITUATION AND MOVE- MENT OF THE PERIHELION POINT. THE MEAN, TRUE, AND EXCENTRIC ANOMALIES. EQUATION OF THE CENTRE. 192. Figure of the earth's orbit about the sun similar to that which the sun may be supposed to describe about the earth - 152 193. The apparent velocity of the sun when greatest and least; the least and greatest angles subtended by the sun's diameter ; and the ratio between the least and greatest distances of the sun from the earth - _ - 152 Ip4. Approximation to the figure of the earth's orbit by a graphic construction. The orbit elliptical. The areas described by the radii vectores proportional to the times - - 1 53 195. The line of apsides - - _ _ - 153 196. The angular velocity in an ellipse varies inversely with the square of the distance _ - _ - 154 197. Manner of determining the instants when the sun's longitudes differ by exactly 180 degrees - - _ - 154 198. The progression of the perigeum - . - 154 199- The length of the anomalistic year . - _ 155 200. Method of determining the perihelion point of the earth's orbit and the instant when the earth is in that point - 155 201. Manner of determining the excentricity approximative^ - 156 202. The mean, true, and ex centric anomalies - - 156 203. Investigation of the mean, in terms of the true anomaly - 157 204. Investigation of the true, in terms of the excentric anomaly ] 58 205. Investigation of the radius vector in terms of the excentric and true anomalies - - _ _ - 158 206. Equation of the centre, and determination of its greatest value for the earth's orbit - . . -159 207. Determination of the excentricity of the orbit in terms of the greatest equation of the centre > _ ifio 208. Nature of the solar tables - . . " 162 CONTENTS. Xvil CHAP. IX. THE ORBIT OF THE MOON. THE FIGURE OF THE MOON'S ORBIT. PERIODICAL TIMES OF HER REVOLUTIONS. THE PRINCIPAL INEQUALITIES OF HER MOTION. HER DISTANCE FROM THE EARTH EMPLOYED TO FIND AP- PROXIMATIVELY THE DISTANCE OF THE LATTER FROM THE SUN. Art. Page 209. Manner of determining by observation the nodes of the moon's orbit, and its obliquity to the ecliptic - - 164 210. Process for determining the obliquity of the moon's orbit, and the position of the line of nodes, from two observed right ascensions and declinations - 165 211. The obliquity of the moon's orbit to the ecliptic is variable, and her nodes have retrograde motions - - 166 212. Manner of determining the angular movement of the moon in her orbit - - - 166 213. Indication of the means employed in finding her distances from the earth, the figure of her orbit, and the equation of the centre or first inequality - 167 214. Manner of finding the time of a synodical revolution of the moon by a comparison of eclipses - 16? 215. Her mean motion found to be accelerated - - - 168 216. Determinations of the moon's sidereal and tropical revolutions 168 217. Time in which her nodes perform a revolution - - l6'9 218. Duration of an anomalistic revolution ; and determination of her mean daily tropical movement - - - 169 219 The moon's movements subject to considerable inequalities - 170 220. The moon's evection, or second inequality - 170 221. The moon's variation, or third inequality - - - 172 222. The moon's annual equation, or fourth inequality - - 172 223. Variations of the moon's latitude and radius vector. The constant of the paraUax - - 172 224. Elementary method of finding an approximation to the earth's distance from the sun, that of the moon from the earth being given - - 173 CHAP. X. APPARENT DISPLACEMENTS OF THE CELESTIAL BODIES ARISING FROM THE FIGURE AND MOTION OF THE EARTH. THE EFFECTS OF PRECESSION, ABERRATION, AND NUTATION. 225. Annual increase in the longitudes of fixed stars, and move- ment of the equinoctial points - - - 174 226. Investigation of a formula for the precession in declination - 174 227. Investigation of a formula for the precession in right as- cension - - - - - -176 228. Discovery of the aberration of light - - 176 a XV111 CONTENTS. Art. Pa * e 229. Manner in which the phenomenon takes place - 177 230. Maximum value of the aberration depending on the motion of the earth in its orbit 231. Explanation of aberration in longitude and latitude - 179 232. Investigation of a formula for aberration in longitude - 179 233. Formula for aberration in latitude - 180 234. Formula for aberration in right ascension - - 180 235. Formula for aberration in declination - - - 181 236. Effect of aberration on the apparent place of the sun - 182 237. The diurnal aberration of light - - 183 238. Discovery of nutation - - 184 239. Explanation of the phenomenon so called : its apparent de- pendence on the place of the moon's node - - 184 240. Nature of the path described by the revolving pole - - 185 241. Investigation of a formula for the lunar equation of preces- sion, and of obliquity - - - - - 186' 242. Formula for the equation of the equinoctial points in right ascension - - - -187 243. Formula for the lunar nutation in polar distance - - 187 244. Formula for the lunar nutation in right ascension - - 188 245. Solar nutation - - - - - -188 CHAP. XL THE ORBITS OF PLANETS AND THEIR SATELLITES. THE ELEMENTS OF PLANETARY ORBITS. VARIABILITY OP THE ELEMENTS. KEPLER'S THREE LAWS. PROCESSES FOR FINDING THE TIME OF A PLANET'S REVOLUTION ABOUT THE SUN. FOR- MTJLJS FOR THE MEAN MOTIONS, ANGULAR VELOCITIES AND TIMES OF DESCRIBING SECTORAL AREAS IN ELLIPTICAL ORBITS. MODI- FICATIONS OF KEPLER'S LAWS FOR BODIES MOVING IN PARA- BOLICAL ORBITS. MOTIONS OF THE SATELLITES OF JUPITER, SATURN, AND URANUS. IMMERSIONS ETC. OF JUPITER*S SATEL- LITES. THE ORBITS OF SATELLITES ARE INCLINED TO THE ECLDPTIC. SATURN'S RING. 246. Method of finding approximative^ the distance of a planet from the earth - - - - 189 247. Manner of determining the trace of a planet's visible path - 190 248. The elements of a planet's orbit - - 191 249. Method of finding the longitude of a planet's node. Proof that the sun is near the centre of a planet's orbit - 191 250. Manner of finding the inclination of a planet's orbit to the ecliptic - - 192 251. Manner of finding a planet's mean motion by observation - 192 252. Manner of determining, by rectangular co-ordinates, a planet's radius vector, its heliocentric distance from a node, and its heliocentric latitude - 193 253. Manner of determining the same elements from observations at conjunction and opposition - - - - 194 CONTENTS. xix Art. Page 254. Kepler's three laws stated as results of observation - - 195 255. The semi-transverse axis of a planet's orbit found from those laws, the time of a sidereal revolution about the sun being given - 195 256. The time of a revolution about the sun found by successive approximations, from two or more longitudes of the sun and as many geocentric longitudes of the planet - - 196 257. The place of the perihelion, the semi- trans verse axis and the excentricity found by means of three given radii vectores with the heliocentric distances from a node - - 197 258. Method of finding the mean diurnal motion of a planet in an elliptical orbit - 198 259- The angular velocity in different parts of the same orbit varies inversely as the square of the radius vector - 198 260. Investigation of the sectoral area described about the focus by the radius vector of an ellipse - 199 261. Time in which the radius vector of a planet describes about the sun an angle equal to a given anomaly - - 201 262. The laws of Kepler require modification for bodies moving in parabolical orbits - - - - - 201 263. In parabolical orbits the squares of the times of describing equal angles, reckoned from the perihelion, vary as the cubes of the perihelion distances - - - 201 264. In elliptical orbits the sectoral areas described in equal times vary as the square roots of the parameters ; and in para- bolical orbits such areas vary as the square roots of the perihelion distances - - 203 265. The sectoral areas described in equal times in a circle and in a parabola (the perihelion distance in the latter being equal to the radius of the circle) are to one another as 1 to v"2 - 203 266. Investigation of the sectoral area described about the focus by the radius vector of a parabola - - 204 267. Time in which the radius vector of a parabola describes about the focus a sectoral area corresponding to a given anomaly 204 268. In any parabola the angular velocity varies inversely as the square as the radius vector - - - -205 269. Nature of the observations to be made for determining the elements of a comet's orbit - - 205 270. The satellites of Jupiter, Saturn,, and Uranus - 206 271. Visible motions of Jupiter's satellites - 206 272. Their disappearances behind the planet, or in its shadow, and their subsequent emersions - 207 273. The satellites of Jupiter and Saturn revolve about their primaries from west to east. Their orbits inclined to the ecliptic - - 208 274. Manner of finding the times of revolution about the primary planet - 208 275. Manner of finding the inclinations of the orbits, and the places of the nodes - - 209 a 2 XX CONTENTS. Art. Page 276. Manner of finding the inequalities of their motions 277. The satellites of the planets revolve on their axes in the time of one revolution about their primaries - - 210 278. Uncertainty in estimating the instants of immersion and emersion - - - - - - 210 279- Saturn's ring, its figure and position : times when it becomes invisible - - - - 211 280. Manner of finding the inclination of the ring - 211 281. Manner of finding the positions of its line of nodes - 212 282. Romer's hypothesis concerning the progressive motion of light - - 212 CHAP. XII. ROTATION OF THE SUN ON ITS AXIS, AND THE LIBRA- TIONS OF THE MOON. 283. Apparent paths of the solar spots,, and the manner of de- termining those paths - - 214 284. Method of finding the geocentric longitude and latitude of a solar spot - - 215 285. The inclination of the plane of the sun's equator to that of the ecliptic determined by rectangular co ordinates - 215 286. Determination of the time in which the sun revolves on its axis - - 216 287. Appearance of the moon's surface - 217 288. Nature of her librations - 217 289 Manner of finding the positions of the moon's spots and of her equator - _ .. - - 218 290 Process for determining the height of the lunar mountains - 219 CHAP. XIII. THE FIXED STARS. REDUCTION OF THE MEAN TO THE APPARENT PLACES PROPER MOTIONS. ANNUAL PARALLAX. 291. Formulae for reducing the mean to the apparent places of the fixed stars, and the converse - - - - 220 292. Certain stars have proper motions . - - 221 293. The apparent magnitudes of stars, and their variability - 221 ?9^. Certain stars are double, and have revolving motions - 222 295. Manner of determining approximatively the orbits of re- volving stars --____ 223 296. Investigation of the inclinations of the apparent to the true orbits --._._ 224 297. The parallaxes of the fixed stars uncertain - - 227 298. Manner of determining that parallax by meridional observa- tions ----._ 228 299- Method employed by Bessel to ascertain the parallax by double stars - . _ 231 CONTENTS. XXI Art. Page 300. Expression of the parallax of a star in terms of its maximum value - 231 301. Investigation of the parallax in a line joining the two which constitute a double star - - 232 302. Bessel's determination of the constant of parallax for 61 Cygni - - - 233 CHAP. XIV. TIME. SIDEREAL, SOLAR, LUNAR, AND PLANETARY DAYS. EQUATION OF TIME. EQUINOCTIAL TIME. REDUCTIONS OF ASTRONOMICAL ELEMENTS TO THEIR VALUES FOR GIVEN TIMES. HOUR ANGLES. 303. Value of a sidereal day, and definition of sidereal time - 234? 304. Error and rate of a sidereal clock found by the observed transits of stars - - 2 85 305. Apparent solar day defined ; its length variable - - 235 306. Difference between a mean solar, and a mean sidereal day - 236 307- Lengths of a lunar, and of a planetary day - 237 308. Reduction of mean sidereal, to mean solar time, and the converse ------ 237 309. Reduction of sidereal time at Greenwich mean noon to sidereal time at mean noon under a different meridian - 238 310. Equation of time defined. Formula for its value - - 238 311. Equinoctial time - - 240 312. Manner of finding the mean time at which the sun or a fixed star culminates - - - - - 241 313. Manner of finding the time at which a planet or the moon culminates - - - _ 7 - - 242 314. Manner of finding the angle at the pole between the horary circles passing through the places of the sun, a fixed star, the moon and a planet, at two times of observation, the interval being given in solar time - 244 315. Reduction to solar time, of the angle at the pole between the meridian of a station and a horary circle passing through the sun, a fixed star, the moon or a planet ; with an example - 245 CHAP. XY. INTERPOLATIONS PRECISION OF OBSERVATIONS. 316. Reduction of astronomical elements in the Nautical Almanac to their values for a given time, and at a given station, by simple proportions, or for first differences - - 248 317. Investigation of a formula for first, second, and third dif- ferences ; with an example for finding the moon's latitude 249 318. Horary motions found by first and second differences - 251 319. Formula for several interpolations between two values of an element ___--- 252 xxii CONTENTS. Art. Pa 8 e 320. The instant of witnessing a phenomenon uncertain - - 253 321. Personal equation - 253 322. Signification of weight - - 253 323. Formula; for the limits of probable error in a mean result of observations - - - - 254 324. Formula for the degree of precision in an average - - 255 325. Relation between precision and weight - - 255 326'. Weight due to the sum or difference of two observed quan- tities - - - 256 327. Manner of finding the relative values of instruments or observations - - - - - 256 328. The most probable values of elements are obtained by com- bining the equations containing the results of many ob- servations _----_ 257 329. Method of grouping equations advantageously for the deter- mination of correct results - - - 258 330. Method of least squares - - 260 CHAP. XVI. NAUTICAL ASTRONOMY. PROBLEMS FOR DETERMINING THE GEOGRAPHICAL POSITION OF A SHIP OR STATION, THE LOCAL TIME, AND THE DECLINATION OF THE MAGNETIC NEEDLE. 331. Importance of celestial observations at sea - - 263 332. The data and objects of research in nautical astronomy - 263 333. Errors which may exist in the observations ; with examples illustrating the manner of correcting the observed altitudes and the declinations of the sun, moon, &c. - - 264 334. (Prob. I.) To find the latitude of a ship or station by means of an observed altitude of the sun when on the meridian. Examples for the sun, the moon, and a planet - 266 335. Method of finding the latitude of a ship by the observed time in which the sun rises or sets - - 268 336. Method of finding the latitude in the arctic regions by meridional altitudes above and below the pole - - 269 337. (Prob. II.) To find the hour of the day by an observed altitude of the sun ; the latitude of the station and the sun's declination being given - - - - 270 338. Method of determining the hour of the night by a fixed star, and by a planet, with examples - - - 2? 2 339. (Prob. III.) To investigate a relation between a small varia- tion in the altitude of a celestial body and the correspond- ing variation of the hour angle - 276 340. (Prob. IV.) By an altitude of the sun, a fixed star or a planet, to compute its azimuth and determine the variation or declination of the magnetic needle ; the latitude of the station and the sun's declination being given. Example, from an observed altitude of the sun - - 277 341. Method of finding the direction of a meridian line on the CONTENTS. XX111 Art. Page ground by the azimuths of the sun and of a fixed ter- restrial object - 279 342. (Prob. V.) Having the latitude of a station, the day of the month, &c. ; to 'find the sun's amplitude - - 280 343. Method of finding the declination of the needle, and the hour of sun-rise, by the amplitude of the sun ; with an example 281 344. Method of finding the effects of iron on the compass of a ship, Application of Barlow's plate - - 283 345. (Prob. VI.) To determine the hour of the day and the error of the watch by equal altitudes of the sun - 285 346. Investigation of the change in the hour angle caused by a variation of the sun's declination - 287 347. Investigation of the change in the sun's azimuth caused by a variation of his declination j and a method of finding the direction of a meridian line by equal altitudes of the sun with the corresponding azimuths. Example - - 288 348. Method of finding the time of apparent noon from equal al- titudes of the sun, the ship changing its place in the in- terval between the observations - 29 1 349. Method of finding the error of a watch by equal altitudes of a fixed star or planet on the same or on different nights - 292 350. (Prob. VII.) To find the latitude of a station by means of an altitude of the sun observed in the morning or afternoon, or by the altitude of a star ; with examples - 293 351. Investigation of a formula for the difference between the ob- served and meridional zenith distances of the sun, the latter being near the meridian ; with an example - 296 352. Formula for determining the latitude of a station by an al- titude of the pole star ; with an example - 298 353. Littrow's formula for finding the latitude of a station by three altitudes of the sun or a star near the meridian ; with an example - 299 354. (Prob. VIII.) To find the latitude of a station having the al- titudes of the sun observed at two times in one day ; with an example* - - 300 355. Method of correcting at sea an observed zenith distance of the sun on account of the change in the ship's place in the interval between the observations - 303 356. Conditions favourable to the accuracy of the computed latitude in this problem - 304 357* Methods of determining the latitude by two altitudes of a celestial body with the interval in azimuth ; and by simul- taneous altitudes of two fixed stars . - - 304 358. (Prob. IX.) To find the altitude of the sun, having the latitude of the station and the hour of the day ; with examples in illustration when the body observed is the sun, a star, and the moon - - - 307 359- Principles on which are founded the methods of finding terrestrial longitudes - 310 * This example has been accidently placed at the end of art. 357. xxiv CONTENTS. Art. p age 360. (Prob. X.) To find the longitude of a station or ship by means of an observed angular distance between the moon and sun, or between the moon and a star ; with an example - 311 361. The nature and use of proportional logarithms * '-' - 314 362. Method of finding the longitude by the moon's hour angle - 315 363. Employment of chronometers for finding the longitudes of ships or stations - - 316 364. Methods of finding the error and rate of a chronometer - 3l6 365. Method of finding the longitude of a station by observed transits of the moon over the meridian - - - 318 366. Method by observed culminations of the moon and certain stars - - 319 367. Method by fire signals - - 320 CHAP. XVII. ECLIPSES. OUTLINES OF THE METHODS OF COMPUTING THE OCCURRENCE OF THE PRINCIPAL PHENOMENA RELATING TO AN ECLIPSE OF THE MOON, AN ECLIPSE OF THE SUN FOR A PARTICULAR PLACE, THE OCCULTATION OF A STAR OR PLANET BY THE MOON, AND THE TRANSITS OF MERCURY AND VENUS OVER THE SUN'S DISK. THE LONGITUDES OF STATIONS FOUND BY ECLIPSES AND OC- CULTATIONS. 368. Formula? for computing the angular semidiameter of the earth's shadow and penumbra - - 324 36'9. Method of finding the time of the commencement, the greatest phase and the end of an eclipse of the moon - 326 370. Method of finding the hour angle of the sun and moon at the instant of true conjunction in right ascension - - 328 371. Process of finding the approximate Greenwich time of the apparent conjunction of the sun and moon in right as- cension, at the station. Elements, from the Nautical Almanac, &c. - 328 372. Method of finding the difference between the apparent de- clinations of the sun and moon at the approximate time of apparent conjunction in right ascension - 331 373. Formulae for the apparent relative horary motions ; and method of finding the time of commencement, greatest phase and end of an eclipse of the sun - 332 374. Conditions under which an occultation of a star by the moon may take place - 333 375. Approximate Greenwich time of apparent conjunction of the moon and a star in right ascension at the station - 333 376. Formulae for the horary motions of the moon and a star ; and method of finding the times of the phenomena - - 335 377- Elements to be taken from the Nautical Almanac for the purpose of finding the times at which Mercury and Venus enter upon, or quit the sun's disk - 336 CONTENTS. XXV Art. Page 378. Process for finding the instants of ingress and egress for a spectator at the centre of the earth - 337 379- The parallaxes of the sun and planet being given ; to compute the instants of ingress and egress for a spectator at a given station - - 338 380. The instants of ingress and egress being given ; to compute the parallaxes of the sun and planet - 339 381. Use of observed eclipses of the sun - 340 382. Process for determining the longitude of a station from an observed eclipse of the sun ; with an example - - 340 383. Process for finding the longitude of a station from an oc- cultatioa of a fixed star or planet by the moon - - 345 384. Method of finding the longitude of a station by observed im- mersions or emersions of Jupiter's satellites - - 348 CHAP. XVIII. GEODESY. METHOD OF CONDUCTING A GEODETICAL SURVEY. MEASUREMENT OF A BASE. FORMULAE FOR VERIFYING THE OBSERVED ANGLES AND COMPUTING THE SIDES OF THE TRIANGLES. MANNER OF DETERMINING THE POSITION, AND COMPUTING THE LENGTH OF A GEODETICAL ARC. 'PROPOSITIONS RELATING TO THE VALUES OF TERRESTRIAL ARCS ON THE SUPPOSITION THAT THE EARTH IS A SPHEROID OF REVOLUTION. THE EMPLOYMENT OF PENDULUMS TO DETERMINE THE FIGURE OF THE EARTH. INSTRUMENTS USED IN FINDING THE ELEMENTS OF TERRESTRIAL MAGNETISM. FORMULAE FOR COMPUTING THOSE ELEMENTS AND THEIR VARI- ATIONS. EQUATIONS OF CONDITION EXEMPLIFIED. 385. Probability that the earth's figure is not that of a regular solid - 350 386. Certainty that it differs little from a sphere or a spheroid of revolution - - 350 387 The geodetic line or arc of shortest distance on the earth defined - 351 388. Terrestrial azimuth defined. The normals passing through two points near one another may be supposed to be in one plane - - 351 389. The first notions concerning the figure of the earth erroneous 352 390. Brief outline of the method of conducting a geodetical survey, and of determining the figure of the earth - 353 391. Details of the process employed in measuring a base - 355 392. Reduction of the base to the level of the sea - 357 393. Advantageous positions for the stations - - 358 394. Advantages of indirect formulae for the reductions - - 358 395. Formula for a reduction to the centre of a station - - 359 396. Formula for verifying the observed angles of a triangle on the surface of the earth, supposed to be a sphere - - 360 b XXVI CONTENTS. Art. Pa 8 e 397. Formula for reducing an observed angle to the plane of the horizon - -36 398. Methods of computing the sides of terrestrial triangles - 363 399. Formulae for computing a side of a triangle in feet or in seconds - - - - - 363 400. Formula for the reduction of a spherical arc to its chord - 364 401. Rule for the reduction of a spherical angle to the angle between the chords of its sides - - 365 402. Theorem of Legendre for computing the sides of terres- trial triangles - - 365 403. Manner of determining the position of a meridian line by observations of the pole-star - - 367 404. Rule for computing the azimuth of the pole-star at its greatest elongation from the meridian - - 368 405. Method of determining the azimuth of a terrestrial object - 368 406. Determination of a meridional arc by computing the in- tervals on it between the sides of the triangles - - 369 407- Method of reducing computed portions of the meridian to the surfaces of the different plane triangles ; and outline of the process for determining a meridional arc by com- puting the intervals between the perpendiculars let fall on it from the stations - 370 408. Processes for measuring a degree of a great circle perpendi- cular to the meridian, and a degree on a parallel of geo- graphical latitude _ - 373 409. Method of finding the latitude of a station by observed transits of a star near the prime vertical - 376 410. Investigations of formulae for determining the differences between the latitudes and longitudes of two stations - 378 411. Proof, from the lengths of the degrees of latitude, that the earth is compressed at the poles - - - 381 NOTE. The following propositions relate to the values of terrestrial arcs on the supposition that the earth is a spheroid of revolution. 412. (Prop. I.) Every section of a spheroid of revolution when made by a plane oblique to the equator is an ellipse - 382 413. (Prop. II.) The excess of the angles of a terrestrial triangle above two right angles is, very nearly, the same whether the earth be a spheroid or a sphere - - - 383 414. (Prop. III.) To investigate the radius of curvature for a vertical section of a spheroid - - - - 385 415. (Prop, IV.) To investigate the ratio between the earth's axes from the measured lengths of two degrees on a me- ridian . 387 416. (Prop. V.) To investigate the law of the increase of the degrees of latitude from the equator towards the poles - 388 417- (Prop. VI.) To determine the radius, and length of an arc on any parallel of terrestrial latitude - - - 388 418. (Prop. VII.) To find the ratio between the earth's axes, from the measured lengths of a degree on the meridian and on a parallel of latitude - - - - 389 CONTENTS. XXV11 Art. Page 419. (Prop. VIII.) To find the distance in feet, on an elliptical meridian,, between a vertical arc perpendicular to the me- ridian and a parallel of latitude, both passing through a given point. Also, to find the difference in feet between the vertical arc and the corresponding portion of the parallel circle - 390 420. (Prop. IX.) To investigate an expression for the length of a meridional arc on the terrestrial spheroid ; having, by ob- servation, the latitudes of the extreme points, with as- sumed values of the equatorial radius and the eccentricity of the meridian - - - - 391 421. Table of the measured lengths of a degree of latitude in different places. Presumed value of the earth's ellipticity 391 422. The triangles in a geodetical survey gradually increase as they recede from the base - - 392 423. Methods of determining the stations in the secondary triangles 393 424. (Prob. I.) To determine the positions of two objects and the distance between them, when there have been observed, at those objects, the angles contained between the line joining them and lines imagined to be drawn from them to two stations whose distance from each other is known - 39^ 425. (Prob. II.) To determine the position of an object, when there have been observed the angles contained between lines imagined to be drawn from it to three stations whose mutual distances are known. Use of the station-pointer - 395 426. (Prob. III.) To determine the positions of two objects with respect to three stations whose mutual distances are known, by angles observed as in the former propositions ; and some one of the stations being invisible from each object 398 427. Investigation of a formula for correcting the computed heights of mountains, on account of the earth's curvature 399 428. The height of the mercury in a barometer applicable to the determination of the heights of mountains - 400 429. Law of the densities of the atmosphere in a series of strata of equal thickness - - _ - - 401 430. Formulae for the height of one station above another in at- mospherical and common logarithms - 402 431. Nature of mountain barometers. Formula for finding the relative heights of stations by the thermometrical barometer 404 432. The vibrations of pendulums in different regions serve to determine the figure of the earth - 406 433. Pendulums whose lengths vary only by changes of temperature, Kater's pendulum - 407 434. Manner of making experiments with detached pendulums - 407 NOTE The following propositions relate to the corrections which are to be made previously to the employment of the results of expe- riments. 435. (Prop. I.) To reduce the number of vibrations made on a given arc, to the number which would be made in the same time on an infinitely small arc - - 409 XXviii CONTENTS. Art. Page 436. (Prop. II.) To correct the length of a pendulum on account of the buoyancy of the air - r.-> - - 410 437. (Prop. III.) To correct the length of a pendulum on account of temperature - - >' - 411 438. (Prop. IV.) To correct the length of a pendulum on account of the height of the station above the sea - - 412 439. Formulae for determining the length of a seconds pendulum at the equator, the ellipticity of the earth, and the varia- tions of gravity - 413 440. The elements of terrestrial magnetism - - - 413 441. The usual variation compass, and dipping needle - - 414 442. Mayer's dipping needle - - - 415 443. Formulas for finding the dip, or inclination, of a needle when its centres of gravity and motion are not coincident - 416 444. Correspondence of the theory of magnetized needles to that of pendulums. Formula for correcting the vibrations of a needle on account of temperature - - 417 445. Instruments used for the more delicate observations - 419 446. The declination magnetometer. Formula for correcting the observed declination on account of torsion - - 419 447. Formulas for 'determining from observation the horizontal intensity of terrestrial magnetism - 420 448. The horizontal force and vertical force magnetometer. For- mulas for determining the ratios which the variations of the horizontal and vertical intensities bear to those compo- nents - - 421 449. Manner of expressing the intensity of terrestrial magnetism 423 450. Equations of condition exemplified, for obtaining the most probable values of elements. For the angles which, with their sum or difference, have been observed. For the transits of stars. For the length of a meridional arc. For the length of a seconds pendulum. Equations of condition to be satisfied simultaneously - - 423 PRACTICAL ASTRONOMY AND GEODESY. CHAPTER I. THE EARTH. PHENOMENA OF THE CELESTIAL BODIES. FORM OF THE EARTH, AND ROTATION ON ITS AXIS. APPARENT MOVEMENT OF THE STARS. REVOLUTION OF THE MOON ABOUT THE EARTH. HYPOTHESIS OF THE EARTH'S ANNUAL MOTION. PHASES OF THE MOON. APPARENT MOVEMENTS OF THE PLANETS. THE CIRCLES OF THE SPHERE. 1. THAT the surface of the earth is of a form nearly spherical may be readily inferred from the appearance presented at any point on the ocean by a ship when receding from thence ; for, on observing that the line which bounds the view on all sides is accurately or nearly the circumference of a circle, and that when a ship has reached any part of this line she seems to sink into the water, the spectator recognizes the fact that she is moving on a surface to which the visual rays from that circumference are tangents. These rays may be imagined to constitute the surface of a cone of which the eye of the spectator is the vertex ; and the solid with which, at every part of its surface, a cone is in contact on the periphery of a line which is accurately or nearly a circle (that is, the solid whose section when cut any where by a plane is accurately or nearly a circle) is (Geom. 1. Prop. Cylind.) accurately or nearly a sphere. The like inference may be drawn from the appearance presented on all sides of a spectator on land, the curve line which bounds his view being the circumference of a circle except where inequalities of the ground destroy its regularity. 2. The plane of the circle which terminates the view of a spectator is designated his visible or sensible horizon. A plane B 2 CELESTIAL PHENOMENA. CHAP. I. conceived to pass through the spectator and the sun at noon, perpendicularly to the horizon, is called his meridian ; and, on the supposition that the earth is a sphere or spheroid, this plane will pass through its centre. Its intersection with the surface of the earth or with a horizontal plane, which, to the extent of a few yards in every direction about the spectator, may be considered as coincident with that surface, is called a meridian line : of this line, the extremity which is nearest to '$& Arctfc;r^c>ii^Qf the earth is called the north point, and that whicli is opposite* to it, the south point. A line imagined ; t^4^ throiigh lh0- > spectator perpendicularly to the plane of tte' horizon,' aricT to be produced above and below it towards the heavens, is denominated a vertical line ; its upper and lower extremities are designated, respectively, the zenith and nadir. Every plane which may be conceived to pass through this line is said to be a vertical plane, but that which is at right angles to the plane of the meridian is called the prime vertical: it cuts the plane of the horizon in a line whose extremities are called the east and west points ; the former being that which is on the right hand of the spectator when he looks towards the Arctic regions of the earth, and the latter, that which is on his left hand when in the same position. 3. Now, if a spectator were at any season of the year to land on the shores of Spitzbergen, the stars which are visible would appear to describe about him circles nearly parallel to his horizon. In the British Isles certain stars towards the north indicate by their movements that they describe during a day and a night the circumferences of circles whose planes are very oblique to the horizon and wholly above it, while others describe arcs which are easily seen to become smaller portions of a circumference as they rise more remotely from the northern part of the horizon ; and a few may be observed which rise and set near the southern point, describing, during the time they are visible, curves which ascend but little above that plane. About the mouth of the Amazon, and in the islands of the Indian Ocean, the spectator would see the stars rise and set perpendicularly to the horizon, each of them describing half the circumference of a circle above it. If the spectator were to transfer himself to the southern regions of the earth he^ would see phenomena similar to those above mentioned exhibited by the stars which are situated in that part of the heavens ; while on directing his eye towards the north, the stars which before were seen to ascend to considerable heights above the southern part of the horizon, would be either invisible or would be seen but for a short time, the places of rising and setting being near the northern point. CHAP. I. MOTIONS OF THE SUN AND MOON. 3 4. These circumstances indicate a general revolution of all the stars about an axis passing through the earth perpen- A, dicularly to the planes of the circular arcs apparently de- scribed by them: but it is very improbable that so many different bodies should perform the revolution in the same time, preserving their apparent distances unchanged ; and, on the ground that the like phenomena would result from a rotation of the earth about the same axis, the latter hypo- thesis is adopted by astronomers. In the present age a certain star ( Polaris) nearly indicates the northern ex- tremity of the axis in the heavens, and an arc supposed to be traced through two bright stars (y and a of the southern 3 cross) tends directly towards the opposite extremity. 5. The stars appearing to be at the same distance from the spectator, they are, for convenience, imagined to be attached to the concave surface of a hollow sphere of which his eye is the centre; and hence the general movement of the stars about the earth is sometimes called the diurnal rotation of the celestial sphere. The sun appears daily to rise and set: the moon also, when visible, exhibits the like phenomena; but these two celestial bodies, and certain stars which are not always visible, have proper movements which prove them to be quite distinct from the other bodies of the universe. 6. At a certain season, in any part of the world, a very distinguishable cluster of stars, which may have been observed to rise very near the eastern part of the horizon, will be on the meridian of the observer a short time before the rising of the sun : on continuing to observe the cluster, its apparent elongation from the sun will constantly increase, and, in two or three months from the time of the first observation, it will appear at morning dawn to be setting in the west. During several months it will not be visible, but it will afterwards appear near the east a little while before the sun is there ; and at the end of one year from the first observation it will be again seen on the meridian near the time of his rising. The like phenomena are exhibited by all the stars which rise in or near the east ; and from this change in the sun's position with respect to such stars, it may be inferred either that those stars, and with them all the others, have been $[ carried towards the west independently of the diurnal move- ft ment, or that the sun has, during the year, moved eastward about the earth. The moon is observed to change her place T) ^ J in like manner; for if on any night her distance eastward from some star be remarked, on the following night the dis- tance in that direction will be sensibly increased : the distance B 2 4 CELESTIAL PHENOMENA. CHAP. L, will go on continually increasing and, in about a month, the moon will be in the same position as at first with respect to the star. That an annual and monthly, as well as a diurnal revolution of all the stars should take place simul- taneously, is highly improbable; andGt may therefore be ! inferred that the moon revolves about the earth from west to east : the phenomena above mentioned may seem to indicate a like movement of the sun; but since they will be the same whether the sun revolve about the earth in one year, or the earth about the sun in the same time, the latter hypo- thesis, which alone is consistent with the laws of general attraction among the great bodies of the universe, may be immediately adopted. 7. It is well known that the sun's angular elevation above the horizon at noon experiences, from mid-winter to mid- summer, a continual increase, and, from mid-summer to mid- winter, a continual decrease. Now, by observing what groups of stars appear to rise and set very near the sun during a year, the trace of his apparent route in the heavens ;..^5 may be distinguished ; and hence it may be readily ascertained in what groups the sun is at the times when his elevations above the horizon are the least and the greatest. These groups, or constellations, are on opposite sides of the celestial sphere with respect to the earth; and it may be perceived that a mean between the greatest and least elevations of the sun at noon is nearly equal to the elevation of that luminary at the noon of the day which is the middle of the interval between mid-winter and mid-summer, or between mid-summer and mid- winter : it may be observed also, that the greatest elevation of any star, which, rising in the east and setting in the west, appears to describe daily about the earth a path whose plane is perpendicular to the axis of the diurnal rotation, is equal to that mean elevation of the sun ; and it may from these circumstances be presumed, that the path of the sun in his apparent yearly revolution about the earth, or the path of the earth about the sun, is a plane curve having a certain inclination to the plane last mentioned. The angular elevations of the points which the moon appears to occupy in the celestial sphere are sometimes greater than the greatest, and sometimes less than the least elevations of the sun ; and it may from thence be inferred that the moon's orbit has a certain inclination to that in which the earth revolves about the sun. 8. Since the angular distance of the sun below the plane passing through the earth perpendicularly to the axis of rotation is at noon, on the day of mid-winter, equal to the angular CHAP. I. THE MOON S PHASES. 5 distance of the sun above that plane at noon on the day of midsummer, and that on the two days of the year which are equally distant from those days the sun is in the plane per- pendicular to the axis of rotation ; it follows, if the earth be supposed to revolve annually about the sun, that the axis of the diurnal rotation must on those four days be in positions A parallel to one another ; also, since the changes in the angular distance of the sun from that plane take place gradually, it may be inferred that the axis of rotation continues always strictly or nearly parallel to itself. 9. If the moon be at first observed near the west about the time that the sun is setting, her angular distance from that luminary will on every succeeding evening be found to have increased, and at the end of about fourteen days, she may be observed rising near the east, when the sun is on the western side of the horizon : she is then said to be in opposition to the sun : from that time, she appears to approach the sun towards the east, the intervals between the times at which the two luminaries rise continually diminishing ; and, at the end of about a month from the first observation, 'the moon is in conjunction with the sun. After remaining invisible for a few days, she re-appears in the west at a small distance from the setting sun, and the like phenomena are repeated. 10. During this revolution of the moon about the earth the form exhibited by the outline of her face gradually changes. At the time of new moon, when, in the western part of the horizon, she first emerges from the sun's rays, she assumes the form of a slender crescent of light, and this crescent daily increases in breadth, till, at the time of oppo- sition, it becomes a complete circle: afterwards the breadth diminishes till the moon is about to become invisible from her proximity to the sun in the east, when the figure is again that of a slender crescent. These phases are exactly such as are presented by a globular body enlightened on half its surface by the sun, the circle bounding the light and shadow being in different positions with respect to the observer; and hence it is inferred that the figure of the moon is exactly or nearly that of a sphere. It happens occasionally that, at the time of opposition, the centre of the moon is exactly or nearly in the direction of a line drawn from the centre of the sun through that of the earth, and produced towards her ; and then, the shadow of j) the earth falling on the moon, the inhabitants of the side of the earth which is nearest to her observe her to be eclipsed, or deprived of the light which she would have received from the sun: it happens also, occasionally, that at the time of B 3 6 CELESTIAL PHENOMENA. CHAP. I. conjunction the centre of the moon is exactly or nearly in a line joining those of the earth and sun ; in which case, the moon intercepting the rays of light coming from the sun towards the earth, the sun, to an inhabitant of the earth who may be situated near the direction of the line, on the side nearest to the luminary, is observed to suffer an eclipse. 11. Attentive and continued observations of the heavens show that some of the stars have movements independent of that general revolution which all of them appear to perform daily about the earth. These are the planets, which are ten in number, though only six can be seen by the unassisted eye, and their designations in the order of their distances from the sun are as follow : Mercury, Venus, Mars, Vesta, Juno, Pallas, Ceres, Jupiter, Saturn, and Uranus; the Earth, which is also a planet, and is situated between Venus and Mars, being omitted in the enumeration. Two of the planets, Mercury and Venus, when visible, appear on the same side of the meridian as the sun is ; and the former but a short time before his rising or after his setting : if first seen nearly in conjunction with the sun in the west, these planets then gradually recede from him towards the south, Mercury to an angular distance not exceeding 28 J degrees, and Venus to a distance not exceeding 48 degrees: they afterwards appear to return towards him, and after having been for some days invisible, they may be seen in the east before sun- rise; at first they appear to recede from that luminary towards the south; and subsequently, the greatest angular distances or elongations being equal to those which were attained in their former positions, they return towards it. After being again for a time invisible, they re-appear in the west as before, and the like phenomena are repeated. In the interval between the disappearance in the west and the next appearance in the east, both planets are occasionally, by the aid of the telescope, seen to pass like dark spots across the disk of the sun ; the telescope moreover shows that each of these, like the moon, assumes the form of a crescent, a semi- circle, an ellipse, and nearly a complete circle; the several phases succeeding each other in regular order. The in- ferences are that these planets are globular, and that they revolve about the sun within the orbit of the earth, Mercury being that which is nearest to him. 12. The other planets are seen at times nearly in conjunction with the sun, and at other times diametrically opposite to him in the heavens, and it is therefore inferred that they revolve about the sun in orbits, on the exterior of that which is described by the earth. The comets also, which occasionally CHAP. I. MAGNITUDE OF THE EARTH. 7 appear in the heavens, are observed to have such movements as indicate that they, like the planets, revolve about the sun. All the planets, moreover, are seen to move in different directions with respect to the fixed stars : sometimes they appear to recede from certain of these towards the west, sometimes towards the east, and, again, to remain for a time stationary, or nearly so. The telescope shows that their disks are nearly circular, or segments of circles, and from the move- ments of the spots which have been observed on most of their surfaces, it is inferred that they are globular bodies, which, like the earth, constantly turn on axes of rotation. The motions of the spots observed on the sun show that this luminary has a similar movement on an axis. The planets Jupiter, Saturn, and Uranus are, by the aid of the telescope, observed to be accompanied by satellit.es, which revolve about them as the moon revolves about the earth ; and Saturn is, moreover, accompanied by a ring which revolves in its own plane about the planet. 13. The stars called fixed have, from the earliest ages, been reduced into groups under the figures chiefly of men and animals ; and representations of such groups, or constellations as they are called, may be seen on any celestial globe. A certain zone of the sphere of stars, extending several degrees northward and southward of the sun's apparent annual path, is called the zodiac, and twelve groups of stars immediately about that path bear the name of the zodiacal constellations. The designations of these are as follow : Aries ( v )> Taurus ( s ), Gemini ( n ), Cancer ( ), Leo ( ft), Virgo ( nji), Libra (^), Scorpio (rri), Sagittarius ( $ ), Capricornus (yf) 5 Aquarius (/?), and Pisces ( X ). 14. An approximate knowledge of the magnitude of the earth may be, and very early was, obtained by the aid of a simple trigonometrical proposition, from the measured length of the shadow cast at noon by a column or obelisk erected at each of two places, lying in a direction nearly due north and south of each other. Thus, it being assumed that the earth is a sphere, and the sun so remote that the rays of light which fall upon the earth at the two stations A and B may be con- sidered as parallel to one another, let the plane of the paper represent that of a terrestrial meridian, whose circumference passes through the stations, and let Aa, B, be the obelisks there set up. Then, if Sam, sbn be two parallel rays pro- ceeding from the sun at noon, Am, BT?, which may be con- sidered as straight lines, will denote the lengths of the shadows ; and in the triangles a Am, bftn, right angled at A and B, the lengths of Aa and Am, B and Bra being known, B 4 CELESTIAL PHENOMENA. CHAP. I. the angles Aam, vbn may be computed. But c representing the centre of the earth, if EC be \ s \ s drawn parallel to s or sb, the angles ACE, BCE will be respectively equal to Aam, ~&bn ; therefore the difference between these last angles is equal to the angle ACB. Hence, the arc AB being measured, the following pro- portion will give the length of the earth's circumference : ACB (in degrees) : 360::AB : circumference ( 24850 miles, nearly.) 15. The processes by which the earth's form and mag- nitude are with precision determined, as well as those which are employed in finding the magnitudes of the sun, moon, and planets, the distances of the moon from the earth, and of the earth and planets from the sun, will be presently explained. It is sufficient to observe here, that since the planetary bodies, when viewed through a telescope, present the appearance of well-defined disks, subtending, at the eye of the spectator, angles of sensible magnitude, while the stars called fixed, though examined with the most powerful instruments, are seen only as lucid points ; it will follow that the sun, the earth, and the planets constitute a particular group of bodies, and that a sphere supposed to encompass the whole of the planetary system may be considered as infinitely small when the imaginary sphere of the fixed stars is represented by one of any finite magnitude. Hence, in describing the systems of circles by which the apparent places of celestial bodies are indicated, it is permitted to imagine that either the earth or the sun is a point in the centre of a sphere representing the heavens : and, in the latter case, the earth and all the planets must be supposed to revolve in orbits whose peripheries are at infinitely small distances from the sun. 16. If the earth's orbit (supposed to be a plane) be pro- duced to the celestial sphere, it will there form the circum- ference of a circle which is called the trace of the ecliptic (let it be E L * ) : this is represented on the common celestial globes ; and on those machines, the representations of the zodiacal stars near which it appears to pass will serve, when the stars are recognised, as indications of its position in CHAP. I. CIRCLES OF THE SPHERE. 9 the heavens. A line passing through the sun at c (the centre of the sphere) perpendicularly to the plane of the ecliptic, meets the heavens in the points designated p and q, which are called the poles of the ecliptic. Now, if planes be supposed to pass through p and q, these planes will be perpendicular to that of the ecliptic, and they will cut the celestial sphere in the circumferences of circles which are called circles of celestial longitude : these are also represented on the celestial globes. 17. The trace of the ecliptic in the heavens is imagined to be divided into twelve equal parts called signs, which bear the names of the zodiacal constellations before mentioned. They follow one another in the same order as those constel- lations, that is, from the west towards the east ; and the movement of any celestial body in that direction is said to be direct, or according to the order of the signs : if the move- ment take place from the east towards the west, it is said to be retrograde, or contrary to the order of the signs. 18. It has been shown in art. 7. that the path (the ecliptic) of the earth about the sun is inclined to the plane which is perpendicular to the axis of the diurnal rotation; the axis p g must, therefore, be inclined to the latter axis. Now, the centre of the earth being at C infinitely near, or in coinci- dence with that of the sun, agreeably to the above supposition, let P Q be the axis of the diurnal rotation ; then the plane A B passing through c perpendicularly to p Q and produced to the heavens will be the plane last mentioned : it will cut the surface of the earth (which is assumed to be a sphere or spheroid) in the circumference of a circle called the terres- trial equator, and that of the sphere of the fixed stars in the circumference A B of the celestial equator. The plane of this circle will cut that of the ecliptic in a line, as C 9 which is called the line of the equinoxes, of which one extremity v in the heavens is called the point of the vernal equinox. If planes pass through p Q they will be perpendi- cular to the equator, and their circumferences in the celestial sphere form what are called circles of declination: such is the circle P R Q which is made to pass through s, the supposed place of a star. Of these circles that which passes through the line of the equinoxes is called the equinoctial colure, and that which, being at right angles to the former, passes through p, is called the solsticial colure. The last-mentioned planes will cut the surface of the earth in the circumferences of circles, if the earth be a sphere, or in the perimeters of ellipses if it be a spheroid ; and these are called the meridians of the stations, or remarkable points which they pass through on 10 CELESTIAL PHENOMENA. CHAP. I. the earth. This is, however, only the popular definition of a terrestrial meridian : if from every point in the circum- ference of a circle of declination in the celestial sphere lines be let fall in the directions of normals, or perpendiculars, to the earth's surface, a curve line supposed to join the points in which the normals meet that surface will be the correct terrestrial meridian ; and if the earth be not a solid of revo- lution this meridian is a curve of double curvature. If any point, as s, be the place where a perpendicular raised from any station on the surface of the earth meets the celestial sphere, s will be the zenith of that station ; R will express its geographical longitude, and R s its geographical latitude. The arc c T on the plane of the ecliptic is designated the longitude of any star s, through which and the axis p q the plane of a circle is supposed to pass ; and the arc T s, or the angle T c s, is called the latitude of such star : p s or q s is called its ecliptic polar distance. A plane passing through any point, parallel to the ecliptic E L, will cut the celestial sphere in the circumference of a circle which is called a parallel of celestial latitude. The arc c R on the plane of the equator, is designated the right -ascension of any star S, through which and the axis P Q the plane of a circle of declination is supposed to pass; and the arc RS or the angle R c s is called the decli- nation of such star : P s or Q s is called the polar distance. A plane passing through any point, parallel to that of the equator A T B, will cut the celestial sphere in the circum- ference of a small circle which is called a parallel of decli- nation. 19. The ecliptic and the circular arcs perpendicular to it form one system of co-ordinates: the equator and its per- pendicular arcs form another system ; and the knowledge of the number of degrees in the arcs R and R s, whether obtained by direct observation, or from astronomical tables, is sufficient to determine the place s of a star in the celestial sphere. It may be necessary to observe that the system of the equator and its perpendiculars is continually changing its position by the annual movement of the earth about the sun ; but on account of the smallness of the orbit when compared with the magnitude of the celestial sphere, and the axis p Q being always parallel to itself (art. 8.), omitting certain deviations which will be hereafter mentioned, that change of position creates no sensible differences, except such as depend on the deviations alluded to, in the situations of the stars with respect to these co-ordinates. 20. A third system of co-ordinates is formed by a plane CHAP. I. AZIMUTH AND AMPLITUDE. 11 supposed to pass through the centre of the earth parallel to the plane of the horizon of a spectator, as before mentioned, and by planes intersecting one another in the line drawn through that centre and the station of the observer, that is, perpendicularly to his horizon. The first plane, or its trace in the heavens, is called the rational horizon of the observer, and the circles in which the perpendicular planes cut the celestial sphere are called azimuthal or vertical circles : their circumferences evidently intersect each other in the zenith and nadir. An arc of the rational horizon intercepted between the meridian of a station and a vertical circle pass- ing through a celestial body is called the azimuth, and an arc of a vertical circle between the horizon and the celestial body is called the altitude of that body : also the arc between the body and the zenith point is called the zenith distance ; and a plane passing through any point, parallel to the hori- zon, will cut the celestial sphere in a small circle which is called a parallel of altitude. This last system of co-ordinates is that to which the places of celestial bodies are immediately referred by such observations as are made at sea; and it is also generally employed by scientific travellers who have occasion to make celestial observations on land. 21. An arc of the horizon intercepted between the east or west point and the place of any celestial body at the instant when, by the diurnal rotation, it comes to the circum- ference of the horizon, (that is, the instant of rising or setting,) is called the amplitude of that celestial body. 12 PROJECTIONS OF THE SPHERE. CHAP. II. CHAP. II. PROJECTIONS OF THE SPHERE. NATURE OF THE DIFFERENT PROJECTIONS EMPLOYED IN PRACTICAL ASTRONOMY AND GEOGRAPHY. PROPOSITIONS RELATING TO THE STEREOGRAPHICAL PROJECTION IN PARTICULAR. EXAMPLES OF THE ORTHOGRAPHICAL, GNOMONICAL, GLOBULAR, AND CONICAL PROJECTIONS. MERCATOR'S DEVELOPMENT. 22. THE trigonometrical operations which occur in the inves- tigations of formulas for the purposes of practical astro- nomy, require the aid of diagrams in order to facilitate the discovery of the steps by which the proposed ends may be most readily gained; and representations of the visible heavens or of the surface of the earth, with the circles by which the astronomical positions of celestial bodies or the geographical positions of places are determined, are par- ticularly necessary for the purpose of exhibiting in one view the configurations of stars or the relative situations of ter- restrial objects. The immediate objects of research in practical astronomy are usually the measures of the sides or angles of the triangles formed by circles which, on the surface of the celestial sphere, connect the apparent places of stars with each other, and with certain points considered as fixed : and as, for exhibiting such triangles, the formation of diagrams on the surface of a ball would be inconvenient, mathematicians have invented methods by which the surface of a sphere with the circles upon it can be represented on a plane, so that the re- markable points on the former may be in corresponding positions on the latter ; and so that with proper scales, when approximative determinations will suffice for the purpose contemplated, the values of the arcs and angles may be easily ascertained. 23. These are called projections of the sphere, and they constitute particular cases of the general theory of projections. The forms assumed by the circles of the sphere on the plane of projection depend upon the position of the spectator's eye, and upon that of the plane ; but of the different kinds of pro- jection which may be employed for the purposes of astronomy and geography, it will be sufficient to notice only those which follow. CHAP. II. DIFFERENT KINDS OF PROJECTIONS. 13 The first is that in which the eye is supposed to be upon the surface of the sphere, and the plane of projection to pass through the centre perpendicularly to the diameter at the extremity of which the eye is situated. This projection is described by Ptolemy in his tract entitled " The Plani- sphere," and its principles are supposed to have been known long before his time : it was subsequently called the Stereo- graphical Projection, from a word signifying the repre- sentation of a solid body. A modification of this projection was made by La Hire, in supposing the eye of the spectator to be at a distance beyond the surface of the sphere equal to the sine of 45 degrees (the radius being considered as unity) ; and this method, which has been much used in the formation of geographical maps, is sometimes called the Globular pro- jection. The second is that in which the eye is supposed to be infinitely distant from the sphere, and the plane of pro- jection to be any where between them, perpendicular to the line drawn from the eye to the centre of the sphere : it is employed by Ptolemy in his tract entitled " The Analemma," and it has been since called the Orthographical Projection. It may be here observed, that when from any given point, or when from every point in a given line or surface, a straight line is imagined to be drawn perpendicularly to any plane, the point, line, or surface, supposed to be marked on the plane by the extremities of the perpendiculars, is said to be an orthographical projection of that point, line, or surface. In the third projection the eye is supposed to be at the centre of the sphere, and the plane of projection to be a tangent to its surface : this projection is called Gnomonical, from a corre- spondence of the projecting point, or place of the eye, to the summit of the gnomon or index of a sun-dial. It was not used by the ancients. A modification of this projection was proposed by Flamstead, but it has not been adopted; the method which is now distinguished by the name of that astronomer consists in placing the projecting point at the centre of the sphere, and projecting a zone of its surface on the concave surface of a hollow cone in contact with the sphere on the circumference of a parallel of latitude or declination, or on the concave surface of a hollow cylinder in contact with the sphere on the circumference of the equator. The demonstrations of the principal properties relating to the different projections, and the rules for constructing the representations of circles of the sphere on a plane surface, constitute the subjects of this chapter. 14 PROJECTIONS CHAP. II. DEFINITION. 24. The eye of the spectator being upon or beyond the surface of the sphere, the circle on whose plane, produced if necessary, that surface is represented, is called the Primitive Circle ; its plane is called the plane of projection. Cor. 1. The projecting point being in the direction of a diameter of the sphere perpendicular to the plane of pro- jection, either extremity of the diameter is one of the poles of the primitive circle, and it is evident that the centre of the latter is the point in which the pole opposite to the pro- jecting point is projected. Cor. 2. Since every two great circles of the sphere intersect one another in a diameter of the sphere, it follows that the projection of any great circle intersects the cir- cumference of the primitive circle in two points which are in the direction of a diameter of the latter ; and that when any two great circles are projected, the line which joins their points of intersection will pass through the centre of the primitive circle. Note. The chords, sines, tangents, and secants of the angles or arcs, which, in the projections of the sphere, are to be formed or measured, may be most conveniently taken from the scales on a sector ; the arms of the instrument being opened, so that the distance between the chord of 60, the sine of 90, the tangent of 45, or the secant of 0, may be equal to the radius of the primitive circle. PROPOSITION I. 25. In any projection, if the plane of a circle of the sphere pass through the projecting point or the eye of the spectator, the representation of its circumference on the plane of pro- jection will be a straight line. For imagine lines to be drawn from all points in that cir- cumference to the projecting point, and to be produced in directions from thence, if necessary, they will be in the plane of the circle (Geom. 1. Planes) ; and the plane of projection is cut by the same lines, that is, by the plane in which they are. But the intersection of two planes is a straight line, therefore the representation of the circumference is a straight line. Q. E. D. PROPOSITION II. 26. If from a point on the exterior of a sphere lines be drawn to the circumference of any circle of the sphere, whose plane does not pass through the point, and those lines, pro- CHAP. II. OF THE SPHERE. 15 duced if necessary, cut a plane passing through the centre of the sphere perpendicular to the diameter in which produced is the given point ; the figure projected on the plane will, except when the circle is parallel to the plane of projection, be an ellipse. Let c be the centre, and let A G B D, supposed to be per- pendicular to the plane of the paper, and having its centre in the line AFB, represent the circle of the sphere ; also let the plane of projection passing through the line Q C T, perpendicularly to the plane of the paper, cut the plane of the cir- cle in some line, as DFG : this line will be per- pendicular to AB and to QT. (Geom., Planes, Prop. 19.) Let E, in the produced diameter PCE at right angles to QT, be the given point ; and through that diameter imagine a great circle APT, in the plane of the paper, to be described : this circle will be perpendicular to the planes of projection and of the circle AGED. Now, if lines be drawn from E to every point in the cir- cumference A G B D, they will constitute the curve surface of a cone whose base A G B D is a circle making any angle with its axis, that is, with a line drawn from E to the centre of A G B D : let this surface be produced if necessary, and let it be cut by the plane of projection ; the section N G M D of the oblique cone is the projection of the circle A G B D, and it is required to prove that it is an ellipse. In the triangles AFN, MFB (Plane Trigonometry, Art. 57.), FN : FA :: sin. FAN : sin. FNA, and FM : FB :: sin. FBM : sin. FMB. ; whence FN.FM : FA.FB :: sin. FAN sin. FBM : sin FNA sin. FMB. Let the two last terms of the proportion be represented by p and q respectively ; then FN.FM : FA.FB::/> : q, and FA.FB -FN.FM: but A G B D being a circle, F A . F B F G 2 (Euc. 35.3.); there- fore i- F N . F M = F G 2 . But again, with the same circle of the P 16 PROJECTIONS CHAP. II. sphere, the four angles above mentioned are constant ; there- fore -2- is constant, and the ratio of F N . F M to r G 2 is constant ; P which, by conic sections, is a property of an ellipse : conse- quently the figure N G M D is an ellipse. Q. E. D. If the given circle]of the sphere be in the position A s B v, so that it does not intersect the plane of projection, and is not parallel to it ; the projected figure is an ellipse. Let a plane parallel to that of the circle cut, in a s b d, the oblique cone formed by lines drawn from E to the circumference of ASBV, so that it may intersect the plane of projection ; this section of the cone will be a circle. For, K being the centre of the circle ASBV, let any plane E K s passing through the axis E K cut the section a s b d in k s ; this line (Geom. Planes, Prop. 14.) will be parallel to K s, and the triangles E K s, E k s will be similar to one another : therefore EK : E A : : K s : k s. But the ratio of E K to E k is constant, and K s is constant ; therefore k s is constant, and a s b d is a circle. Now \ngmd represent the projection of A s B v, and dfg be the intersection of this projected figure with the plane of the circle a s b d, we shall have, reasoning as in the former case, a constant ratio between nf. fm and/*? 2 : consequently, the projected figure n g m d is an ellipse. Cor. 1. When the projecting point is on the surface of the sphere, as at E' in the preceding figure, the projection of any circle of the sphere, as A G B D not passing through E', is a circle. For draw A R parallel to Q T ; then (Euc. 30. 3.) the arc E'A will be equal to the arc E'R, and consequently (Euc. 26. 3.) the angle E'AR to the angle ABE'; but E'AR = AN'T (Euc. 2 9. 1.); therefore FN'A = FBM', and the opposite angles at F being equal to one another, the angle F A N' = F M' B. It follows that, in the present case, the products corresponding to those represented by p and q above are equal to one another, and FN'. F M' becomes equal toF G 2 , which is a property of a circle (Euc. 35. 3.) ; therefore when E is on the surface of the sphere, &c. Scholium. If the plane of a circle of the sphere were parallel to the plane of projection, it is evident, whether E be CHAP. II. OF THE SPHERE. 17 on the surface of the sphere, or within it, or on its exterior, that lines drawn from thence to every part of the circum- ference would form the convex surface of an upright cone having a circular base ; and the surface of this cone, produced if necessary, being cut by the plane of projection, the section, that is, the projected figure, will (Geom., Cor. 7. Def. Cyl.) be a circle. The observation will evidently hold good whether the plane of projection pass through the centre of the sphere or be elsewhere situated. Cor. 2. When the point E is infinitely remote, so that all lines drawn to it from the circumference of any circle of the sphere may be considered as parallel to one another, those lines will constitute the convex surface of a cylinder whose base is a circle forming any angle whatever with the axis, that is, with a line drawn through the centre of the circle parallel to the lines before mentioned. In this situation of the eye the section of the cylinder made by the plane of projection, that is, the projected figure, will, except when the circle of the sphere is perpendicular to the axis, be an ellipse. For, in the first figure to this proposition, if A G B D be the given circle of the sphere, the points A and B, the extremities of its diameter, will be projected respectively in the points N", M", where perpendiculars from A and B meet Q T : therefore, in the triangles A F N", B F M", the angles at N" and M" are right angles ; and since the opposite angles at F are equal to one another, we have (PI. Trigon., Art. 56.) A F cos. F = F N", and F B cos. F = F M" ; F N". FM" consequently, A F. F B = . COS. 2 F But AF. F B = F G 2 ; therefore '' F M ' = F G 2 , COS. 2 F, that is, F N". F M" has a constant ratio to F G 2 , which is a property of an ellipse. Therefore the section N" G M" D is an ellipse. 27. Scholium. The orthographical projection of an ellipse, on any plane, is also an ellipse. For let it be supposed that the figure D A G B is an ellipse : then by conic sections, t being the semi-transverse, and c the semi-conjugate axis, c 2 -5- A F. F B = F G 2 ; or the rectangle A F. F B has a constant ratio to FG 2 . Now from the second corollary we have AF.FB F N". F M" , C 2 , ^ C FN." FM" = - ; therefore -^ AF.FB (= F G 2 ) = - T : cos. 2 F p t 2 cos. 2 F hence the rectangle F N." F M" has still a constant ratio to F G 2 , or the projected figure N" G M" D is an ellipse. C 18 PKOJECTIONS OF THE SPHERE. CHAP. II. PROPOSITION III. 28. If a plane touch a sphere at any point, and from the centre of the sphere lines be drawn through the circumference of a small circle whose plane is perpendicular to the tangent plane ; those lines if produced will meet the latter in points which will be in an hyperbola. Let c be the centre of the sphere, and D the point at which the tangent plane D G is in contact with it : let a a" be part of the circumference of a small circle whose plane is perpendicular to DG, and let DE be part of the circumference of a great circle parallel to the small circle ; also let the plane of this great circle meet the tangent plane, to which it is perpendicular, in Dr. Again, let a plane pass through CD perpendicular to the planes of the great and small circle, it will also be perpendicular to the tangent plane ; and in it let the point a be situated : then a line drawn from c through a will meet the tangent plane in some point as A. Let, also, a plane pass through c and any other point is the centre, and the line DP the radius of the projection (art. 32.): thus the re- quired circle P m s may be described. It is obvious that if the given point were, as at P', not in the cir- cumference of the primitive circle, the centre of the re- quired projection might be found thus : describe an arc of a circle with p' as a centre, and with a radius equal to the secant of the angle which the circle makes with the plane of the primitive; and also an arc from c the centre of the primitive, with a radius equal to the tangent of the angle (the radius of the primitive being supposed to be unity) : the intersection of these arcs would be the centre of the required projected circle, and the secant of the angle would be its radius. PROPOSITION XIV. 39. Through any given point to describe the projection of a great circle of the sphere, making a given angle with the projection of a given great circle. Let MAB be the primitive circle, and MEN the projection of the given great circle, also let P be the given point. Find (art. 35.) H, a pole of MEN, and about H as a pole describe a small circle a b, (art. 36.) at a distance from H equal to the arc, or number of degrees by which the given angle is expressed ; or at a dis- tance equal to the supplement of that arc, if the latter were greater than a quadrant. About p as a pole, describe the projection QR of a great circle (art. 35.) ; and if the data be such that the construction is possible, this circle will touch or cut the small circle a b : let it cut the circle in p and q. Then, either from p or q (suppose from p) as a pole describe the projection p T of a great circle, it will pass through p and be the projection which is re- quired. CHAP. II. STEREOGRAPHICAL PROJECTION. 29 For let it intersect the circle M E N in E : then, since p is a projected pole of PET, and H is a projected pole of MEN, an arc of a great circle drawn from p to H would measure the distance between the poles of the circles MEN and PET; therefore it would measure the angle M E p, or that at which the circles are inclined to one another. But the circle a b was described about H at a distance equal to the measure of that angle ; therefore either the angle M E p or the angle M E T is equal to the given angle, and P E T is the required projection. If through a given point, as p, it were required to describe the projection of a great circle, making with the projection N P S of a great circle perpendicular to the plane of projection any given angle, the construction might be very conveniently effected in the following manner. Through p describe the projection of a great circle at right angles to N S ; this will pass through A and B, the poles of N S, at the ex- tremities of the diameter A c B at right angles to NS, and let R be its centre. Then, considering A p B as a new primitive circle, through R draw the line R X at right angles to N S ; and draw P D making the angle R P D equal to the complement of the given angle, that is, equal to the angle which the circle whose projection is required makes with the plane of the circle whose projection is A p B. The point D is the centre, and D p the radius (art. 32.) of the circle E p F, the required pro- jection. PROPOSITION XV. 40. To describe the projection of a great circle making, with two projected great circles, angles which are given. Let M Q N be the primitive circle, c its centre, and M p N, Q p R, the projections of two given great circles. Find (art. 35.) p a projected pole of M p N, and q a projected pole of Q P R : about p as a pole (art. 36.) describe the pro- jection of a circle at a distance equal to the angle which the re- quired projection is to make with the circle M p N, and about q as a pole, a circle at a distance equal to the angle which the 30 PROJECTIONS OF THE SPHERE. CHAP. II. required projection is to make with Q P R. Then, if the data be such that the construction of the problem is possible, the circles about p and q will either touch in some point, or cut each other in two points ; let them cut each other in H and K, and about either H or K (suppose H) as a pole describe (art. 35.) the projection x Y of a great circle. This will be the circle required, and the angles at s and T will be equal to the given angles. For, since p is a pole of M p N and H a pole of x S Y, the arc of a great circle which measures the distance between p and H will measure the angle p s x or P s T, between the circles M P N and x s Y : and for a like reason the arc of a great circle between q and H will measure the angle p T s or p T Y. But the distances between p and H, q and H are by- construction equal to arcs which measure those angles ; therefore the angles at S and T are equal to those which were given. PROPOSITION XVI. 41. To measure an arc of a projected great circle. If the given arc, as A B, be anywhere on the circumference of the primitive circle, it may be measured by a scale of chords on which the chord of 60 is equal to the radius of the primitive. And if the given arc, as c D, be on the projection of a circle at right angles to the plane of projection, and one extremity of the arc be at the centre or pole of the pri- mitive, it may be measured on a scale of tangents ; the number of degrees found on the scale being doubled, because C D (art. 33. 1 Cor.) is equal to the tangent of half the arc which it represents. If the arc, as M N, be on the circumference of a projected circle which does not coincide with the primitive, it may be measured in the following manner. Find (art. 35.) P a projected pole of s M N A, and draw the lines P M m and PNTZ, cutting the primitive circle in m and n ; the arc m n, measured by a scale of chords, is the value of the arc of which M N is the projection. For p M m and p N n being straight lines, are the projections of two circles which pass through the pro- jecting point, or the pole of the primitive ; and since they pass through p, a projected pole of s M N A, the circles which they represent must pass through a pole of the circle CHAP. II. STEREOGRAPHICAL PROJECTION. 31 of which S M N A is the projection. Therefore (art. 30. Schol.) the arcs on the sphere, which are represented by MN and m n, are equal to one another. If the lines P' M m', p' N n' had been drawn from the exterior pole P', the arc rnf n' would have been that which is equal to M N on the sphere. PROPOSITION XVII. 42. To measure the angle contained by the projections of two great circles of the sphere. If the given angle be at the centre of the primitive circle, it may be measured on that circle by the chord of the arc by which it F is subtended. If it be at the cir- cumference of the primitive and be contained between that circle and the projection p m s of a great circle which is inclined to it ; find the centre R, or the pole p of the circle PmS; then (art. 32. Schol.) C R measured on a scale of tangents, or (art. 34.) Gp measured on a scale of tangents and the number of degrees doubled, will give the number of degrees in the angle of which AP m is the projection : or again, C m measured on a scale of tangents, and the number of degrees doubled, will give the complement of that angle. If the angle be contained between the projections, as P Q S, M Q N of two great circles, the angular point not being at the centre or circumference of the primitive ; find the projected pole p of P Q S, and the projected pole q of M Q N, and draw the lines Q p E, Q q F, meeting the primitive circle in E and F ; then E F measured by a scale of chords will give the value of the angle of which M Q P is the projection. For p and q being the projected poles of P Q s and M Q N, an arc of a great circle joining them will measure the angle p Q M between the circles ; and arcs of projected great circles joining Q and p, Q and q 9 being the projections of quadrants, Q is the projected pole of a great circle passing through^ and q. But, by the Scholium in art. 30., and as in the last proposition, E F measures the arc of which one joining p and q is the projection ; therefore it measures the angle p Q M. PROPOSITION XVIII. 43. A line traced on the surface of a sphere representing the earth so as to make constantly a right angle with the meridian circles is either the circumference of the equator or 32 ORTHOGRAPHICAL AND CHAP. II of some parallel of latitude ; but a line traced so as to make any constant acute angle with the meridian circles will be a spiral of double curvature : and if such a curve be represented on a stereographical projection of a hemisphere of the earth, the equator being the primitive circle, the projected curve will be that which is called a logarithmic spiral. For let MN be the plane of the equator, and p its pole ; and let P, fb, PC, &c. be the projections of terrestrial meri- dians making equal angles a P b, b\>c, &c. with one another; those angles being, by supposition, infinitely small. Let ABC &c. represent the projection of the curve line which on the sphere makes equal angles with the me- ridians : then the arcs AB, BC, &c., being infinitely small, may be con- sidered as portions of great circles of the sphere ; and consequently, by the principles of the stereo- graphical projection, the angles PAB, PBC, &c. on the paper are also equal to one another. But all the angles at P are by construction equal to one another ; therefore all the triangles, considered now as rectilineal, will be similar to one another. Hence AP PB PB PC, PB : PC :: PC : PD, &c. Thus all the radii PA, PB, PC, &c. are in geometrical pro- gression, while the angles APB, A PC, APD, &c., or the arcs a b) ac, ad) &c. are in arithmetical progression : hence, by the nature of logarithms, the arcs ab, ac, ad, &c. may be con- sidered as logarithms of the radii P B, p c, P D, &c., and the curve line ABCD, &c. may be considered as a logarithmic spiral. Scholium. The curve en the sphere, or in the projection, is usually called a loxodromic line, and on the earth it is that which would be traced by a ship if the latter continued to sail on the same course, provided that course were neither due east and west, nor due north and south. 44. If a hemisphere be projected orthographically on the plane of the equator, that circle will be the pri- mitive, and its pole will be the centre of the projection ; the terrestrial meridians, or the declination circles, the planes of which pass through the projecting point, will be straight lines diverging, as in the annexed diagram, CHAP. II. GNOMONICAL PROJECTIONS. 33 from the centre or pole P, and the parallels of terrestrial latitude, or of decimation, the planes of which are parallel to the plane of projection, will be the circumferences of circles whose radii, or distances from the centre, will evidently be equal to the sines of their distances on the sphere* in latitude or declination, from the pole. 45. If a hemisphere be projected orthographically upon the plane of a terrestrial meridian, or of a circle of declination, such circle will be the primitive ; the projecting point wilj be upon the produced plane of the equator in a line passing through the centre of the projection per- pendicularly to its plane. The equator AB, and the declination circle PS, which pass through that centre, as well as the paral- lels of latitude or of declination, being all perpendicular to the plane of projection, will be straight lines ; and if, as in the figure, the primitive circle represent the solsticial colure, the ecliptic will also be a straight line, as E Q, crossing the equator in c, the centre of the projection, which will then represent one of the equinoctial points. The straight line, PCS, will be the equinoctial colure ; and, except this and the primitive circle, the meridians or declination circles, being the projections of circles inclined to the primitive, will (2 Cor. art. 26.) be ellipses. The distances of the parallel circles from the equator will be evidently equal to the sines of their latitudes or declina- tions ; and the distances CM, CN, &c., cm, en, &c., of the declination ellipses measured on the equator from its centre, or on each parallel of latitude or declination from its middle point, in the line P s, will be the sines of their longitudes or right ascensions ; the radius of the equator and of each parallel being considered as unity. 46. When a portion of the surface of a sphere is projected gnomonically, and the plane of pro- jection is a tangent to the sphere at one of the poles of the equator, the terrestrial meridians, or the circles of declination, and it may be added, every great circle of the sphere, since all pass through the projecting point, are represented by straight lines ; the declination circles intersecting each O D 34 PROJECTIONS OF THE SPHERE. CHAP. II. other, as at P, in the centre of the projection. The parallels of latitude or declination being the bases of upright cones of which the projecting point is the common vertex, are represented by circles having P for their common centre ; and their radii are evidently equal to the tangents of their dis- tances on the sphere from the pole of the equator. 47. When the plane of projection is in contact with the sphere at some point, as C, on the equator, the terrestrial meridians or circles of declination are represented by straight lines, as P S, M N, &c., since their planes pass through the projecting point ; their dis- tances from the centre, c, of the projection being equal to the tangents of the longitudes, or right ascensions, reckoned from that point on AB, which represents a portion of the equator. The parallels of de- clination, m n, p q 9 &c. are (art. 28.) hyperbolic curves ; and their distances from the equator, measured on a declination circle, PCS, passing through the centre of the projection, are equal to the tangents of their latitudes, or declinations, on the sphere. 48. By placing the projecting point at a distance from the surface of the sphere equal to the sine of 45, the radius of the primitive circle being unity, La Hire has diminished the distortion to which the surface of the sphere is subject in the stereographical projection, and on that account it is more con- venient than the latter for merely geographical purposes. The most important circumstance in the globular projection, as it is called, is, that on a great circle whose plane passes through the projecting point, an arc equal in extent to 45 degrees, or half a quadrant, when measured from the pole opposite the projecting point, is represented by half the radius of the primitive circle. In proof of this theorem, let the plane of projection pass through the diameter M N perpen- dicularly to the plane of the paper : let E be the projecting point in the direction of the diameter B A, perpen- dicular to MN, and at a distance from A, equal to the sine of 45 ( = ,/ ^), the semidiameter of the sphere being unity ; and let B p be half the quad- CHAP. II. GLOBULAR PROJECTION. 35 rantal arc M p B. Draw the line EP cutting MC in p ; then Cp will be equal to the half of c M, or Cp will be equal to J. Let fall PR perpendicularly on AB; then the angle p c B, being 45 degrees, P R and c R (the sine and cosine of 45) are each equal to V^, andER = 2 V% + 1 ; also EC = A/4 -f 1. Now the triangles ERP and EC/? being similar to one another, ER : EC :: RP : Cp, that is 2A/J + 1 : A/i + 1 :: V% : cp', therefore Multiplying both the numerator and denominator of the fraction by 2 V% 1, the value of Cp becomes -|, or Cp is equal to half the radius of the primitive circle. If lines were drawn from E to any other point in the arc M B so as to intersect c M, the distances of the intersections from C would be nearly, but not exactly, proportional to the corresponding arcs on MB; but, for ordinary purposes in geography, it is usual to consider them as such. Therefore, in representing a hemisphere of the earth on the plane of a meridian, the projections of the oblique meridians and of the parallels of latitude, which are respectively at equal distances from one another on the sphere, are usually made at equal distances from one another in the representation. The oblique meridians, and the parallels of latitude which, in the projec- tion, should be portions of ellipses, are usually represented by portions of circles from which, in maps on a small scale, they do not sensibly differ. 49. The conical projection is used only in representing a zone of the sphere ; the concave surface of the cone being supposed to be in contact with its surface on the circumfer- ence of a parallel of latitude, and the geographical points to be projected upon it by lines drawn from the centre. Thus let the segment APS represent part of any meridian of the sphere, P the pole of the equator, the latter passing through the diameter AB perpendicu- larly to the paper ; and let M N be the radius of the parallel of latitude which is at the middle of the zone to be pro- jected. Let VM produced be a tan- gent to the circle A PBS at the point M, and let it be the side of the cone which touches the sphere on the cir- cumference of the parallel circle MR. Then, the projecting point being at c, if the planes of the meridians be produced they will cut T> 2 36 PROJECTIONS OF THE SPHERE. CHAP. II. the surface of the cone in straight lines converging to v ; and if lines be drawn from c through the circumferences of the parallels of latitude on the sphere, to meet the surface of the cone, the projected parallels will be circles. Now if the surface of the cone be cut in the direction of a line from v, as VMD, and laid on a plane surface, it will take the form of a sector of a circle, as MRM', the meridians on the conical surface will still be straight lines diverging from v, and the projected parallels will be arcs of circles having v for their common centre : their distances from M are equal to the tangents of the cor- responding arcs on the sphere, the tangents being measured from M both towards, and from Y, and the radius of the sphere being considered as unity. In order to find the angle subtended at v by M R M' on the developed conical surface, let I represent the length of a degree on a circle whose radius is unity; then 360.Z.MN, in the pre- ceding figure, will be equal to the circumference of the parallel of latitude MR on the sphere ; and if d denote the number of degrees in the angle subtended by M R M' in the development, MV./. f? will express the length of the arc MRM'. But the circumference of the parallel and the arc of development were coincident when the cone encompassed the zone of the sphere : therefore, 360 . /. M N = M v . L d, or 360. MN = MV.d; whence d 360. JM. V But the triangle VMC in the preceding figure, being right angled at M, and MN being perpendicular to PC, MV : MN :: CM : CN (Eucl., Cor. 8. 6.), that is MV : MN :: radius : sine of the latitude of M : consequently d = 360 x sine of the latitude of the parallel of contact, or of the middle of the. zone to be developed. Thus a map of a zone of the earth may be readily con- structed. Such a map, however, though more correct than one formed by any of the projections before described, will become ^sensibly erroneous if it extend more than 15 or 20 degrees in latitude. 50. If the concave surface of a cylinder be in contact with a sphere on the circumference of the equator, and, the eye being at the centre, if the planes of the meridians be produced CHAP. II. CONICAL PROJECTION. 37 to meet the cylinder, also if the circumferences of the paral- lels of latitude be projected as before upon the cylinder ; then, the surface of the cylinder being extended on a plane, the projected meridians and parallels will be straight lines at right angles to one another. A map so formed will be nearly the same as that which is called a plane chart ; and if it extend but a few degrees on each side of the equator, it will consti- tute, like the conical development above mentioned, a nearly accurate representation of the earth's surface. 51. Instead of making the convex surface of the cone a tangent to the sphere, it has been proposed to make a frustum of a cone cut the sphere on the circumferences of two paral- lels of latitude and extend on the exterior to certain distances northward and southward from those parallels : then, adopting the conditions assumed by the Rev. P. Murdoch (Phil. Trans., 1758) the distortion which exists in a projection upon the surface of a tangent cone, though small, will be diminished. These conditions are that the surface of the conical frustum shall be equal to that of the spherical zone, which it is in- tended to represent, and that its length shall be equal to the meridional arc between the extremities of the zone. Let PMA (fig. to art. 49.) be a quadrant of a terrestrial meridian, a and b, the northern and southern extremities of the zone to be developed, M the middle point in latitude : join c, M, and imagine V'D, the line by whose revolution about P c the cone is to be formed, to be drawn perpendicularly to CM, meeting the polar semidiameter c P produced in v'. From E the intersection of V'D with CM make EF, ED each equal to the arc Ma or Mb, and let fall am, bn, and EG perpendicularly on PC: then (Geom., 10. Cyl.) the rectangle DF. circum. EG is equal to the convex surface of the conical frustum, and the rectangle mn. circum. CM is equal (1 Cor. 14. Cyl.) to the surface of the zone ; therefore, conformably to the hypothesis, DF circum. EG = mn circum. CM, or (Euc. 16. 6.) DF : mn :: circum. CM : circum. EG; or again (Geom. 10. Circ.) DF : mra :: CM : EG. But D F is equal to the given arc a b ; and, supposing the radius c M of the sphere to be unity, m n is the difference between the sines of the latitudes of a and Z>, the given extremities of the zone. Let the difference between the sines of the latitudes be represented by d, then EG = . Again, in the right- arc ab angled triangle EG C,CG : CE(::CN : CM) : : sin. lat. M : radius. D 3 38 PROJECTIONS OF THE SPHEKE. CHAP. II. And the triangle V'EC being right angled at E is similar to CGE (Euc. 8. 6.); therefore CG : CE :: EG : EV'. or sin. lat. M : radius _ . V / B _ / - arc a b ~ Varc a b sin. lat. , MJ Thus EG and E v' being obtained, the angle subtended at v' on the map, when the cone produced by the revolution of V'D is extended on a plane, may be found as before, and the map may be constructed. 52. A modification of Flamstead's principle has been adopted in France, for geographical maps, by which a portion of the earth's surface from one of the poles to the equator, and con- taining 90 degrees of longitude, is represented with great cor- rectness. The scale of the first of the two preceding diagrams being enlarged to any convenient magnitude, imagine M in that dia- gram to be a point at 45 degrees from the equator ; then MV, a tangent at M on any meridian, will be the tangent of 45. Let the conical surface produced by the revolution of VM about the axis PC be extended on the paper ; and since it is proposed to exhibit a portion of that surface containing only 90 degrees of longitude, the expression d = 90 x sin. lat. M (d 3 art. 49., being the angle subtended at v by one quarter of the circumference of the parallel of contact when developed) gives d - 63 38 -4'. Therefore make the angle M v M' in the annexed figure equal to that value of d and bisect it by the line v A. With v as a centre, and a radius equal to that of the sphere (which may be considered as unity) describe the circular arc MNM'; this will be the parallel of the 45th degree of latitude on the map, and the arc will contain 90 degrees of longitude on that parallel. Make NP and NA each equal to the length of an arc of 45 degrees on the sphere ; then P will represent the pole, and A will be a point on the equator. Di- vide P A into equal parts, each representing an interval of one, five, ten, or any convenient number of degrees, and through the points of division describe arcs of circles from v as a common centre ; these will represent the parallels of latitude. On each of these parallels, as d d', are to be set from its intersection with PA, spaces, as ab.bc, &c., ab f , b'c', &c., each equal to the chord of one, five, ten, or fifteen degrees, the length of the chord being CHAP. II. MERCATOR'S PROJECTION. 39 computed from the lengths, on the sphere, of the degrees of longitude on the respective parallels ; then curves traced, as in the diagram, from p to the equator BC through the ex- tremities of the chords, will represent the several meridians. 53. The facilities afforded by a map on which the circles of the sphere are represented by straight lines, in laying down on paper the course of a ship at sea, and in determining by a geometrical construction the differences of latitude and of lon- gitude between the points of departure and arrival, on a given course, led to that modification of the plane chart which was proposed by Mercator about the year 1550, or by Wright in England some years later. This consists in representing the meridians by straight lines parallel to one another, and at distances equal to the lengths of one, two, or any convenient number of degrees on the equator, the circumference of this circle of the sphere being conceived to be extended in a right line, which will be per- pendicular to all the meridians : the parallels of latitude are also represented by straight lines, but their distances from the developed equator exceed the corresponding distances from that circle, on the sphere, in the same proportion as the lengths of the degrees of longitude on the several parallels are (in consequence of the parallelism of the meridians) increased on the chart, when compared with their lengths on the sphere. Thus the loxodromic curve, which cuts all the parallel meridians at equal angles, is a straight line, and, in the pro- jection, the angle contained between a line joining two points representing the places of a ship, or of two stations on the earth, and the meridian line passing though one of them is correctly equal to the angle of position, or the bearing of one of those points from the other. 54. In order to determine the distance of any parallel of latitude from the equator, let it be ob- served that the radius MN of such parallel circle on the sphere is equal to the cosine of AM, its latitude (the radius of the sphere being unity), and that the length of a degree on the circumference of any circle varies with the radius of the circle; therefore the length of a degree on any parallel of latitude is to the length of a degree on a meridian circle as the cosine of the latitude of the parallel is to radius. But, in the projection, the length of a degree on each of the different parallels of latitude is equal to the length of an equatorial degree ; therefore the cosine of the latitude of any parallel is to radius D 4 40 PROJECTIONS OF THE SPHERE. CHAP. II. as the length of an equatorial degree is to the length of a degree on the meridian, at the place of the parallel ; or radius is to the secant of the latitude of the parallel as the length of an equatorial degree is to the length of a degree on the meridian, at the place of the parallel : hence it is evident, since radius and the length of the equatorial degree are constant, that, on the projection, the distance of any parallel of latitude from the equator may be denoted approximatively by the sum of the secants of all the degrees of latitude from the equator to the parallel (the length of an equatorial degree being considered as unity). But using the processes of the differential calculus, a correct expression for the distance of any parallel of latitude from the equator, on the projection, may be investigated in the following manner. Let / represent the distance of any parallel of latitude on the sphere from the equator, and L the corresponding distance on the pro- jection : let also d I represent an evanescent arc of the meridian, and d L the corresponding arc on the projection, then, cos. Z : radius (= 1) :: d I : dL, and diL = - -. cos. / But the arc being circular, if x represent an abscissa, as c N, andy (= ^(\ # 2 )) the corresponding ordinate M N, we shall have dl (= S(dx*+df)) = and cos. I = Vl- therefore d L = I-* 2 ' The second member being put in the form - - + , we 1 ~i~ x 1 x have, on bringing to a common denominator, and equating the numerators, A Ax + B + Bar = 1; from which, on equating like powers of x, we obtain A = B == J ; consequently the last equation for diL becomes which, being integrated, gives L = i ( n yP- log- (1 + #) hyp. log. (1 #) j + const, or, L = J hyp. log - ~ + const. ; or again, L = hyp. log. (- -)*+ const. But when x = 0, L rr ; therefore there is no constant, and CHAP. II. MERCATOR'S PROJECTION. 41 L= hyp. log. ([)'. Now, in the figure, M N = x/(l x 1 ) and S N 1-f a; ; con- sequently, S N l-f# - , or tang. L S M N, is equal to -^, or MN N/(l X) Therefore, L = hyp. log. tan. S M N, or = hyp. log. cotan. p s M ; or again, L= hyp. log. cotan. J PC M, that is, in the projection, the length of a meridional arc measured from the equator, northwards or southwards (the latitude of its extremity being expressed by f) is equal to the hyp. log. of the cotangent of half the complement of /, the radius of the sphere being unity. This theorem was first demonstrated by Dr. Halley. (Phil. Trans., No. 219.) The numbers in the tables of meridional parts which are usually given in treatises of navigation may be obtained from the above formula; but in those tables the length of an equatorial minute is made equal to unity, and consequently the radius of the sphere is supposed to be 3437*75 : therefore the value of L which is obtained immediately from the formula, must be multiplied by this number in order to have that which appears in the tables. For example, let it be required to find the number in the table of meridional parts corresponding to the 80th degree of latitude : Half the colatitude is 5, whose log. cotan. 1-05805, and the logarithm of this number is .... 0-02451 Subtract the log. of modulus (0-43429) . . 1 -63778 Logarithm of the hyp. log. cotan. 5 ... 0-38673 Add log. of 3437-75 . . ... 3-53627 Log. of 8375 (=L) ...... 3-92300 And 8375 is the number in the tables. 55. As an example of the manner in which Mercator's projection is applied, let the distance which a ship has sailed be 100 miles, on a course making an angle of 50 with the meridian of her point of departure ; the latitude of this point being 60. Let A be the point of departure, and draw the straight line PA to represent the meridian. Make the angle PAB equal to 50 ; and, from any scale of equal parts representing geographical miles or equatorial minutes, take 100 for the length of AB, then draw BC perpendicular to PA. The line A c computed by plane trigonometry, or measured on the same scale, will express the difference of latitude ( = 64*28, or 42 PROJECTIONS OF THE SPHEEE. CHAP. II- 1 4-280, and in like manner CB(=:76-6 or 1 1 6 -60 the Departure. Now the latitude of A being 60, that of B or c is 61 4-28', and from a table of meridional parts we have for 61 4-28', the number 4658 : for 60, the number 4527 : the difference ( = 131) is the difference of latitude in the projection ; there- fore make A c, from the same scale, equal to 131, and draw cb parallel to CB. Then cb being computed, or measured on the scale, will be 156*15 or 2 36-15', the difference between the longitudes of A and B. CHAP. III. SPHERICAL TRIGONOMETRY. 43 CHAP. III. SPHERICAL TRIGONOMETRY. DEFINITIONS AND THEOREMS. 56. THE objects principally contemplated in propositions relating to the elementary parts of practical astronomy, are the distances of points from one another on the surface of an imaginary sphere, to which the points are referred by a spectator at its centre, and the angles contained between the planes of circles cutting the sphere and passing through the points, it being understood that the plane of a circle passing through every two points is intersected in two different lines by the planes of the circles passing through those points and a third. Thus the circular arcs connecting three points are conceived to form the sides of a triangle on the surface of the sphere ; and hence the branch of science which comprehends the rules for computing the unknown sides and angles is designated spherical trigonometry. 57. In general, each side of a triangle is expressed by the number of degrees, minutes, &c., in the angle which it sub- tends at the centre of the circle of which it is a part, and each angle of the triangle by the degrees, &c. in the angle at which the two circles containing it are inclined to one another; but it is frequently found convenient to express both sides and angles by the lengths of the corresponding arcs of a circle whose radius is unity, the trigonometrical functions (sines, tangents, &c.) of the sides and angles being also expressed as usual in terms of a radius equal to unity. The latter method is absolutely necessary when any such function is developed in a series of terms containing the side or angle of which it is a function. It is obvious that, in order to render the measures of the sides of spherical triangles comparable with one another when expressed in terms of their radii, those radii must be equal to one another; and therefore such triangles are, in general, conceived to be formed by great circles of the sphere on whose surface they are supposed to exist. In the processes of practical astronomy, it is however often necessary to determine the lengths of the arcs of small circles as indirect means of finding the values of some parts of spherical triangles ; but, before a final result is obtained, these must be converted 44 SPHERICAL CHAP. Ill, into the corresponding arcs of great circles : occasionally also it is required to convert the arcs of great circles into the corresponding arcs of small circles, and the manner of effecting such conversions will be presently explained. 58. When the apparent places of celestial bodies are referred to what is frequently called the celestial sphere, whose radius may be conceived to be incalculably great when compared with the semidiameter of the earth, or with the distance of any planet from the sun, the arcs of great circles passing through such places may be supposed, at pleasure, to have their common centre at the eye of the spectator, or at the centre of the earth or of the sun ; and the first suppo- sition is generally adopted in computations relating to the positions of fixed stars : but, as the semidiameter of the earth is a sensible quantity, when compared with the distances of the sun, moon, and planets from its centre, and from a spec- tator on its surface, it is in general requisite, in determining the positions of the bodies of the solar system by the rules of trigonometry, to transfer those bodies in imagination to the surface of a sphere whose centre coincides with that of the earth. The planets, among which the earth may be included, are also conceived to be transferred to the surface of a sphere whose centre is that of the sun. 59. It is easy to perceive that there must be a certain resemblance between the propositions of spherical trigonometry and those which relate to plane triangles ; and, in fact, any of the former, except one, may be rendered identical with such of the latter as correspond to them in respect of the terms given and required, by considering the rectilinear sides of the plane triangles as arcs of circles whose radii are infinitely great, or, which is the same, by considering them as infinitely small arcs of great circles of a sphere, whose radius is finite. For, in either case, on comparing the spherical triangles with the others, the sides of the former, if expressed as arcs in terms of the radius, may be substituted for their sines or tan- gents ; also unity, or the radius, may be substituted for the cosines of the sides and the reciprocals of the sides for their cotangents. The exception alluded to is that case in which the three angles of a spherical triangle are given, to find any one of the sides; for in the corresponding proposition of plane trigo- nometry, the ^ ratio only of the sides to one another can be determined ; it may be observed, however, that the sides of a spherical triangle, when computed by means of the angles, are also indeterminate unless it be considered that they apper- tain to a sphere whose diameter is given. With this exception, CHAP. III. TRIGONOMETRY. 45 the propositions of plane trigonometry might be considered as corollaries to those of spherical trigonometry for the case in which the spherical angles become those which would be made by the intersections of three planes at right angles to that which passes through the angular points of the triangle, and in which, consequently, they are together equal to two right angles only ; and it is evident that the sum of the angles of a spherical triangle of given magnitude approaches nearer to equality to two right angles as the radius of the sphere increases. PROPOSITION I. 60. To express one of the angles of a spherical triangle in terms of the three sides. Let A c B be a triangle on the surface of a sphere whose centre is o, and imagine its sides to lie in three planes passing through that centre intersecting one another in the lines OA, OB, oc. (The inclinations of the planes to one another, or the angles of the spherical triangle, are supposed to be less than right angles, and each of the sides to be less than a quadrant.) Imagine a plane to pass through oc perpendicularly to the plane A OB, cutting the surface of the sphere in c D : then the triangle ABC will be di- vided into two right-angled triangles ADC, BDC. Next let MCN be a plane touching the sphere at c, and bounded by the planes co A, COB, BOA produced, and let it meet the latter in MN ; also imagine the plane ODC to be produced till it cuts MCN in CP. The plane COP is by supposition at right angles to MON ; and because c o is perpendicular to the tangent plane MCN, the plane COP is perpendicular to the same plane MCN; therefore (Geom. 19. Planes, and 1. Def. PL) MN is perpen- dicular to the plane COP and to the lines OP, CP : hence the plane triangles CPM, CPN, OPM, OPN are right angled at P. Now (Plane Trigon., art. 56.) CP T CP - cos. MCP and = cos. NCP; CM CN also - sin. MCP and = sin. NCP: therefore CM CN C P 2 , P M . P N =: cos. MCP cos. NCP and = sin. M c P sin NC P : CM.CN CM.CN 46 SPHERICAL CHAP. III. consequently by subtraction, o 1 P^ T* ivr T* TV = cos. MCP cos. NCP - sm. MCP sin. NCP, CM.CN = (PL Trigon., art. 32.) cos. (MCP + NCP) or cos. ACB: and in like manner, OP 2 -PM.PN __ cog ^ AQB or cog> AB OM.ON Hence cp 2 PM.PN = CM.CN cos. ACB, and OP 2 PM.PN = OM. ON cos. AB. But the radius of the sphere being unity, c P is the tangent and o P the secant of the angle DOC, or of the arc c D ; like- wise c M is the tangent and o M the secant of C A ; also c N is the tangent and o N the secant of C B ; therefore, subtracting the first of these equations from the last, observing that the difference between the squares of the secant and the tangent of any angle is equal to the square of radius, which is unity, we have 1 sec. AC sec. BC cos. AB tan. AC tan. BC cos. ACB; or, substituting for secant, and S1 ^ for tangent, cos. cos. cos. AB sin. AC sin. BC 1 - cos. ACB; COS. AC COS. BC COS. AC COS. BC whence cos. AC cos. BC = cos. AB sin. AC sin. BC cos. ACB and cos. A C ] In like manner, -, COS. A B COS. A C COS. B C , x and cos. ACB = ; 5 . . . . (a) sm. AC sin. BC COS. CAB --ggglBC-COB.ABCOB.AC ;(|) sm. AB sm. AC A COS. AC COS. AB COS. BC ,\ and cos. ABC = -. ; . . . . (c) sin. AB sm. BC If the radius of the sphere and of the arcs which measure the angles of the triangle, instead of being unity, had been represented by r, we should have had, for the equivalent of any trigonometrical functions, as sin. A, cos. A, &c., the terms sin. A, cos. A - - &c. ; therefore, when any trigonometrical formula has been obtained on the supposition that the radius is unity, it may be transformed into the corresponding formula for a radius equal to r, by dividing each factor in the dif- ferent terms by r, and then reducing the whole to its simplest form. Thus the above formula (a) would become r cos. ACB __r cos. AB cos. AC cos. BC "" ~ ; aiul li 1S evident that r CHAP. III. TRIGONOMETRY. 47 be introduced in any formula by multiplying each term by such a power of r as will render all the terms homogeneous ; that is, as will render the number of simple factors equal in all the terms. Cor. 1. When the angle ACB is a right angle, its cosine vanishes ; and radius being unity, the formula () becomes COS. AB = COS. AC COS. BC. If the radius be represented by r, this last expression becomes r cos. AB =. cos. AC cos. BC ; . . . (d) and corresponding equations may be obtained from (5) and (c). Cor. 2. Let the terms in the formula (d) be supposed to appertain to the right-angled spherical triangle ABC ; then, on substituting for them their equiva- lents in the complemental triangle BFE(Sph. Geom., 18.), AG, AT, CE, and GE being quadrants and the angles at c, G, and r, right angles ; also the angles at A and E, being measured by the arcs GF and CG re- spectively, we shall have r sin. B F sin. B E F sin. BE.... (e) Again, substituting for the terms in this formula their equivalents in the complemental triangle ENM, the arcs BH, BN, MF and MH being quadrants and the angles at F, H, and N right angles ; also the angles at B and M being measured by the arcs HN and FH respectively, we have r cos. EMN = sin. MEN cos. EN. ... (/) Cor. 3. In any oblique spherical triangle, as ABC (fig. to the Prop.), letting fall a perpendicular c I) from one of the angles, as C, we have, from the equation (cT), in the right-angled triangles ADC, BDC, r cos. AC^COS. AD cos. DC and r cos. BC i ,. . COS. AC COS. AD cos. BD cos. DC; whence, by division, = - , COS. B C COS. B D or cos. AC : cos. BC :: cos. AD : cos. BD ; or, again, COS. A C COS. BD COS. B C COS. AD. PROPOSITION II. 61. The sines of the sides of any spherical triangle are to one another as the sines of the opposite angles. Let ABC (fig. to the Prop., art. 60.) be a spherical triangle, and let c D be a perpendicular let fall from any one of the angles, as C, to the opposite side : then the terms which, in the right- angled triangles A CD, BCD correspond to those in formula 48 SPHERICAL CHAP. III. (e) above (for the right-angled triangle BE F) being substituted for the latter terms, we have r sin. CD = sin. CAD sin. AC, and r sin. c D = sin. c B D sin. B c : whence sin. CAD sin AC = sin. CBD sin. BC, or sin. AC sin. BC : sin. CBD sin. CAD. In like manner, sin. AC sin. AB : sin. ABC sin. ACB, and sin. BC sin. AB : sin. BAC sin. ACB. PROPOSITION III. 62. To express one of the sides of a spherical triangle in terms of the three angles. Let ACB be any spherical triangle, and A'C'B' be that which is called (Sph. Geo., 1.) the supplemental triangle : then, substituting in the formulae (), (), (c), art. 60., terms taken from the latter triangle which are the equivalents of the sines and cosines of the sides, and the cosines of the angles belonging to the first triangle ; observing that while the sides, and the arcs which measure the angles of the triangle ABC, are less than quadrants, their supplemental arcs in the triangle A'B'C' are greater; and therefore that in the substitution the signs of cosines must be changed from positive to negative, and the contrary, we have (radius being unity) for the equation preceding the for- mula (a) cos. B' cos. A' =: cos. c' + sin. B' sin. A' cos. A'B' ; whence cos. A'B' = cos. c' + cos B' cos. A' In like manner cos. B' c' = sm. B' sin. A' cos. A' + cos. B' cos. c' sin. B' sin. c' and cos. A'C' = COS. B' + COS. A' COS. .(c') sin. A' sin. C' These expressions hold good for any spherical triangles, and therefore the accents may be omitted. Cor. 1. When the angle C' is a right angle its cosine is zero, and the formula (') becomes, omitting the accents and introducing the radius, or r, r cos. AB = cotan. B cotan. A. ... (d f ) Substituting for the terms in this formula their equivalents in the complements! triangle B F E (fig. to 2 Cor., art. 60.) we have r sin. BF = cotan. B tan. FE. . . . (e f ) CHAP. III. TRIGONOMETRY. 49 Again, substituting for the terms in this last formula their equivalents in the complemental triangle ENM, we get r cos. E M N tan. M N cotan. EM.... (/'). COR. 2. The equation preceding (a), omitting the ac- cents, is cos. c = sin. B sin. A cos. A B cos. B cos. A. Now, substituting in this equation the value of cos. B from the formula (V); viz. cos. AC sin. A sin. c cos. A cos. c, it becomes cos. c sin. B sin. A cos. AB cos. AC sin. A cos. A sin. c + cos. 2 A cos. c, or transposing the last term of the second member, and sub- stituting sin. 2 A for 1 cos. 2 A (PL Trigon., art. 19.), we get cos. C sin. 2 A = sin. B sin. A cos. AB cos. AC sin. A cos. A sin. c ; whence cos. c sin. A = sin. B cos. AB cos. AC cos. A sin. c. But from art. 61. we have sin. c = --. - ; therefore, sin. AC dividing i the first of these equations by the other, viz. the first and last terms of the former by sin. c, and the middle term by the equivalent of sin. c, we have cotan. c sin. A sin. AC cotan. AB cos. AC cos. A. 63. The corollaries (d), (e), (/), (<*'), (e ), (/') contain the formulae which are equivalent to what are called the Rules of Napier ; and since these rules are easily retained in the memory, their use is very general for the solution of right- angled spherical triangles. The manner of applying them may be thus explained. In every triangle, plane as well as spherical, three terms are usually given to find a fourth ; and, in those which have a right angle, one of the known terms is, of course, that angle. Therefore, omitting for the present any notice of the right angle among the data, it may be said that in right-angled spherical triangles two terms are given to find a third. Now the three terms may lie contiguously to one another (under- standing that when the right angle intervenes between two terms those terms are to be considered as joined together), or one of them, on the contour of the triangle, may be separated from the two others, on the right and left, by a side or an angle which is not among the terms given or required ; and that term which is situated between the two others is called the middle part. The terms which are contiguous to it, E , 50 SPHERICAL CHAP. III. one on the right and the other on the left, are called adjacent parts ; and those which are situated on contrary sides of it, but are separated from it by a side or an angle, are called op- posite parts. Thus, in the solution of a spherical triangle, as ABC, right angled at c, and in which two terms are given besides the right angle to find a third, there may exist six cases according to the position of the middle term with respect to the two .. others, as in the following table, in which the middle part is placed in the third column between the extremes, the latter being in the second and fourth columns. The first column contains merely the numbers of the several cases, and the fifth denotes the corollaries to the preceding propositions, in which are given the formula for finding any one of the three parts, the two others being given. The first three cases are those in which the extreme parts are adjacent, and the other three those in which they are opposite to the middle part. 1. angle A 2. hypot. AB 3. angle A 4. side AC 5. angle A 6. hypot. AB hypot. AB angle A side AC hypot. AB angle B side BC angle B side AC side BC side BC side AC angle A (d') Prop. 3. (/') Prop. 3. (V) Prop. 3. (ifi Prop. 1. (/) Prop. 1. (e) Prop. 1. The two rules which were discovered by Napier for the solution of right-angled spherical triangles in these six cases may be thus expressed, M being put for the middle part, E and E' for the adjacent extremes, and D,D' for the opposite or disjoined extremes : Had. sin. M = tan. E tan. E', and Bad. sin. M cos. r> cos. D'. But, in using the rule, the following circumstance must be attended to : when the middle term or either of the extreme terms is the hypotenuse, or one of the angles adjacent to it, the complement of the value of that term must be substituted for the term itself. Thus if M, E, or D, &c. denote the hy- potenuse or one of the angles ; for sin. M must be written cos. M ; for tan. E, cotan. E ; for cos. D, sin. D, &c. 64. The problems which require for their solution the de- termination of certain parts of an oblique spherical triangle may conveniently, except when three sides or three angles are the data, be worked by the Kules of Napier, or by the formulae in the six corollaries above mentioned, on imagining a per- CHAP. III. TKIGONOMETKY 51 pendicular to be let fall from one of the angular points of the oblique triangle in such a manner that the latter may be di- vided into two right-angled spherical triangles, in one of which there shall be data sufficient for the application of the rules ; and, as a general method, it may be observed that such perpendicular should fall upon one of the sides from an ex- tremity of a known side and opposite to a known angle. Thus, as an example, let the sides AB and AC, with the included angle A, be given, in the oblique spherical tri- angle ABC ; and let the side BC with A the angles at B and c be required. Imagine the arc BD to be let fall from the point B upon the side AC, then in the right-angled triangle ADB, the side AB and the angle at A are given besides the right angle at D; and with these data let the segment AD be found. If the rules of Napier be employed, it is to be observed that the angle at A will be the middle part, and that the hypotenuse AB with the side AD are adjacent extremes ; therefore by the first of the two rules above (as in the formula (/') art. 62.) r. cos. BAD =: cotan. AB tan. AD; thus AD may be found, and, subtracting it from AC, the segment DC may also be found. The side BC will be most readily obtained by forming in each of the triangles ADB, B DC, an equation corresponding to (d) in art. 60., and dividing one by the other ; each of these equations may be formed by the method given in that article, or by the second of Napier's Rules. Using the latter method, and considering AB and BC in those triangles to be the middle parts, in the first triangle let AD, D B, and in the second let B D, DC be the opposite or disjoined extremes ; then by the Rule, r. cos. AB = cos. AD cos. BD and r . cos. BC = cos. DC cos. BD, and dividing the former by the latter, we have COS. AB COS. AD , ; whence cos. AD : cos. DC : : cos. AB : cos. BC, COS. BC COS. DC or cos. DC cos. AB = cos. AD cos. BC . . . . (A), and either from the equation or the proportion the value of B c may be found. The angles ABC and AC B might now be found by Prop. 2., using the proportions sin. BC : sin. A :: sin. AB : sin. c, sin. BC : sin. A:: sin. AC : sin. B; or the angle at c may be conveniently found from the triangle E 2 52 SPHERICAL CHAP. III. BDC, by Napier's first rule, considering the angle at c as the middle part, and the sides BC, DC as the adjacent parts ; there- fore (as at (/) art. 62.) cos. ACB ir cotan. BC tan. DC. Thus all the unknown parts in the triangle ABC are de- termined. 65. The formulae given in the first and third Propositions, and in the second corollary to the latter, are unfit to be im- mediately employed in logarithmic computations, since they contain terms which are connected together by the signs of addition and subtraction ; and, therefore, before they can be rendered subservient to the determination of numerical values in the problems relating to practical astronomy, they must be transformed into others in which all the terms may be either factors or divisors ; for then, by the mere addition or sub- traction of the logarithms of those terms, the value of the unknown quantity may be obtained. The transformations which are more immediately necessary are contained in the two following Propositions ; and those which may be required in the investigations of particular formulae for the purposes of astronomy and geodesy will be given with the Propositions in which those formulae are employed. PROPOSITION IV. 66. To investigate a formula which shall be convenient for logarithmic computation, for finding any one angle of a sphe- rical triangle in terms of its sides. From Prop. I. we have COS. AB COS. AC COS. BC COS. ACB = -. j ; sin. AC sin. BC and subtracting the members of this equation from those of ,1 .-, ,. , ,. T sin. AC sin. BC the identical equation 1 = -. ; . we get sin. AC sin. BC , cos. AC cos. BC + sin. AC sin. BC cos. AB I COS. ACB ; ; , sin. AC sin. BC cos. (AC BC) cos. AB or 1 cos. ACB = W- r sin. AC sm. BC But (Pl.Trigon., art. 36.) 1 cos. ACB, or the versed sine of ACB, is equivalent to 2 sin. 2 ^ ACB ; and, substituting in the numerator of the second member the equivalent of the difference between the two cosines (PL Trigon., art. 41.), ob- serving that AB is greater than AC BC, since two sides of a CHAP. III. TEIGONOMETRY. 53 triangle are greater than the third, the last equation becomes 2 sin. 2 i ACB = 2 sin - i ( AB + AC -B c) sin, J (AB - AC + B o) sin. AC sin. BC But if P represent the perimeter of the triangle, (AB + AC BC) = P BC, and J (AB AC + BC) = % P AC, therefore the formula becomes sin. 2 ^ ACB (or J vers. sin. ACB) = sin. ( p BC) sin. (-J P AC) , v sin. AC sin. BC Again, if to the members of the above equation for cos. ACB (Prop. I.) there be added those of the identical equation sin. AC sin. BC ,. .,, , , , . -, 1 = - ; , there will be obtained sin. AC sin. BC sin. AC sin. BC cos. AC cos. BC + cos. AB 1 + COS. ACB = -. -. , sin. AC sin. BC COS. AB COS. (AC + BC) or 1 + cos. ACB = - r A- sin. AC sin. BC But (PL Trigon., art. 36.) 1 + cos. ACB is equivalent to 2 cos. 2 J ACB; and, substituting in the numerator of the second member the equivalent of the difference between the two* cosines (PL Trigon., art. 41.), the last equation becomes 2 sin. (AC + BC + AB) sin. J (AC + BC AB) m sin. AC sin. BC putting, as before, | p for (AC -I- BC + AB) and J p AB for J (AC + BC AB), sin. 4 P sin. (4- p AB) , N we have cos. 2 * ACB = ^ ^ ' . . . . (n). sin. AC sin. BC Thirdly, dividing the formula (i) by (n), member by mem- ber, we get sin. (ip BC) sin. (ip AC) , N tan. 2 J ACB n >4_ _Z_^ x2 t / ni \ sin. J P sin. ( P AB) In the above investigations it has been supposed that the radius is unity : if it be represented by r, the numerators of the formulae (i), (n) and (m) must (art. 60.) be multiplied by r 2 . PKOPOSITION V. 67. To investigate formulae convenient for logarithmic computation in order to determine two of the angles of any spherical triangle when there are given the other angle and the two sides which contain it. E 3 54 SPHERICAL CHAP. III. In a spherical triangle ABC, let A B and AC be the given sides, and BAG the given angle; it is required to find the angles at B and c. From (a) and () respectively in Prop. I. we have cos. AB = cos. ACB sin. AC sin. BC + cos. AC cos. BC . . . . (A) COS. BC r= COS. BAG Sin. AC sin. AB + COS. AC COS. AB (k) Multiplying both members of this last equation by cos. AC we get cos. BC cos. AC = cos. BAC sin. AC cos. AC sin. AB -f cos. 2 AC cos. AB, and substituting the second member for its equivalent in (A) there results cos. AB = cos. ACB sin. AC sin. BC + cos. BAC sin. AC cos. AC sin. AB + COS. 2 AC COS. AB, or cos. AB (1 cos. 2 AC) = cos. ACB sin. AC sin. BC -j- cos. BAC sin. AC cos. AC sin. AB. In this last substituting sin. 2 AC for 1 cos. 2 AC, and dividing all the terms by sin. AC, we have cos. AB sin. AC = cos. ACB sin. BC -f cos. BAG cos. AC sin. AB . . . . (m). In like manner, from the formulas (c) and () in Prop. I., and from (k) above, we get cos. AC sin. AB = cos. ABC sin. BC + cos. BAC cos. AB sin. AC .... (ft); and it may be perceived that this last equation can be ob- tained from that which precedes it by merely substituting B for c and c for B. Adding together the equations (m) and (n\ and afterwards subtracting (n) from (m) we have, respectively, putting A, B, and c for the angles BAC, ABC, and ACB, cos. AB sin. AC + sin. AB cos. AC =r sin. BC (cos. c -f cos. B) + cos. A (cos. AC sin. AB -f sin. AC cos. AB), cos. AB sin. AC sin. AB cos. AC = sin. BC (cos. C cos. B) + cos. A (cos. AC sin. AB sin. AC cos. AB). But (PI. Trigon., art. 41.) cos. c + cos. B = 2 cos. J (B -f c) cos. | (B c), and cos. c cos. B = 2 sin \ (B + c) sin. J (B c) : also (PI. Trigon., art. 32.) sin. AC cos. AB cos. AC sin. AE = sin. (AC AB): therefore, after transposition and substitution, the last equa- tions become sin. (AC -f AB) (1 cos. A) = 2 sin. BC cos. t (B + c) cos. \ (B c), CHAP. III. TRIGONOMETRY. 55 and sin. (AC AB) (1 4- cos. A) = 2 sin. BC sin. (B + c) sin. J (B c). But again (PI. Tr., arts. 35, 36.) sin. a = 2 sin. -J a cos. ^ a, 1 cos. A = 2 sin. 2 J A, and 1 -f cos. A = 2 cos. J A : therefore the last equations may be put in the form 2 sin. (AC + AB) cos. J (AC + AB) sin. 2 J A = sin. BC cos. J (B + c) cos. J (B c) . . . (/>), and 2 sin. ^ (AC AB) cos. J (AC AB) cos. 2 \ A m sin. BC sin. -J (B -f c) sin. ^ (B c) . . . (q). Dividing (^) by (p) we have sin. Jr(AC.-AB)cos. J(AC-AB) sin. J (AC + AB) cos. (AC + AB) tan. |(B + c) tan. J (B c). . . . (r). Now from Prop. II. we have sin. B : sin. c :: sin. AC : sin. AB; whence sin. B -f sin. c : sin. B sin. c :: sin. AC + sin. AB : sin. AC sin. AB, -, sin. AC H- sin. AB sin. B -f sin. c and -. - -. = -. - ; - , sm. AC sin. AB sm. B sin. c or (PI. Trigon., art. 41.), sin. ^( AC + AB) cos, j- (AC AB) _ sin. |(B + c) cos. ^ (B c) t cos. \ (AC + AB) sin. \ (AC AB) ~ cos. \ (B + c) sin. (B c) ' or again, sin. \ (AC + AB) cos. \ (AC AB) _ tan. \ (B + c) , ^ cos. i (AC + AB) sin. i (AC AB) ~ tan. \ (B c)* " Multiplying (f) by (s) we have whence = tan. H- + 0) . Again, dividing (r) by (s) and extracting the roots, sin. \ (AB AC) , , N / N < cotan. i CAB = tan. J (B c) ---- (n). sm. \ (AB + AC) Thus from the formulae (i) and (n) there may be obtained the values of the angles ABC and ACB in terms of the angle CAB and of the sides AB and AC, which contain it. 4 56 SPHERICAL CHAP. III. PROPOSITION VI. 68. To investigate formulae convenient for logarithmic computation, for determining two sides of any spherical tri- angle when there are given the other side and the two ad- jacent angles. In a spherical triangle ABC, the angles at B and c and the side BC being given ; it is required to find the sides AB and AC. From the formulae (a f ) and (&') in Prop. III., omitting the accents, after multiplying both members of the latter by cos*. A, and substituting as in the last Proposition, there will be ob- tained the equation cos. AB sin. A=COS. C sin. B-f-cos. BC cos. B sin. c . . . . (m') Also from the formulae (c') and (&') in Prop. III., or, which is the same, writing c for B and B for c in the last equation, there is obtained cos. AC sin. A = cos. B sin. c + cos. BC cos. c sin. B. . . . (V). Adding together (n f ) and (m'\ and afterwards subtracting (n') from (m r ) we get, on transforming, as in Art. 67., sin. A cos. ^ (AB -f AC) cos. \ (AB AC) = sin. (B -f c) cos. 2 \ B c . . . . ( p'\ and sin. A sin. \ (AB + AC) sin. \ (AB AC) sin (B c)sin. 2 | BC . . . . (cf). Then, dividing (q') by (p f ) there is obtained sin. \ (AB + AC) sin. j- (AB AC) _ sin. \ (B c) cos. \ (B c) cos. \ (AB + AC) cos. (AB AC) ~~ sin. J~I(B + c) cos. \ (B + c) tan. 2 J BC ____ (r f ). This last equation being first multiplied by (s), and afterwards divided by (5), in the last Proposition, there will be obtained after the necessary reductions ta, AB + A o = tan. Thus the values of AB and AC may be separately found. The formulae (i) and (n) in Props. V. and VL, were disco- vered by Napier, and are frequently designated " Napier's Analogies." 69. The investigation of formulae expressing the relations CHAP. III. TRIGONOMETEY. 57 between an angle and its orthographical projection on a plane inclined to that of the angle may be made as follows. If the vertices of the given and projected angles are to be coincident, the sides containing the given angle and its pro- jection may be conceived to be in the planes of two great circles of a sphere whose centre is the angular point, and which intersect each other in a line passing through that point. Thus, let acb be the given angle, then, zc being any line passing through c, if the planes of two circles pass through zc and the lines c, cb, and meet a plane passing through c perpendicu- larly to zc, the intersections of the circles with the latter plane will be in the lines CA, CB, and the angle ACB will be the orthographical projection of acb. Now the area b, of a great circle, measures the angle acb; and the arc AB measures the projected angle ACB; there- fore the arcs A a, B&, or the angles aCA, &CB being given, their complements za, zb are known; and in the spherical triangle azb, with the three sides z#, zb, ab, the angle azb may be computed by one of the formulae (), (), or (c), Prop. I., or by one of the formulae (i), (n), or (in) Prop. IV., and consequently its equal AC B is found. If one of the sides, as cb, were coincident with a side, as CB, of the reduced angle, since zb would then be a quadrant, its sine would be equal to radius, and its cosine to zero : therefore, one of the formulae (), (&), or (c), Prop. I. would give z cos. ab cos. ab , ,. , . .., cos. azb = , or = , (radius being unity), sin. za cos. A a ^ or cos. AC a : cos. acb :: rad. (= 1) : cos. ACB. A particular formula for the reduction of the angle acb y when A a and B are small arcs, will be given in the chapter on Geodesy (art. 397.). If the arcs A a, B#, or the angles AC a, BC# be equal to one another, each of them may be considered as the inclina- tion of the plane ac b to ACB, and the reduction may be made thus : imagine a plane, as a c'b, to pass through a b parallel to ACB, and the straight lines, or chords, ab, AB to be drawn ; then the angle a c'b will be equal to ACB, and the triangles a c'b, ACB will be similar to one another : therefore c'b : CB :: chord ab : chord AB, or as 2 sin. ^ arc. ab : 2 sin. ^ arc. A B. But c'b : CB :: sin. zb : rad., or as cos. &b : radius; therefore sin. zb : rad. :: sin. ^ arc. ab : sin. ^ arc. AB, or sin. zb : rad. :: sin. acb : sin. ^ ACB. 58 SPHERICAL CHAP. III. 70. If it were required to find the length of an arc, as AB of a great circle from the given length of the corresponding arc a b 9 of SL small circle, having the same poles (of which let z be one), and consequently (Sph. Geom., 1 Cor. 1 Def.) parallel to it, the sectors ACB, ac'b being similar to one another, we have c'a : CA :: ab : AB. But za is the distance of the small circle from its pole z; and, the radius of the sphere being unity, c'a is the sine of that distance ; therefore the above proportion becomes sin. za : 1 (= radius) :: ab : AB; ab ab whence AB = . = . sin. z cos. A a Conversely ab = AB sin. za AB cos. A a. If the chord of the arc a b were transferred from A to B' on the arc AB, we should have chord a b = C'a . 2 sin. % a c'b, and chord A B' = c A . 2 sin. ACB' : but chord ab = chord AB'; therefore C'a . 2 sin. | ac'b - c A . 2 sin. ACB', and c'a : CA:: sin. J ACB 7 : sin. J ac'b, or cos. Aa : rad. (= 1) :: sin. \ ACB' : sin. % ac'b. When the arc& is small, we have, nearly, the angle ACB' = ACB' a c'b cos. A; or. conversely, ac'b = . cos. A a 71. When two great circles, as PA, PB, make with each other a small angle at P, their point of intersection, and when from any point as M, in one of these, an arc Mj9 of a great circle is let fall perpendicularly on the other ; also from the same point M, an arc M^, of a small circle having p for its pole, is described ; it may be required to find approximative^ the value of pq, and the difference between the arcs M^ and M/>. Let c be the centre of the sphere, and draw the radii CM, Cp ; also imagine a plane MC'^ to pass through M^ perpendicularly to PC, and let it cut the plane MC/? in the line MN. Then, since the plane MC' which it meets in that plane ; and the angle M C f q is CHAP. III. TRIGONOMETRY. 59 equal to BPA or BCA, the inclination of the circles PA, PB to one another. Now, if the radius of the sphere be considered as unity, we shall have MC' =: sin. PM, and cc' = cos. PM ; then c'N (= MC' cos. MC'N) = sin. PM cos. P, also C'N ( cc' tan. pep) =: cos. PM tan. pp; therefore sin. PM cos. P cos. PM tan. Pp, cos. PM , cos. PM sin. Pp and cos. P = - tan. Pp, or = - -. sin. PM sin. PM cos. Pp Subtracting the first and last members from unity, we have cos. PM sin. Pp sin. (PM PC) 1 cos. P = 1 -- r -= - . sin. PM cos. Pp sin. PM cos. P/? But 1 cos. p=2 sin. 2 Jp (PL Trigon., art. 36.), and since P is supposed to be small, the arc which measures the angle may be put for its sine ; therefore 1 cos. P becomes (P being expressed in seconds so that ^ P sin. V may represent sin. p) equal to Jp 2 sin. 2 1". Also, for sin. (PM PJO) may be put (PM Pp) sin. 1", or pq sin. 1", and in the denominator, pp may be considered as equal to PM; therefore, for the deno- minator there may be put sin. PM cos. PM, or its equivalent i sin. 2 PM. Thus we obtain, approximatively, i P 2 sin. 2 1" = f FM ~ '2- sin. 1", or lp 2 sin. 2PM sin. \"pq sm. 2PM (in seconds). Again, since M N = sin. M_p, on developing Mp in terms of its sine (PL Trigon., art. 47.), neglecting powers higher than the third, we have MJO = MN -t- |MN 3 : , . MN . M<7 MN . M<7 And, since -- . = sin. -., or - = sin. ~ * , on de- MC X MC r sm. PM sin. PM veloping as before, we get M<7 MN MN 3 . MN 3 -i - ^ = -T + 4- . o , Or Mff = MN + i -53 --- sm. PM sm.PM 6 sm. 3 PM 6 sm. 2 PM Subtracting from this the above equation for MJO, we obtain sm. 2 PM / sm. 2 PM = - MN 3 cotan. 2 PM; or expressing M/? in seconds, and putting Mp sin. 1" for MN, we have (in arc) Mg' Mp = M/? 3 sin/'l^ cotan. 2 PM, and, in seconds, = M 3 sin. 2 \" cotan. 2 PM. 60 ASTRONOMICAL INSTRUMENTS. CHAI>. IV CHAP. IV. DESCRIPTIONS OF THE INSTRUMENTS EMPLOYED IN PRACTICAL ASTRONOMY. THE SIDEREAL CLOCK. MICROMETER. TRANSIT INSTRUMENT. MURAL CIRCLE. AZIMUTH AND ALTITUDE CIRCLE. ZENITH SECTOR. EQUATORIAL INSTRUMENT. COLLIMATOR. REPEAT- ING CIRCLE. REFLECTING INSTRUMENTS. 72. THE longitudes and latitudes of celestial bodies are not, now, directly obtained from observations, on account of the difficulties which would attend the adjustments of the instru- ments requisite for such a purpose : but on land, particularly in a regular observatory, the positions of the sun, moon, planets, and fixed stars are generally determined by the me- thod which was first practised by Homer or La Hire. This consists in observing the right ascensions by means of a transit telescope and a sidereal clock, and the declinations by means of a circular instrument whose plane coincides with that of the meridian: the longitudes and latitudes, when required, are then computed by the rules of trigonometry. It will be proper therefore, in this place, to explain the na- ture of the instruments just mentioned, their adjustments and verifications, and the manner of employing them. It is not intended, however, to describe at length the great instru- ments which are set up in a national observatory, but merely to indicate them, and to explain the natures and uses of such as, being similar to them and of more simple construction, may without great risk of injury be transported to foreign stations, where regular observatories do not exist, in order to be employed by persons charged with the duty of making celestial observations, either for the advancement of astro- nomy itself or in connection with objects of geodetical or physical inquiry. 73. The Sidereal Clock is one which is regulated so that the extremity of its hour-hand may revolve round the circum- ference of the dial-plate in the interval of time between the instants when, by the diurnal rotation of the earth, that inter- section of the traces of the equator and ecliptic which is de- signated the vernal equinox, or the first point of Aries, appears successively in the plane of the geographical meridian, on the same side of the pole. This interval is called a sidereal day, CHAP. IV. SIDEREAL CLOCK. 61 being very nearly that in which a fixed star appears to revolve about the earth. The dial of the clock having its circumference divided into twenty -four equal parts, the hour- hand is made to indicate xxiv, or hours at the instant that the first point of Aries is in the meridian ; and if the clock were duly adjusted, it is evident that the hour, minute, and second which it might express at any moment would be equivalent to the angle, at that moment, between the plane of the meridian and a plane passing the earth's axis and the equinoctial point above-mentioned. Thus, the clock would show the right ascension of the mid-heaven, or meri- dian, at that instant, right ascensions being generally expressed in time. It follows, therefore, that if a telescope were accurately placed in the plane of the meridian, and a star were to ap- pear to be bisected by a vertical wire in the middle of the field of view, the time shown by the sidereal clock at that instant would express the apparent right ascension of the star. On the other hand, if it were required to make the clock express sidereal time, it would be only necessary, at the moment that a fixed star whose apparent right ascension is given in the Nautical Almanac is observed to be bisected by the wire of the telescope, to set the hands to the hour, mi- nute, &c. so given ; for then, the clock being supposed to be duly regulated, hours will be indicated by it when the first point of Aries is on the meridian. The different species of time, and the processes which are to be used in reducing one species to another, will be explained further on. (Chap. XIV.) 74. The Micrometer is an instrument by which small angles are measured ; and it is employed for the purpose of ascertaining the angle subtended at the observer's eye by the diameter of the sun, the moon, or a planet, or by the distance between a fixed star and the moon or a comet when they are very near each other, or for any like purpose in practical astronomy. It consists of a brass tube with lenses, constituting the eye-piece of a tele- scope and carrying a frame, A B, containing two perforated plates: to one of these is a tached a very fine wire, or spider thread, pq, and to the other a similar wire st, 62 ASTRONOMICAL INSTRUMENTS. CHAP. IV. the two wires being parallel to one another. There is be- sides, usually fixed to one of the plates, a fine wire mn at right angles to laothpq and st. The plates are capable of a rectilinear motion in the direction AB, perpendicular to the axis of the telescope, by means of screws at A and B, and thus the wires may be made to approach to, or recede from, one another, retaining always their parallelism. When they have been moved till the object, or the space between two objects, is comprehended between them, the angle subtended at the eye by such object or space is ascertained by a scale b in the field of the telescope : one revolution of each screw car- ries its wires through an extent equal to one of the gradua- tions on the scale ab, and this extent is subdivided by the graduations on the circular heads c and D of the screws. The angular value of each graduation depends on the magnifying power of the telescope to which the micrometer is applied, and it must always be determined by observation : this may easily be effected, when the micrometer is attached to the telescope of a circle revolving in azimuth like the horizontal limb of a theodolite, by means of a small and well-defined terrestrial object. 75. Having made one of the wires bisect the object, move the other by causing its screw to make any number of revo- lutions; then the azimuth circle being turned till this last wire appears to bisect the object, the difference between the two readings on the circle when the object was bisected, being divided by the number of revolutions, will give the value of an angle subtended at the eye by the positions of the micro- meter wire at the commencement and end of one revolution of the screw: and as one revolution of a screw carries its wire over a space equal to one of the intervals between two divisions on the scale, the angular value of a revolution is, of course, the equivalent of such interval. 76. The value of a revolution may be obtained by measur- ing between the wires the apparent diameter of the sun, if the field of view is sufficiently extensive ; in this case the micrometer screw must be turned till the two wires are tan- gents to the upper and lower edges of the sun's disk. Then, the angle subtended by that diameter being given in the Nautical Almanac, on dividing this angle by the number of revolutions, the result is the angle corresponding to one revolution. It may be found, also, by observing the transit of the pole-star upon one of the wires when the star is on, or very near the meridian, and again after having displaced the wire by several revolutions of the screw ; now t (in seconds) being the time in which the pole-star passes from the CHAP. IV. MICROMETER. 63 wire in the first, to the same wire in the second position, 15 t (since by the rotation of the earth on its axis in 24 hours every point in the heavens appears to describe about that axis an angle or arc equal to 15 degrees in an hour, 15 minutes in one minute of time, &c.) will express in seconds of a de- gree the angle subtended at the centre of the star's parallel of declination by the interval between the two places of the wire, and being multiplied by the cosine of the star's declina- tion (art. 70.) it will express the angle subtended at the centre of the equator by an equal interval on that great circle. But, as will be hereafter explained, this as well as the other stars has constantly a movement in right ascension, that is, in the same direction as the earth revolves on its axis, the amount of which, during the time t, may be found from the Nautical Almanac ; and, whether the star be above or below the pole, an increase of right ascension will increase the time in which, by the diurnal rotation, it appears to pass between the two positions of the wire : therefore that change of right ascension (in arc) must be subtracted from the above product in order to express the angle corresponding to the number of the screw's revolutions. 77. If the micrometrical apparatus be placed in a frame which is capable of being turned about the optical axis of the telescope, and be provided with a graduated circle, as EF, whose centre is in that axis, so that the position of the mi- crometer wires with respect to the plane of any vertical, or horary circle, may be known at the time that the microme- trical angle is observed, the instrument is called a Position Micrometer. It is usually attached to an equatorial instru- ment or to a telescope having an equatorial movement. The manner of using a micrometer for the measurement of small angles will be explained in the description of the equatorial (art. 119.). 78. The micrometer microscope, which is attached to the rim of an astronomical circle, consists of a system of lenses similar to those of an ordinary microscope, and the image of the graduations of the circle is by the disposition of the lenses made to fall at the place of a wire or of a pair of wires crossing each other at an acute angle near the eye-end of the tube. Now the smaller kind of astronomical circles are divided into spaces each equal to 15 minutes of a degree, but those of a larger kind into spaces equal to 10 minutes or 5 minutes ; and across the field of view in the microscope is a scale or a plate divided into spaces, each of which is equal to one minute. The micrometer screw, by one complete revolution, moves the wire through one of the spaces on the plate, and its cir- 64 ASTRONOMICAL INSTRUMENTS. CHAP. IV. cular head being divided into 60 parts (for example) a portion of a revolution, which carries one of these parts under the fixed index, evidently causes the wire to move through a space equal to one second of a degree. The object glass of the microscope has a small motion in the direction of the axis, in order that the magnifying power may be varied ; and that by such motion the image of one of the intervals on the circle (suppose it to be one-twelfth of a degree), or of the number of parts on the scale, into which it is divided, may be increased or diminished till so many revolutions of the micrometer screw will cause the wire to move exactly through the extent of that interval ; the tube containing the eye-glasses being adjusted so as to afford distinct vision. 79. As it is difficult to adjust the microscopes so that five revolutions of the micrometer screw shall carry the wire over exactly one of the five-minute spaces on the circle (if the latter be so graduated), it is preferred to observe the number of revolutions and the part of a revolution made by the screw while the wire passes over the space ; then if, for example, the number of revolutions, instead of five, be 5 4, the value of one revolution will be , or 0'* 926 nearly, and the whole 5*4 number of minutes and parts of a minute which are indicated by the revolutions of the screw being multiplied by that value will give the correct value of the reading. The number of re- volutions of the screw, which in this case are 5 *4, made while the wire moves along one of the divisions on the circle, is called a run of the micrometer screw. 80. In using the micrometer microscope, after the object has been put in contact with the wire in the field of the tele- scope, the number of whole degrees, and of the quarter, sixth, or twelfth parts of a degree are read on the circle ; and if a division-line should exactly coincide with the wire in the microscope when that wire is at the zero point of the micro- meter scale, the reading is complete : but if a division do not so coincide the wire must, by the micrometer screw, be moved up along the scale till it coincides with the next divi- sion-line on the circle ; then the number of revolutions, and the part of a revolution which the screw has made must, when reduced as above, be added to, or subtracted from the degrees, &c. which are read on the circle, according to the direction in which the latter has been turned in taking the angle. 81. The transit instrument is a telescope whose tube con- sists of two cylindrical parts united near the middle of its length to the opposite sides of a portion which is either cubical or globular. At right angles to the tube is an axis of brass CHAP. IV. TRANSIT TELESCOPE. 65 consisting of two conical arms which, at their larger extre- mities, are joined on opposite sides to the cubical or globular portion just mentioned : the smaller extremities of the cones are made cylindrical, and equal to one another; and when the telescope is mounted for service, each of these extremities lies in a notch cut at the top of a moveable vertical plate of brass, (corresponding to what in a common theodolite is called a Y,) which enters into a fixed plate of the like metal. The latter either rests on a stone pier or forms the head of the stand supporting the telescope ; and each notch has the form of two inclined planes whose surfaces are tangents to the cylindrical pivot : one of the notched plates is capable, by means of a screw, of being elevated or depressed for the pur- pose of rendering the axis of motion horizontal ; and the other may, in like manner, be moved a small way in azimuth, in order that the optical axis of the telescope, (the line joining the centres of all the lenses,) may be brought accurately into the plane of the meridian. 82. At the focus of the object glass are fixed three, five, or seven parallel wires, besides one which crosses them at right angles : the latter is intended to be always in a horizon- tal position ; and the others, when the telescope lies horizon- tally between its supports, are in vertical positions. The diaphragm, or perforated plate to which the wires are attached, is capable of a small movement by means of screws which pass through the sides of the telescope, in order that the intersection of the horizontal with the central wire at right angles to it may be made to fall exactly in the optical axis or line of collimation, as it is called, of the telescope. One of the arms of the axis of motion is hollow, and a lamp being attached to the pier, or side of the stand on which that arm F 66 ASTRONOMICAL INSTRUMENTS. CHAP. I V rests, light passes through the arm into the telescope where it falls upon a perforated plane mirror placed at an angle of 45 degrees with the optical axis, and from thence it is reflected to the eye of the observer. By this means the wires are rendered visible in the night, the light from the lamp being regulated so as not to efface that of the star, which is seen through the aperture in the mirror. The extremities of the telescope are sometimes capable of being removed from the body and interchanged with one another ; and in this case the reflecting mirror is capable of being turned on an axis so that its inclination to the optical axis of the telescope may be reversed, and thus the light from the lamp may be thrown at pleasure towards either extremity: by this contrivance in- accuracies in the form of the cylindrical pivots may be com- pensated, since different parts of their surfaces may be made to rest on the notches or inclined planes at the tops of the piers or supports. 83. At the extremity of the other arm is usually fixed, in a vertical position, a graduated circle which turns with the telescope upon the axis of motion, and is furnished with an index which is attached to the pier or stand, and consequently is immoveable, except for adjustment. The bar which carries the index sometimes extends across the circle in the direction of a diameter, . and at each extremity is a vernier for sub- dividing the graduations of the circle : a spirit level attached to the bar serves to indicate the exact horizontality of the line joining the opposite indexes. The use of this circle is to obtain roughly the altitude of a celestial body at the time of the transit : or, the meridian altitude being previously com- puted, the observer is enabled by the circle to direct the tele- scope to the apparent place of the body on the meridian, preparatory to the observation of the transit. 84. The horizontality of the axis of motion is usually ascertained by means of a spirit level, which is either suspended below that axis from angular _ hooks passing over the cylin- rrrffl Ul t" 1 " ' ' ' drical pivots, or is made to stand above, the feet being notched so as to rest tangentially on those pivots; and the annexed dia- gram represents one of the latter kind. The spirit level may be adjusted, and the axis upon which the transit telescope turns may be rendered horizontal, by methods similar to those which are employed in levelling a common theodolite ; but in order to avoid the risk of injuring the instrument by too frequent reversions, it is usual to have a graduated scale adapted to the level so that its middle point, which for sim- CHAP. IV. TRANSIT TELESCOPE. 67 plicity may be the zero of the graduations, is vertically above the middle of the bubble or column of air, and to determine, by the difference of the readings at the extremities of the line, the error, if any exist, in the horizontality of the axis of motion. When the level is in a horizontal position, the bubble of air is stationary at the middle of the upper surface of the spirit in the tube ; but on giving the latter a small inclination to the horizon, the air will move towards the higher end, till the force of ascent is in equilibria with the adhesion of the water to the glass ; and it is assumed that the space which the centre or either extremity of the air bubble moves through is proportional to the angle of inclination. In such angle the number of seconds which correspond to each graduation of the scale may be determined by the maker of the instrument, or the astronomer may determine it himself by placing the level on the connecting bars of a graduated circle which is capable of turning in altitude, and reading the number of minutes of a degree through which the circle is turned, in order to make the air bubble move under any convenient number of graduations on the scale. Now the length of the column of air in the tube will vary with changes in the temperature of the atmosphere, and the axis of the spirit tube may not be parallel to a plane passing through the points on which rest the feet of the level : there- fore, though the latter plane were horizontal, the middle of the air might deviate from the zero of the scale. If the number of the graduation above each extremity of the air column be read, the excess of the greater number above half the sum of the two numbers will indicate the inclination of the axis of the spirit tube to a horizontal line, the axis being highest at that end of the scale towards which the air has moved ; then, reversing the level on its points of support, and reading the numbers as before, if the excess of the greater number above half the sum of the two should be the same as before, it is evident that the points of support will be in a horizontal plane. But, should the said excess not be the same in thecontrary positions of the spirit level, half their sum, if the two excesses are on the same side of the centre of the scale, or half their difference, if on opposite sides, will be the number of graduations expressing the inclination of the plane passing through the points of support to one which is horizon- tal. Thus, for example, let the first excess be equal to 2.4 divisions of the scale (which, if each division correspond to 5 seconds of a degree, will be equivalent to an inclination of 12") towards the west; and the second excess equal to 0.8 divisions, or 4 ", also towards the west ; then half their sum, F 2 68 ASTRONOMICAL INSTRUMENTS. CHAP. IV. or 1.6 divisions ( = 8") will be the true quantity by which the west end of the axis is too high : therefore that end must be lowered by means of the screw under the support till the west end of the column of air has moved 1.6 divisions of the scale nearer the middle of the level, when the axis will be in a horizontal position. If the excesses of the readings on the scale above their mean values, in the direct and reversed posi- tions of the level, had been in opposite directions, one towards the west and the other towards the east, half the difference between the excesses would have been the quantity by which the higher end of the air column must have been moved towards the middle, in order to render the axis of motion horizontal. When the level is placed above the axis of the telescope's motion, there should be attached to it horizontally a small level at right angles to its length, in order by the bubble of air in the latter being in the middle, to ensure the vertical position of a plane passing through the axis of the telescope's motion and that of the principal level. 85. The deviation of the axis of motion from a horizontal position being ascertained, the error occasioned by it in the time of the observed transit of a star may be thus determined. Let N z N' represent the plane of the meridian, NrN x half a great circle of the sphere, in which the telescope is supposed to turn, the line NON' in the plane of the meridian being parallel to the horizon, and o being the place of the observer ; let also the angle zoz or the arc zz measure the deviation of the tele- scope from z, the zenith, or the inclination of the axis of the telescope to the horizon, and let P be the pole of the world. Then, for a star which passes the meridian in or very near the zenith, the angle subtended at P by the arc zz, or the arc ab of the equator equivalent to the angle ZPZ expressed in seconds of a degree will (art. 70.) be equal to z z '- , or to the number of seconds in zz divided by the sin. z p cosine of zb, the latitude of the station. But, for a star which passes the meridian on any other parallel of declina- tion, let Sp represent a portion of such parallel between the meridian NZN' and the circle NZN': then we have (art. 70.) sp very nearly equal to zz sin. N'S, or to the product of the seconds in zz by the cosine of zs, the star's meridional distance from z ; and the angle subtended at P by sp, or the arc 1) c of the equator equivalent to s PJO, equal to -. , that CHAP. IV. TRANSIT TELESCOPE. 69 sin. N's COS.(PSTPZ) is, the angle SP is equal to z* ; ortozz > ~. sin. PS sm. PS This value being divided by 15, since by the rotation of the earth on its axis every point, as s, in the heavens appears to describe about P an angle or arc equal to 15 degrees in one hour, &c., will give the number of seconds which must be added to, or subtracted from the observed time of the transit, according as the circle N2N' is eastward or westward of the meridian, in order to have the corrected time of the transit. 86. The horizontality of the axis may be verified without a spirit-level, on observing by reflexion from mercury the transit of a star near the zenith or the pole, at the first and second wires ; then turning upwards quickly the object end of the telescope and observing the same star by direct view at the fourth and fifth wires (the number of wires being supposed to be five). The intervals between all the wires being equal to one another, if the difference between the times at which the star was observed to be bisected by the first and fifth wire be equal to four times, and the difference between those at which it was observed to be bisected by the second and fourth be twice the difference between the times of being bisected by the first and second, or by the fourth and fifth, it will follow that the axis of motion is correctly horizontal. The error, if any exist, in the horizontality of the axis may be obviated by observing the transit of a star directly on one night, and by reflexion on the next : but in this method it is supposed that the rate of the clock is well known, and that the telescope suffers no derangement between the times of the observations. 87. In order to make the central or meridional wire in the telescope pass through the optical axis, the telescope must be directed so that the wire may coincide with, or bisect some conspicuous object, as a white disk on a post or building at a considerable distance ; then, the pivots of the horizontal axis being reversed on their supports, if the wire still coincide with the object the adjustment is complete; otherwise, the wire must be moved towards the object as much as half the observed deviation, by means of the screw in the side of the tube, and the other half by turning the axis a small way in azimuth ; employing for this purpose the screw which moves the notched plate at the top of the pier or support of the telescope. It may be necessary to repeat this operation several times before the deviation, or the error of collimation as it is called, is quite corrected. 88. If the transit telescope be provided with a micrometer, the error in the position of the optical axis may be determined in the following manner : By turning the screw of the mi- F 3 70 ASTRONOMICAL INSTRUMENTS. CHAP. IV. crometer till the wire which was previously in coincidence with the meridional wire of the telescope is made to bisect a well-defined terrestrial mark, ascertain the number of seconds by which that meridional wire deviates from the mark, and having reversed the axis of the telescope, ascertain, in like manner, the deviation in this position. Half the difference, if there be any, between the deviations, is the true deviation of the line of collimation from a line perpendicular to the axis on which the telescope turns. This deviation is supposed to be expressed in divisions of the micrometer scale, and if these denote seconds of a degree, on dividing by 15 as above mentioned, the deviation will be expressed in seconds of time. The observed time of the transit of an equatorial star must be corrected by the quantity of the deviation so found ; but for a star having north or south declination, that deviation must (art. 70.) be divided by the cosine, or multiplied by the secant of the declination, in order to obtain the corre- sponding arc of the equator, or angle at the pole, which con- stitutes the error of the transit for such star. It must be observed, that if the line of collimation deviate from the south either eastward or westward, it also deviates (the telescope being turned on its axis) from the north towards the same part ; and that when the telescope is reversed on its supports, the deviation is from the south or north towards the contrary parts : if the transit of a star below the pole be observed, the correction which, for a transit above the pole, north or south of the zenith, would be additive, must then be subtractive, and the contrary. At the Royal Observatory, Greenwich, the error of collimation is found by means of a telescope placed, for the purpose, in the north window of the Transit Room. See Collimator, art. 125. 89. When it is required to ascertain that the central or meridional wire is in a vertical position, the optical axis being horizontal, it is only necessary to observe that the wire ex- actly covers or bisects a terrestrial object, as the white disk before mentioned, while the telescope is turned on the pivots of its horizontal axis ; or a star which passes the meridian near the pole may be made to serve the same purpose as the terrestrial mark. If the mark or star should appear to de- viate from the wire near the upper and lower parts of the field of view, the eye-piece with the wires must be turned on the optical axis of the telescope till the object remains bisected in the whole length of the wire. The horizontality of the wire which should be at right angles to the meridional wire and those which are parallel to it is ascertained by placing it on an equatorial star when the latter enters the field of view, and observing that the star is bisected by the wire during the CHAP. IV. TRANSIT TELESCOPE. 71 time of its passage across that field. It is scarcely necessary to remark, that the wires in the eye-piece of the telescope must be adjusted by drawing them towards or from the eye till they appear well defined ; and till, at the same time, a star which may be observed near one of them does not change its position on moving the eye towards the right or left hand. The coincidence of the horizontal wire with a diameter passing through the optical axis of the telescope, is of small importance when it is not intended to use the transit instru- ment for the purpose of obtaining altitudes ; but, in the event of this being one of the objects contemplated, the adjustment must be carefully made, or the error accurately determined ; the process which is to be employed will be explained in the description of the altitude and azimuth circle (art. 106.). 90. The advantage of having one wire or more parallel to, and on each side of that which is in the plane of the meridian, is, that the unavoidable inaccuracy in estimating the time at which a star, on its transit, appears to be bisected by a wire, may be almost wholly corrected by using a mean of the times at which the bisections take place on all the wires. If the distances between the parallel wires were precisely equal to one another, an arithmetical mean of the times (the sum of all the times divided by the number of wires) might be con- sidered as the correct time of the transit at the middle wire ; but, on account of the inequalities of those distances, a mean of the times at which any star appears to be bisected by the several wires is to be taken for the time of the transit at an imaginary wire situated near the central wire ; and the dif- ferences between the times of the transit at the several wires and at this imaginary wire may then be taken. These dif- ferences being multiplied by the cosine of the star's declination (art. 70.), will give the corresponding distance for a star supposed to be in the plane of the equator. 91. In a small transit telescope having five wires, it was found, for a star supposed to be in the plane of the equator, that the differences between the times of the transit at each wire, and at the imaginary mean wire, were as follow : First wire + 51".601 Second + 25".590 Third + 0".110 Fourth - - 25".890 Fifth - 51".410 When, therefore, the transit of a star has not been observed at all the wires of a telescope; if it be required to obtain, from such observations as have been made, the time of transit at the imaginary mean wire, one of the following processes F 4 72 ASTRONOMICAL INSTRUMENTS. CHAP. IV. may be used : From a table formed as above, take the number corresponding to each wire at which the transit has been observed, divide it by the cosine of the star's declination, and add the quotient, according to its sign, to the observed time of transit ; the result is the time of transit at the mean wire : then a mean of these separate means will be the required time of transit at the imaginary mean wire. Or, from a table as above, take the number corresponding to each wire at which the observation has been made ; then take a mean of these, and divide it by the cosine of the star's declination. Take a mean of the observed times of transit, and add to it, according to its sign, the quotient just found ; the result will be the time of transit at the imaginary mean wire. It must be ob- served, however, that the signs of the numbers in the table formed as above are adapted to the culminations of stars above the pole : those signs, and also the order of the wires, must be reversed when the transits are observed below the pole. 92. The equatorial differences in a table such as that which has been given above, is generally found from the ob- served transits of stars near the pole, as a Polaris and S Ursae Minoris ; and these stars describing very small circles about the pole, their paths are sensibly curved in the passage from the first to the last wire : therefore, though the intervals between the wires were equal, the observed times of passing such equal intervals would not be equal. Thus let abc be part of the arc described by a star in consequence of the diurnal rotation, and , b, c, places of the star when bisected by three of the wires, of which let the straight line drawn through c be the middle wire : it is manifest (the intervals am, bn, perpendicular to the middle wire being the distances of the wires at a and b from the mean wire) that the observed in- tervals between the time of transit at a and c, b and c must be reduced in the ratio of the arcs ac, be to the lines am, bn; that is, of (= 0.00436m) to sin. 15m, m being the minutes of time in the observed intervals between the transits, and TT the half circumference of a circle whose radius is unity. 93. In order^to obtain the time of transit at the mean wire, when a planet is observed, the number given in the above table, after being divided by the cosine of the planet's de- clination, must be increased or diminished by the product of that quotient multiplied by the increase or diminution of the planet's right ascension (in time) during one second of time, as a correction on account of the variation in the planet's right CHAP. IV. TRANSIT TELESCOPE. 73 ascension while it is passing between the wire at which the transit was observed and the mean wire. When the moon is observed, the number in the table, after being divided by the cosine of her declination, must, in like manner, be in- creased on account of the increase of her right ascension in such interval of time : but the time in which this celestial body passes between two wires in a telescope is further af- fected by the small variation of her parallax in that time ; and a correction on this account will be presently given (art. 161.). The variations of the right ascensions may be taken from the Nautical Almanac; for the moon, in the pages containing the moon's daily right ascensions and declin- ations ; and for a planet, in the columns of geocentric right ascensions. 94. When it is required to place a transit telescope in the plane of the meridian, an approximate knowledge of the position of a line on the ground in a north and south direc- tion must first be obtained : this may be done in various ways, and one of the most simple depends on observing a star through the telescope of a well-adjusted theodolite at the instants when, in the course of the same night, it has equal altitudes ; for the arc apparently described by the star above the ho- rizon being a segment of a circle, if a picket be driven into the ground in the direction of the telescope when it is made to bisect the horizontal angle between its positions at the two times of observation, a line traced on the ground from the picket to a point vertically under the centre of the theodolite will be nearly in the direction of a meridian line. This me- thod, or one similar to it, must be put in practice when the observer is unacquainted with the longitude of his station, and when his watch is not regulated so as to show the time of an observation correctly, or when, though possessing a surveying compass, he may not know the declination of the needle. Little accuracy can be expected in a first operation, but the process may be repeated several times during one night with the same star, and a mean of the bisecting lines will be very nearly in the true direction of the meridian. 95. After a transit telescope has thus been set up very nearly due north and south, when also the horizontality of the axis of motion, the correctness of the line of collimation, and the intervals between the wires at the focus of the object-glass have been ascertained, it is necessary to have the means of making the optical axis of the telescope describe accurately a great circle of the sphere coincident with the plane of the meridian, and also of determining the amount of any acci- dental deviation of the axis from that plane. Two methods 74 ASTRONOMICAL INSTRUMENTS. CHAP. IV. are. commonly employed for this purpose, it being supposed that the azimuthal deviation does not exceed a few seconds of a degree : one of them depends on the observed transits of two stars which differ considerably in altitude, and the other upon the observed transits- of a circumpolar star above and below the pole. With respect to the first method, let a hemisphere of the heavens be projected on the plane of the observer's horizon, and let that horizon be represented by the circle A w M E, of which the centre z is the projection of the zenith. Draw the diameter p z M for the meridian, in which let P be the pole of the equator, and let AZA'be the projection of the vertical circle in whose plane the optical axis of the telescope moves: let also s be the place of a star when it is seen in the telescope, and through it draw the horary circle P S ; then (art. 61.) we have in the triangle P z s, sin. z S sin. S P z = sin. P z S . : sm. p s or, since the angle at P and the supplement of the angle at z are very small, we have, employing the number of seconds of a degree in p and z instead of the sines of those angles, J sin. (P s T P z) -, or P = z - V - nearly : 1 QlT- T> CJ 9 sin. z s sin. P s 3 sin. p s and, in order that the angle P may be expressed in seconds of sidereal time, the second member of the equation must be divided by 15. The formula may be represented by P = z n for the first of two stars s and s', which enter the telescope, and by p' = z n' for the second. Now, for the first star, let t be the time of the transit, observed on a sidereal clock, T the true time, or the right- ascension of the star, given in the Nautical Almanac ; and for the second, let t' and T' be the corresponding times. Let e represent the error (supposed to be unknown) of the clock by which the times were observed : then t e T (= P, in time) = z . n, and, f e T' ( = p', in time) = Z . n' : hence (f- T')-(*-T) = Z ('-), and z =^ / ~ T ~~ T) . Thus the azimuthal deviation z ( = A! z M) is found. In using the formula, a deviation of the telescope, that is, of the circle CHAP. IV. TRANSIT TELESCOPE. 75 AZA', towards the south-west and north-east of the true meridian (as in the figure) is indicated by the value of z being positive : and a deviation towards the south-east and north- west by z being negative. This rule holds good whether the upper or the lower star culminate first, and whatever be the positions of the two stars with respect to the zenith and the pole. The value of z thus determined is expressed in seconds of sidereal time, and it must be multiplied by 15 in order to reduce it to seconds of a degree. It may be observed also, that if the star pass the true meridian (as in the figure) before it is seen in the telescope, the value of z n or of z n' (the cor- rection of the time of transit) must be subtracted from the observed time in order to give the time of the transit over the meridian p z. On the contrary, the correction must be added if the star is seen in the telescope before it comes to the true meridian. The difference between this time of transit over the meridian, and the calculated right ascension of the star, in the Nautical Almanac, is the error of the sidereal clock. Note. In the second equation for p, the lower sign is to be used when the star comes to the meridian below the pole. 96. For the second method, let a hemisphere be projected as before, and let s s' be the places of a star at the times of its observed transits above and below the pole p. Then the angle SPS', expressed in time, measures the least of the two intervals of time between the transits, and, letting fall P t perpendicularly on the vertical circle z s s', the angle s p t will be equal to half that interval, which consequently is known from the observed transits. Now (art. 62. (df) ) we have in the right-angled spherical triangle P s t, r. cos. PS= cotan. PS cotan. PS. But PS being nearly equal to a right \ / angle, we may, for cotan. tp s, put the number of seconds in the complement (to 6 hours) of half the least interval of time ; let this number, after being multiplied by 1 5, be represented by D : also the angle P s t being very small, we may write for its cotangent, that is, for ^ '- , sin. P s t the term (p s t being expressed in seconds of a degree), p s t and then the above equation will become cos. P S = - , or = . Again, in the spherical triangle PZS, we COS* Jr S have (art. 61.), 76 ASTRONOMICAL INSTRUMENTS. CHAP. IV. sin. P z : sin. p s z ( sin. p s t) : : sin. p s : sin. P z s, or, sin. p z : P s t : : sin. p s : z ; from which proportion, after substituting the above value of PS, we obtain, T , D sin. PS D tan. P S z (m seconds) r , or = ^ , cos. P s sin. P z sin. p z the azimuthal deviation required. 97. In a regular observatory, a large circle of brass attached to the east or west face of a wall, or stone pier, is used for obtaining the altitudes of celestial bodies above the horizon, or their distances from the zenith ; or again, their distances from the pole of the equator. Such an instrument, called a mural circle, is generally of considerable dimensions (6 feet diameter), and it turns upon a horizontal axis, part of which enters into the supporting wall or pier ; either its side or its edge is graduated, and six micrometer microscopes attached to the face of the wall at the circumference of the circle, and at nearly, or exactly, equal distances from each other, are used in reading the subdivisions of the degrees by which the required altitude or distance is expressed. A telescope is made to turn with the circle on the horizontal axis, in making the observation ; but it is also capable of being turned independently, on the same axis, and of being made fast to the circle in several different positions, with respect to the zero of the graduations, in order that the angular admeasurement may be read on any part of the cir- cumference at pleasure. 98. In general the horizontality of the axis of motion is verified by means of plumb-lines, or spirit-levels, or by observing both by direct view and by reflexion the transits of a star at the several wires in the eye-piece of the telescope as was mentioned in the account of the transit instrument ; and the line of collimation is made perpendicular to the horizontal axis by the aid of meridian marks previously set up towards the north and south of the telescope by means of a transit instrument, which admits of being reversed on its supports. The deviation of the plane of the circle from the meridian may be found by observing the transits of two stars differing considerably in altitude, or of a circumpolar star when at its greatest and least elevation ; but great accuracy of adjustment in this respect is evidently of less importance for an instrument which is intended to give altitudes or declinations only than for one by which transits are to be observed. CHAP. IV. MUIIAL CIRCLE. 77 99. If it be intended to obtain at once, from observation, the polar distances of celestial bodies there must be previously found what is called the place of the pole on the circle, that is, the number of the graduation at which the index stands when the optical axis of the telescope is directed to the pole in the heavens. For this purpose, the altitudes of a circum- polar star at its greatest and least elevations must be obtained by the instrument, and corrected on account of the effects of refraction: then, since the pole is equally distant from the two corrected places of the star, half the sum of the altitudes so corrected will evidently be the required polar point. A mean of the like observations on several circumpolar stars will give the polar point with sufficient accuracy; and this point is to be considered as the zero of the graduations when the polar distance of any celestial body is subsequently to be obtained. 100. If the angular distances of celestial bodies from the zenith are to be immediately obtained from observation with a mural circle, there must be previously found what is called the horizontal point, that is, the number of the graduation at which the index stands when the optical axis of the telescope is in a horizontal position. For this purpose, the altitude of a star on the meridian must be observed by direct view, and also by reflexion in a trough of mercury ; both observations being made, and the altitudes read on the circle while the star is in the field of the telescope : the number of the graduation corresponding to the middle point between the places of the index at the two observations will evidently, since both observations are equally affected by refraction, be the required horizontal point, or that which is to be considered as the zero in subsequently determining the altitude of any celestial body ; and a mean of many such observations will give it with sufficient accuracy. The following is, however, considered as a more correct process for determining the horizontal point. On any night let the altitudes or zenith distances of two stars be succes- sively observed by direct view, and on the next night let one of them, suppose the first, be observed by direct view, and the other by reflexion from mercury : then, from a table of refractions, find the change which, on account of differences in the density and temperature of the air, may in the interval have taken place in the altitude of the first star ; and from the Nautical Almanac find the difference, if any, in its de- clination. Let the amount of these changes be applied as a correction to the zenith distance of that star on the first night, and let the difference between this corrected zenith 78 ASTRONOMICAL INSTRUMENTS. CHAP. IV. distance and the observed zenith dis- tance of the same star on the second night be considered as a change in the zenith distance of either star on account of a derangement of the instrument. Let BB" represent this change, and let it be applied by addition or subtraction to the zenith distance (suppose ZB) of the second star on the first night, so that ZB" may be considered as the zenith distance of that star on the second night. Now let ZA' be the zenith distance of the same star when observed by reflexion on the second night; then B^A' will represent the double altitude of that star, and the middle point between B" and A' will be the place of the horizontal point on the circle. Several pairs of stars should be observed in like manner, and a mean of the results taken. 101. In using the mural circle for the purpose of ob- taining by observation the declination of a star, if, when the star is bisected by the horizontal wire it is not on the central or meridional wire, but on one of those which are parallel to it, two corrections will be required : one of them depending on the change in the star's polar distance between the time of the observation, and the time at which the star was, or would be on that central wire ; and the other on the distance be- tween the parallel of declination apparently described by the star and the great circle passing through it, of which the horizontal wire is considered as a part. The first correc- tion is readily found from the variation of the declination, or polar distance of the celestial body, given in the Nautical Almanac ; and the second may be obtained from the formula |p' 2 . sin. 2PM sin. 1" (art. 71.), in which p is (in seconds of a degree) the equatorial distance from the middle wire to that at which the observation was made, and P M is the star's north polar distance. 102. If the declination or polar distance of the upper or lower limb of the moon is to be obtained from observation when that limb is not entirely enlightened, there will be re- quired a correction which may be thus determined. Let PM be the meridian, s the sun, c the centre of the moon at the time of culminating, and C S an arc of a great circle passing through the moon and sun : the moon's disk having a crescent form, apa'q, let a be the point which is in con- tact with the horizontal wire of the telescope at the time of the observation. Imagine am to be a horizontal line passing through a, and let b be the nearest extremity of a vertical CHAP. IV. MURAL CIRCLE. 79 diameter of the moon ; then b in is the correction required: this is manifestly equal to Ca versin. acb, in which ex- pression c# is the semidiameter of the moon, the angle acb is equal to the complement of MCS ; and this last angle may be found sufficiently near the truth by means of a common celestial globe. In a similar manner may the correction be found when the moon is oval, or gibbous, as ap'a'q: in that case s' being the supposed place of the sun, the required correction b'm' is equal to c a versin. a' c b f ; and the angle a' c b' is the com- plement of PCS', which may be found as in the other case by means of a celestial globe. 103. The Greenwich mural instrument consists of two circles which, being on the same horizontal axis, act as counterpoises to each other, and neither plumb-lines nor spirit-levels are employed in their adjustment. They are brought as near as possible to the plane of the meridian, and each is provided with six microscopes nearly equally distant from one another at the circumference. Each also is pro- vided with an artificial horizon of mercury, showing as much as possible of the reflected meridian. The meridian circle, which was made by Reichenbach for the observatory at Gottingen, in 1820, serves at once for a transit instrument, and for measuring altitudes : its telescope is five feet long, and it has, at the focus of the object-glass, seven vertical and two horizontal spider threads. The hori- zontal axis is about three feet long, and it carries, on one side, two concentric circles, three feet in diameter, whose outer surfaces are nearly in one plane : the exterior circle revolves with the telescope, and carries the graduations ; and the in- terior circle would turn on the same axis, were it not that it may be made immovable by means of a clamp attached to the adjacent pier : on this circle are four indices, with verniers, each of which is at forty-five degrees from a vertical line passing through the centre. A suspended level serves to place the axis in a horizontal position ; and the zero of the graduations is at the top, or zenith point. The instrument is capable of being reversed on the axis, in order, by making the observations with it in opposite positions, to eliminate the error of collimation : and in observing the circumpolar stars, the zenith distances, both at the superior and inferior cul- mination, are taken by direct view, and also by reflexion. 104. The most useful instrument in an observatory which 80 ASTKONOMICAL INSTRUMENTS. CHAP. IV. may be established for temporary purposes in connection with astronomy and geodesy is the Azimuth and Altitude Circle, since it possesses the properties of a transit telescope and a mural circle together with those of a theodolite. The altitude circle, which carries a telescope, is capable of being fixed in the plane of the meridian; and when necessary, it can be turned so as to allow the observer to obtain, at the same time, the azimuth and altitude of a celestial body or of a terrestrial object. The annexed figures represent two views of the instrument ; in the first is seen the edge, and in the second the face of the vertical, or altitude, circle : the lower part consists of a circular brass frame ABD, which may be from 20 to 24 inches in diameter, and is capable of having its upper surface made parallel to the horizon by means of three screws, two of which are seen at c and c' ; these rest on a firm table, or on the top of a stone pedestal, and within the ring A B is the graduated circle which turns on the vertical axis of the whole instrument. The ring is capable of turning in azimuth as much as 4 or 5 degrees, in order that a telescope E, which is attached to the frame, may be directed accurately to a mark made on some distant object in the direction of the meridian, and it carries two micrometer microscopes, r, G, one at each extremity of a diameter, by which the minutes and seconds in the observed azimuthal angle are read. A vertical pillar H of brass, in which the radii of the horizontal circle are fixed, carries a stage K with four pillars, two of which, as MN, on each side, are connected together by a horizontal bar PQ, and on these CHAP. IV. ALTITUDE AND AZIMUTH CIRCLE. 81 bars, in notches cut in the form of double inclined planes, rest the two extremities of the horizontal axis of the vertical circle RS whose diameter may be the same as that of the former circle. The vertical circle is graduated, and between the two brass pillars on one side of the instrument are fixed, one above the other, at the opposite extremities of a diameter of the circle, two micrometer microscopes F', G', by which the minutes and seconds in the observed altitude or zenith distance are read. This circle carries a telescope TY in the direction of a diameter ; and in the focus of the object glass is placed a net- work, consisting of three or five wires in horizontal, and as many in vertical positions ; one arm of its horizontal axis is hollow, and a lamp is applied between the pillars on one side of the circle for the purpose, as in the transit instrument, of enlightening the wires when a star is observed by night. A brass ball, suspended in a vessel of water by a fine wire which is attached to the upper part of the instrument, serves to show when the axis of the azimuthal motion is in a vertical position by the suspending wire appearing, during a revolution of the instrument horizontally, to bisect a small dot on a white disk which is fixed below the rim of the vertical circle : this adjustment is effected by successive trials, in part by means of the three screws which, passing through the horizontal frame, rest on the table or pedestal, and in part by a small movement of the disk itself. The axis of the vertical circle is capable of a small movement by means of a capstan-headed screw which raises or depresses the double inclined plane in which one end of the axle rests ; and its horizontality is shown by a spirit level which may be made to rest on the cylindrical pivots of the axle. 105. In the telescope, each of the wires which should be parallel to the horizon may have its position in that respect verified by causing a distant and well-defined terrestrial mark to be bisected by it, and observing that it continues to be bisected while it remains in the field of view, on the whole instrument (previously adjusted) being turned in azimuth; or by directing the telescope to an equatorial star when on the meridian, and observing that while the star appears to move across the field of view it is bisected by the wire. Each of the wires, which should be perpendicular to the last, may have its position in this respect verified by observing that a terrestrial mark continues to be bisected by it while the telescope is moved up or down upon the horizontal axis. 106. In this instrument it is of importance that the middle wires, both that which is parallel to the horizon and that which is at right angles to it, should pass through the optical G 82 ASTRONOMICAL INSTRUMENTS. CHAP. IV. axis of the telescope : the position of the latter wire may be verified, as in the transit telescope, by making it bisect some well-defined mark ; and then, having reversed the pivots of the vertical circle on their supports, so that the graduated face of the circle which, before, may have been towards the east, shall now be towards the west, observing that the wire is neither to the right nor left of the centre of the mark. Or the error of collimation in azimuth may be found by means of a micrometer, as explained in the description of the transit telescope (arts. 87, 88.). In order to verify the position of the horizontal wire, or find the deviation in altitude of the point of intersection from the optical axis (commonly called the error of collimation in altitude or the index error of the instrument), the apparent zenith distances of a star must be observed when on the meridian, with the graduated face of the vertical circle in contrary positions ; and then half the difference between those distances will be the value of the error. Let the circle ZDN (fig. to art. 100.) be in the plane of the meridian passing through the optical axis of the telescope, and let ZCN be a vertical line passing through c, the centre of the altitude circle in the instrument; also let A be the true or required place of the intersection of the wires at the focus of the object glass when a star on the meridian is at s in the line AB produced: then if &B' be the observed position of the optical axis, or a be the place of the inter- section when the latter appears to coincide with the star s, the arc aN or ZB' will express the apparent distance of the star from the zenith. Next, let the instrument be turned half round in azimuth ; then the line C A will be in the position CA', so that NA is equal to NA', and the line ca will be in the position c'; and if the circle be turned on the horizontal axis passing through C till the star S appears as at first to coincide with the intersection of the wires, CA' will be in the position CA and a' will be at b at a distance beyond A equal to A! a! or A a ; the apparent distance of the star from the zenith will then be Z>N or ZB". Hence it follows that half the sum of the apparent zenith distances z B', ZB" will be equal to ZB, the true zenith distance; or half the difference between the apparent zenith distances will be the index error of the instrument in altitude. It will be most convenient to obtain the first apparent zenith distance when the star is on the central horizontal wire just before it comes to the intersection of the latter with the meridional wire, and the second apparent zenith distance when the star is on the same horizontal wire just CHAP. IV. ZENITH SECTOR. 83 after it has passed the intersection ; for thus the star, particularly if near the pole, will not sensibly change its altitude during the observation. A mean of several ob- servations of this kind, on the same or on different stars, will afford a very nearly correct value of the index error. 107. The Zenith Sector is an instrument by which the angular distance of a star from the zenith of the observer is immediately obtained with great accuracy, the star being on, or very near the plane of the meridian of the station, and being one of those which, in that situation, are within a few degrees, or even minutes, of the zenith. Such distances are taken when it is required to ascertain from observation the effects of nutation, or aberration, or the parallax of the fixed stars; and in geodetical operations they serve to determine astronomically the latitudes of the principal stations. The instrument consists of a metal plate, or a system of bars, which is placed in a vertical position, and on which, near the lower extremity, is a graduated arc extending to five or ten degrees, or in some cases to five minutes only, on each side of the zero point : this plate is capable of being turned quite round on a vertical axis, the lower extremity resting on a conical pivot, and the upper extremity turning in a collar at the top of the frame by which the instrument is supported. A telescope is attached by several rings to an index bar which turns on a horizontal axis near the top or at the middle : it is furnished with wires crossing one another as usual in the field of view, and the lower part of the index bar carries a microscope for the purpose of subdividing the graduations by which the observed angular distance is ex- pressed. The zenith point may be determined for this instrument in the same manner as for the mural circle. 108. With all the zenith sectors hitherto constructed it is necessary to make observations on successive nights having the instrument in reversed positions ; but, in order to obviate the inconvenience and delay caused frequently by unsettled weather, the Astronomer Royal has recently caused one to be constructed by which zenith distances may be obtained in one night. The whole stand of the instrument is of cast- iron, and consists of a base AB with two upright piers which are connected together at top by a horizontal bar CD. Three foot screws pass through the base, and rest upon a plate, or tray, which is bolted or screwed to the top o{ the pedestal serving to support the whole instrument. Between the piers is a frame ET of bell-metal about four feet long, which is capable of being placed in direct and re- versed positions ; and in each of its opposite extremities is G 2 84 ASTRONOMICAL INSTRUMENTS. CHAP. IV. sunk a conical hole which receives a corresponding pivot attached to the base and to the upper hori- zontal bar, so that whichever of the two extremities is uppermost the frame can be made to revolve about a vertical axis. On one side of this frame is the bar HK, which carries the telescope NP ; and both bar and telescope have a move- ment of small extent on each side of a vertical line, in a plane parallel to the front of the frame and on ? a horizontal axis which passes A [ through the latter. Near each extremity of the revolving frame is a graduated arc with two micrometer microscopes for the subdivisions. The vertically of the axis of the frame is ascertained by three spirit-levels which are attached horizontally at the back of the frame, and have the power of being reversed when the latter is inverted on the conical pivots. All the corrections due to the want of adjustment in the instrument are to be made by computation, after they have been ascertained by the observed transits of stars with the frame in direct and reversed positions ; the reversions being with respect to the east and west sides and to the upper and lower extremities. 109. In making an observation there must be two ob- servers ; the first sets the telescope as nearly as possible to the zenith distance of the star, and reads the four microscopes ; the second at the same time reads the scales on the levels. As soon as the star has arrived at a convenient part of the field, and before it comes to the centre, the first observer bisects it by the micrometer wire and reads the micrometers : he then turns the instrument half round in azimuth. The second observer now bisects the star by means of the tangent screw, or by the micrometer screw, and then reads the four microscopes ; lastly, the first observer reads the levels. This completes the double observation; and the zenith distance may be obtained by the following rule : Add together the mean readings of the microscopes, their corrections, the mean of the equivalents for the three levels and the equivalent for the micrometer reading : the mean between the two sums for the opposite positions in azimuth will be a quantity cor- responding to a zenithal observation. Such a mean may be obtained for two or more stars in one night ; and the difference between the number of the graduation expressing CHAP. IV. EQUATORIAL. 85 the zenith point and the sum for each star is the true zenith distance of that star. 110. The Equatorial Instrument is occasionally used to obtain by direct observation the right ascensions and polar distances of celestial bodies when not in the plane of the meridian, but it has been of late chiefly employed in determining the differences between the right ascensions and declinations of stars which are visible at the same time in the field of the telescope, and in finding the positions and relative movements of double stars. It consists of two graduated circles CD and EF, of which one is always parallel to the plane of the equator, being fixed at right angles to the mathematical axis AB of a bar, or sys- tem of bars, which axis is in the plane of the meridian, and inclined to the ho- rizon in an angle equal to the lati- tude of the place; it is consequently parallel to the axis of the earth's ro- tation : the other, called the declina- tion circle (to which in general the telescope HK is fixed), turns upon an axis which is always perpendicular to the former; and the circle is consequently in the plane of the meridian when its own axis, whose extremity is seen at G, is parallel to the horizon. But the bar or frame which carries the equatorial circle c D turns upon its axis A B, and carries with it the axis of the declination circle, so that the latter circle is in the plane of a horary circle inclined to the meridian when its axis is inclined to the horizon. The telescope being made to turn with the circle upon this axis, it is evident, when the telescope is directed to a celestial body, that an index fixed to the stand of the instrument may be made to show on the equatorial circle the horary angle of that body, or the angle which the plane of the declination circle makes with the plane of the meridian ; and that an index fixed to the polar axis may be made to show on the declination circle the distance of the celestial body from the pole of the equator. The equatorial circle may be turned round by hand, or by an endless screw working in teeth on the edge of the circle ; G 3 8G ASTRONOMICAL INSTRUMENTS. CHAP. IV. but it is now usual to connect that circle with a clock-work movement by which, when the telescope has been directed to a star, it may be made to revolve about the polar axis with a velocity equal to that of the earth's diurnal rotation; and thus the celestial body will remain in the field of the telescope during the time of making the observation. The hands of the observer are, therefore, left at liberty to turn the micrometer screw at the eye-piece of the telescope when it is required to ascertain, for example, the distances of two stars from each other. The framework MN consists of four bars, or tubes, two on each side of the circle E F, to whose plane they are parallel ; and those which are on opposite sides of the circle are at such a distance from it that the circle with the telescope can turn freely between them in its own plane. The upper extremities of the four bars which form the frame MN are let into a ring at N, and from this ring rise three arms which terminate in the conical pivot at B. The whole apparatus is supported at A and B in conical cups or sockets which receive the pivots ; the lower socket being capable of a small movement in the plane of the meridian, and at right angles to that plane, for the purpose of allowing any derangement of the polar axis A B to be corrected. 111. An instrument like that which is above represented, being supported at the two extremities A and B, is only fit for a fixed observatory, but the following figure represents one in which the polar axis can be placed at any angle with the horizon, and consequently it may be set up in any part of the world. The part MN represents the circular base of the whole instrument, and at each of the opposite extremities of one of its diameters is a pillar or stand BQ ; the upper parts of these two pillars support a bar which turns on a horizontal axis, passing through B in an east and west direction; and, in the diagram, 'it is supposed to be perpendicular to the paper. At right angles to this bar is< fixed a tube in the direction AB, which, turning on the axis just mentioned, and carrying a gra- duated semicircle a b, its axis AB may be placed at any angle with CHAP. IV. EQUATORIAL. 87 a vertical line passing through it; consequently, at any station whose latitude is known, on bringing that vertical line and the axis AB in the plane of the meridian, the latter axis may be made parallel to that of the earth's rotation. Within this tube is a solid cylinder to the top of which is attached a rectangular brass plate at c D, perpendicular to the axis of the cylinder and tube, and capable of being turned upon that axis. The length of the plate, of which CD is one extremity, is about equal to the diameter of the circle MN; and the direction of the length is in the diagram supposed to be perpendicular to the paper. At each of the opposite extremities of this plate, and per- pendicularly to its plane, rises an arm, one of which appears from B to G, and the upper parts of these arms receive the extremities of a bar whose axis is parallel to the length of the plate at CD, or perpendicular to the polar axis AB (in the dia- gram the axis of the bar is also supposed to be perpendicular to the paper). The bar, near one of its extremities, as G, carries the declination circle EF, and, at the opposite extremity, the telescope HK: the circle and the telescope turn together on the axis passing through G ; and at the same time the circle, telescope, and axis turn upon the polar axis AB : thus the circle with its telescope is constantly in a plane parallel to that of some horary circle^ and the telescope may have any movement in decimation. 112. If it be supposed that the polar axis of an equatorial instrument is already nearly parallel to the axis of the earth's rotation, its position with respect to the latter axis may be determined by the following processes. Direct the telescope to any one of the circum-polar stars whose apparent places are given in the Nautical Almanac, when the star is on, or nearly on the plane of the meridian, and (the index of the declination circle being at or near zero when the telescope is directed to the pole of the equator) observe by that circle the star's distance from the zero point ; then turn the instrument half round on the polar axis, and having again directed the telescope to the star, read the distance of the index from zero. Take the mean of these distances, and find the refraction due to the star's altitude : the sum of these values is the corrected instrumental distance of the star from the pole ; and, being compared with the polar distance of the star in the Nautical Almanac, the difference will be the error of the polar axis in altitude. This error, in an instrument constructed like that which is represented in the figure to art. 110., may be corrected by two of the screws at the foot of the axis (the angular movement of the axis depending on a G 4 88 ASTRONOMICAL INSTRUMENTS. CHAP. IV. revolution of the screw having been previously found). Half the difference between the two polar distances read on the declination circle is the index error of that circle : and the index may, by its proper screws, be moved through a space equal to that half difference in order to render the readings equal, in the two positions of the circle. The instrument must now be turned on the polar axis till the plane of the declination circle is at right angles to that of the meridian ; and in this state the telescope must be directed to a star on the eastern or western side. The instrumental polar distance of the star must then be read, and corrected for the effects of refraction in polar distance (which may be computed by a formula proper for this purpose) ; and if the result should not agree with the north polar distance in the Nautical Almanac, the polar axis deviates from its true po- sition, towards the east or west. This deviation must be cor- rected by means of the other two screws at the foot of the axis. In portable equatorials, like that which is represented in the figure to the preceding article, the adjustment of the polar axis is made by the vertical semicircle ab, and by a screw acting on the horizontal axis passing through B : the base MN having been previously levelled by means of the foot screws at M and N. 113. The position of the meridian wire in the telescope with respect to the optical axis is found by bringing the de- clination circle in, or near the plane of the meridian, and ob- serving the transit of a star near the equator. Let both the time of the transit, by the sidereal clock, and the graduation at the index of the equatorial circle be read (it being sup- posed that this circle is divided into hours, &c. like the dial of the clock) ; then, turning the instrument half round on the polar axis in order that the meridional wire, if not correctly placed, may be on the opposite side of the optical axis, ob- serve the transit of the same star : read the time by the clock, and the graduation on the equatorial circle as before. The difference between the times shown by the clock, and the dif- ference between the readings on the circle (which would be equal if the wire passed through the optical axis), being sup- posed to disagree, the wires must be moved through a space equal to half the error by means of their proper screw at one side of the eye-piece. In order that this adjustment may be complete, several successive trials will probably be neces- sary. 114. The axis of the declination circle in the first of the two preceding figures may be made perpendicular to the polar axis by means of a spirit level : thus, bring the declination CHAP. IV. EQUATORIAL. 89 circle in, or near the plane of the meridian, and having made its axis horizontal by means of the level (the latter having been previously adjusted by trying it in direct and in re- versed positions), read the graduation at the index of the equatorial circle ; then turn the instrument half round on the polar axis, and having again made the axis of the de- clination circle horizontal by the level, read the equatorial circle. If these readings should differ by exactly twelve hours, the last-mentioned axis is correct : if not, move the declina- tion circle till the index stands half way between its actual place and that which it ought to occupy in order that the readings may so differ, and make the axis horizontal by means of. its proper screws ; this axis will then be perpendicular to AB. 115. Lastly, the index of the vernier belonging to the equatorial circle should be exactly at the zero of the hours when the plane of the declination circle is in the meridian. Or, when one of the stars whose apparent places are given in the Nautical Almanac appears to be bisected by the middle horary wire in the telescope (the latter being directed to the star), the index of the equatorial circle should show the hour, &c. corresponding to the star's horary angle, that is, the angle between the meridian and a horary circle passing through the star. The star observed for this purpose should be near the equator ; and if its horary angle be computed for the in- stant at which it appears to be bisected by the middle wire, the index of the equatorial circle should point to that angle, or it must be made to do so by its proper screw. 116. In order to use the equatorial instrument for the purpose of obtaining differences of right ascension, and of de- clination ; turn the telescope to one of the objects, suppose a comet : then the clamp screws being fastened, mark the times by the sidereal clock when the comet appears to be bisected by the horary wires, as in a transit telescope, and take a mean of those times ; also bisect the comet by the wire which is perpendicular to those wires. If shortly afterwards a known star should enter the field of the telescope, the whole instru- ment remaining clamped, the time of its transit at the middle wire may be observed on the sidereal clock, and it may be bisected by the perpendicular wire, which, for this purpose, must be moved in polar distance by the screw of the micro- meter in the telescope. The difference between the times of the transits at the middle wire will be the difference be- tween the right ascensions of the comet and star, and the scale of the micrometer will give the difference between their declinations. Should there be no star in the immediate vici- 90 ASTRONOMICAL INSTRUMENTS. CHAP. IV. nity of the comet, the telescope may be moved in declination till any star is found above or below the comet, and the dif- ference of declination must then be read on the circle ; the difference of right ascension being obtained as before from the times of the transits. If the transit and the declination of the star be afterwards observed on the meridian, the star's place, and consequently that of the comet, may be very cor- rectly determined. But if, with the equatorial instrument, a star be observed above, and another below the parallel of the comet's declination ; also, if, after having made the ob- servations on the comet and star, the instrument be turned half round on the polar axis, and similar observations be again made, the mean results will give the place of the comet still more accurately. 117. An equatorial instrument enables an observer readily to find a star in the heavens during the day. For this pur- pose, the right ascension and declination being known, for a given time, by computation, or from the Nautical Almanac, the telescope is clamped in the position which it assumes when the indices of the equatorial and declination circles are set to the computed right ascension and declination respec- tively : then, at the given time, the star will be seen in the field of the telescope. 118. A telescope is frequently mounted on a stand having at its top a cylindrical block of wood, which is cut through obliquely to its mathematical axis, so that the plane of section makes angles with the upper and lower surfaces equal to half the colatitude of the place in which it is to be used ; the upper portion being capable of turning upon that plane about a pin fixed in the lower portion perpendicularly to the same plane. Thus, let A ED, ABD re- present the two portions of the block, and let a plane passing through A D perpendicularly to the paper be the plane of separa- tion: then, if the upper portion be placed in the position which it has in the figure, the superior surface of the block will be in- clined to the inferior surface in an angle equal to the colatitude of the place. If the upper portion be turned till the point A comes to D, those surfaces will be parallel to one another, and both of them may be made parallel to the horizon. To the upper surface is attached, perpendicularly, a brass CHAP. IV. EQUATORIAL. 91 pillar FG, which carries the telescope HK, and a graduated semicircle p Q ; and these turn together on an axis at F. The upper portion of the block being disposed as in the figure, when the pivot on which it turns is in the plane of the me- ridian, its superior surface will be parallel to the equator, and the axis Fa will be parallel to that of the earth's rotation. The rim AE is graduated in hours, &c. like the dial of a clock, and the telescope, while it has a movement on the axis of its pillar FG, is capable, according as the upper surface of the block is parallel to the horizon or equator, of being turned in altitude or declination about the axis passing through F per- pendicularly to that pillar. An index attached to FG, and turning with it on the circle A E, serves to show the movement of the telescope in azimuth or in right ascension, while an- other index, also attached to FG, shows on the circle PQ its movement in altitude or in declination. The handles M and N enable the observer to give slow motions to the telescope by means of endless screws which turn in notches cut in the circumferences of the circles AE and PQ. This cylinder, which is called Smeatorfs Block, from the name of that engineer, enables the telescope to have the move- ments of an altitude and azimuth, or of an equatorial instru- ment ; and a micrometer at the eye-piece of the telescope serves for obtaining, by observation, differences in azimuth and altitude, or in right ascension and declination, when two ce- lestial bodies can be seen in the field at the same time. 119. When a micrometer is to be employed for the purpose of measuring arcs. of small extent, and angles of position, in the heavens, the whole instrument should be turned on the optical axis of the telescope to which it is attached, till the wire mn (fig. to art. 74.) is parallel to the equator; and then the wires p q and s t will be in the planes of two horary circles. The verification of the position of mn is made in the same manner as that of the horizontal wire in the transit telescope (art. 89.), and in this state the index of the micrometer circle E F should be at the zero of the graduations : if not, the error must be ascertained in order that it may be applied as a correction. Then, if (for example) it were required to obtain the angle which a line joining two stars a and b would make with the plane of a horary circle P S passing through one of them ; having turned the part carrying the system of wires upon the optical axis of the telescope, till the wire m n 92 ASTRONOMICAL INSTRUMENTS. CHAP. IV. passes through both the stars, or is estimated to be parallel to such line (as in the figure), the index of the micrometer circle will show the value of the angle Pap, which, being corrected if necessary on account of the index error before mentioned, is the complement of the required angle b a s. The wire mn is not in strictness necessary, for the micro- meter may be turned so that one of the wires pq shall coin- cide with the line ab of the stars ; then the number of the graduation at the index of the micrometer circle being read, that circle, and with it the whole micrometer, may be turned one quarter round upon the optical axis of the telescope, when the wires pq will be perpendicular to the line ab joining the stars. In order to measure the angular distance ab, the whole equatorial instrument must be turned on the polar axis till one of the wires, as pq, passes through the star a (for ex- ample) ; then, turning the micrometer screw which moves the other wire, bring the latter wire to bisect the star b (the first wire continuing to pass through the star a), and read on the screw head belonging to the second wire the graduation which is in coincidence with the index. Again, move the whole equatorial till the first wire bisects the star b, when the second wire will be moved to b' so that a b' will be equal to twice the distance between the stars ; and then, by turning the mi- crometer screw belonging to the second wire, bring the latter wire back to the star a : read the number of revolutions made by this wire, and the parts of a revolution on its screw head ; then half the difference between this and the former reading will be the required distance, in terms of the screw's revo- lutions. 120. The knowledge of the distance ab in seconds of a degree, and of the angle fab, or its supplement, will enable the observer, by letting fall be perpendicularly on p, pro- duced if necessary, to compute a c and b e : the former is the difference between the declinations of the stars a and b, and the latter, when reduced to the corresponding arc of the equator, that is, when divided by the cosine of the star's de- clination (art. 70.), is the difference (in arc) between their right ascensions. If one of the celestial bodies were the moon, whose de- clination at times changes rapidly ; on making the wire m n of the position micrometer a tangent to the path of the moon's upper or lower edge as she appears to move across the field of view, that wire, instead of being parallel to the equator, would, as in the last example, be in some other position, as m n, and CHAP. IV. HELIOMETEK. 93 one of the perpendicular wires, instead of being in the plane of a horary cir- cle P s, would be in the position p q ; so that the observed transit of a star moving in the parallel ef of declina- tion would take place at c: then sc will be the apparent difference of de- clination, measured by the micrometer, between the star and the moon's upper limb ; and the angle PSj9 being given by the micrometer circle, SC the true difference of declination may be found by the formula Sc cos. PS/? sc. (PL Trigon., art. 56.) Again, let the contact of the enlightened edge of the moon at the time of her apparent transit be at d when the upper edge (for example) moves along m n, and let fall dv perpendicularly on PS ; then sd being considered as equal to the moon's semi- diameter, and the angle PSp or DSd being known, we have (art. 61.), PD and PC being the polar distances of the moon and star, and putting small arcs or angles for their sines, , sd sin. P S p , S c sin. p s s P d = -. - ; also, s p c ; : sin. PD sin. PC the sum of these two angles, or the angle c?d is (in seconds of a degree) the whole error in the observed difference between the right ascensions of the moon and star in consequence of the moon's movement in declination. 121. The Heliometer employed by M. Bessel for the pur- pose of ascertaining the parallax of the fixed stars is a telescope about 9 feet long and 6 inches diameter, which is equa- torially mounted, and the whole instrument is turned by clock-work with a movement equal to the diurnal rotation. The object glass is divided into two parts in the direction of a diameter, and the two halves have a parallel movement along the line of separation by means of screws at the eye end of the telescope. The circular head of each of these screws is divided into one hundred parts, and the divisions are great enough to allow a tenth part of each to be estimated. The circle by which the angles of position are obtained is placed at the object end of the telescope, and it has four verniers by which the angles are read to minutes. In observing double stars, one of the halves of the object glass is moved till the four images are brought into one straight line, and all appear at equal distances from one another ; then each star will have moved, apparently, over twice the in- terval between the two, and the measure of this interval is ex- 94 ASTRONOMICAL INSTRUMENTS. CHAP. IV. pressed by the divisions of the micrometer. The half-object glass is then moved in like manner in the opposite direction ; and now the divisions of the micrometer express four times the distance of one star from the other in terms depending upon the unit of the scale. A mean of the readings at the verniers of the position circle expresses the angle of position between a line joining the stars and the hour circle passing through one of them. 122. When a micrometer is used to find the angular posi- tion of a star with respect to one which is with it in the field of view, particularly in observations on double and revolving stars, it is customary to distinguish the place of the star whose position is to be determined by the latter being in one of the quadrants of a circle imagined to be described in the heavens about the other star, considered as situated in the centre. The meridian of the station, if the star is on the meridian, otherwise the hour circle passing through the star, is supposed to cut the said circle in two points n and s, which are the northern and southern extremities of a diameter ; and another diameter pf is supposed to be at right angles to this : the four quadrants are distinguished by the letters nf (north following)) sf (south following), sp (south preceding), and n p (north pre- ceding) ; and they are situated in the manner represented in the figure, which is such a circle described about a star A, and is supposed to be viewed by the naked eye, or in a telescope which does not invert objects. The posi- tion of the line joining the stars A and B would be indicated by the angle /AB in the quadrant nf, and the position of the line AB' by the angle ./AB 7 in the quadrant sf: in like manner the position of a line from A in the other semicircle would be expressed by an angle reckoned from p in the quadrant sp or np. The words following and preceding relate to the position of the star B, in right-ascension, with respect to A. This mode of designating a position has, how- ever, been objected to on account of the mistakes to which it is liable ; and it has been preferred to express the position by an angle as WAB, reckoned from n quite round the circum- ference, in the order of the letters nfsp, or from the north, by the west, the observer being supposed to look southwards. 123. The late Captain Kater invented two kinds of instru- ments which he called Collimators, for the purpose of deter- mining the horizontal and zenith points on astronomical circles, Each kind consisted of a telescope which was either made to rest in a horizontal position on a plate of iron float- ing on the surface of mercury ; or it was fixed in a frame, CHAP. IV. COLLIMATOK. 95 the lower part of which was an iron ring whose plane was perpendicular to the axis of the telescope, and this ring floated on mercury in an annular vessel, so that the telescope being in a vertical position, the observer looked towards the floor, or towards the cieling of the apartment, through the open space in the centre of the vessel. The former was called a horizontal, and the latter a vertical collinlator. 124. In using the horizontal collimator, the vessel, with the telescope floating on the mercury, is placed either on the north or south side of the astronomical circle ; and the telescope of this circle being brought by hand nearly in a horizontal posi- tion, the axes of the two telescopes are made nearly to coin- cide in direction, with the object-glass of one turned towards that of the other. The collimating telescope, which is about twelve inches long, is provided with two wires crossing each other at the focus of the object glass, and care is taken more- over that it shall be in the optical axis : now the rays of light in the pencils which diverge from those wires are, after re- fraction in passing through the object glass, made to proceed from thence parallel to one another ; and consequently, falling in this state upon the object glass of the circle, they form in its focus a distinct image of the wires from whence they diverged. Thus the observer, on looking into the telescope of the circle, sees the wires of both telescopes distinctly ; and, upon making the points of their intersections coincident, it will follow that a line joining those intersections is parallel to the horizon : consequently the graduation at which the index stands on the circle may be considered as the zero of the in- strument when altitudes are to be observed. The vertical collimator is used in a similar manner, and the wires of both telescopes being made coincident, a line joining them is in a vertical position : the index, therefore, stands then at the graduation which must be considered as zero when zenith distances are to be observed. 125. These floating collimators being unsteady, and the optical axes of their telescopes being seldom exactly horizontal or vertical, a telescope mounted on a stand, with its optical axis carefully adjusted, and furnished with a spirit-level which is capable of being reversed on its points of support, has been used for the like purpose. The disposition of the instrument is similar to that which has been described, and the axis of the telescope is made level by means of foot- screws. In order to use a collimating telescope for the purpose of finding the error of collimation in a transit telescope the latter must be furnished with a micrometer: the moveable wire in the micrometer must be brought to coincide with the 96 A8TBONOMICAL INSTRUMENTS. CHAP. IV crossing of the wires in the collimator, and the number of revolutions with the part of a revolution must be read. The same moveable wire must then be brought to the fixed wire in the transit telescope, and the number of revolutions with the part of a revolution again read. These movements and readings are to be repeated several times with the transit telescope in one position ; and the like operations and read- ings must take place with that telescope in a reversed position : then the difference between the two means of the readings, reduced to seconds of time (corrected for the effects of diurnal aberration if thought necessary), is the error of collimation with respect to the middle wire of the transit telescope. The distance (in time) of this middle wire from the place of the imaginary mean wire is supposed to be known (art. 91.), and that distance must be added to or subtracted from the error just found in order to have the exact correction which, on account of imperfect collimation, is to be applied to the mean time of the observed transit of a star ; it being understood that the star is in the equator, otherwise the correction must be divided by the cosine of the stars declination (art. 70.) before it is applied. 126. The Repeating Circle is a graduated instrument carry- ing two telescopes which are parallel to its plane, one on each side : these turn independently of each other on the centre of the circle ; and the whole instrument is supported on a stand, the upper part of which is made to turn round in azimuth, carrying with it the circle and telescopes, and it has a joint which allows the plane of the circle to be placed in any posi- tion which may be required. When the zenith distances of celestial bodies are to be observed, the circle is placed in a vertical plane, and the correctness of its position is determined by a plumb line. Let ZHNO represent the circle in a vertical position in the plane of the meridian (for example), and let s be the place of a celestial body in the same plane. 127. The telescope EF being clamped so that the index is at the zero of the graduations, suppose at E, let the circle be moved in its own plane about c till that telescope is directed to the object s ; and then turn the telescope on the opposite face of the circle till, by the spirit-level attached to it, it is in a horizontal position, as HO. Next turn the circle half round in azimuth so that the telescope EF may lie in the position ef 9 the zero point CHAP. IV. REPEATING CIRCLE. 97 now being at e on the circle, and the telescope HO remaining horizontal but its extremities being reversed. The circle being clamped, turn the telescope at ef about c till the object is again seen in the centre of its field ; it will a second time have the position EF and the number of the graduation at E will express the value of the arc EN 2 :: tang. ANR : tang. ACR. i 114 DIUKNAL PARALLAX CHAP. V. Let the equatorial and polar semiaxes, Cg and Cp, be to one another as 305 to 304, which from geodetical determi- nations (arts. 415. 418.) is the ratio adopted by Mr. Woolhouse in the appendix to the Nautical Almanac for 1836 (p. 58.): C 1 T) ' then we shall have - - 2 =r.9934; therefore if the geographi- Ci^ cal latitude be represented by L, and the geocentric latitude by 1) we have .9934 tang. L mtang. I. The second line in the following table shows, for every tenth degree of geographical latitude, the number of minutes, &c., which should be subtracted from that latitude in order to obtain the corresponding geocentric latitude. 10 20 30 40 45o 50 60 70 80 3 45.5 7 21 9 54 11 16.2 11 27.3 11 17 9 56 7 23 3 56 153. The word parallax, is used to express the angle at any celestial body between two lines drawn from its centre to the points from whence it may be supposed to be viewed : or it is the arc of the celestial sphere between the two places which a body would, at the same instant, appear to occupy if it were observed at two different stations. The celestial arc between the sun and moon, or between the moon and a star, at any instant, will evidently subtend different angles at the eyes of two observers at different stations on the surface of the earth; and in order that a common angle may measure the same arc, it is necessary that each observed angle should be reduced to that which would be subtended by the same arc at the centre of the earth. When the arc measures the altitude of a celestial body above the visible horizon, the correction which must be applied in order to convert the observed angle of elevation to that which would have been obtained if the angular point had been at the centre of the earth, is called the diurnal parallax. This name has been given to it because it goes through all its variations between the times at which the body rises and sets, being greatest when the latter is in the horizon, and least when in the plane of the meridian ; and since every parallax is necessarily in a plane passing through the two points of observation and the object, it is evident that the diurnal parallax will be in a plane passing through the spectator and the centre of the earth ; that is, in a vertical plane. 154. The angular distance between two places to which, in the heavens, a celestial body is referred when it is supposed to be viewed from the sun and from the earth, or from the earth at two different points in its orbit, is called the annual CHAP. V. IN ALTITUDE. 115 parallax. This will be considered further on (art 256.); the investigations which immediately follow relate only to the diurnal parallax. Let s be the place of a spectator on the surface of the earth, supposed at present to be a sphere, and let SB be a section through s, and through c the centre. Let MM' be part of the circumference of a vertical circle passing through the sun, the moon, or any planet ; and let ZH be a quarter of the cir- cumference of the circle when produced to the celestial sphere. Again, let M be the place of the celestial body in the sensible horizon of the spectator, and M' its place when at any altitude ; also let m and TZ, p and q be the points in the heavens to which M and M' are referred when seen from s and c : then the angle SMC, which may be considered as equal to n s m, is called the horizontal parallax, and the angle SM'C or qsp is the parallax in altitude. Let M'S be produced till in A it meets a line CA, let fall perpen- dicularly on it from c ; then, in the right-angled triangles M s C, M'AC, the hypotenuses CM and CM' are equal to one another, and c S, C A represent the sines of the parallaxes : therefore C s : c A : : sin. horizontal parallax : sin parallax in altitude. But in the right-angled triangle CAS, by Trigonometry, cs : CA :: radius (= 1) : sin. ASC (=. cos. M'SM): the angle M'SM expresses the apparent altitude of M'; therefore, rad. : cos. app. alt. of M' : : sin. hor. parallax : sin. par. in alt. Hence, if the horizontal parallax be given (it may be found in the Nautical Almanac) the parallax in altitude may be computed: and as the parallaxes are small, the arcs which subtend them (in seconds) may be substituted for their sines. 155. Since the true altitude of a celestial body M' is ex- pressed by the angle M'CH or (the lines SM and CH being parallel to one another) by its equal M'EM, and since the ex- terior angle M' E M of the triangle s M' E is equal to the sum of the interior and opposite angles ; it follows that the parallax in altitude must be added to the observed altitude of a celestial body in order to obtain the corrected altitude. Let P be the horizontal parallax, p the parallax in altitude, and z the ap- parent zenith distance ; then from the above proportion we have sin. p sin. p sin. z ; I 2 116 DIURNAL PARALLAX CHAP. V. or, if z f express the true zenith distance, sin. p = sin. P sin. (zf + p). It may be observed that the greatest parallax of a celestial body takes place when the body is in the horizon ; and, both above and below the horizon, the parallax diminishes with the sine of the distance from the zenith. The apparent place of a celestial body is always given by observation ; but, in seeking the effect of parallax when the true place of the body is obtained by computation from astronomical tables, it will be convenient first to find the parallax approximatively from the formula p' = p sin. z' (where z f is the true zenith distance), and then to substitute that approximate value in the . formula sin. p = sin. p sin. (z 1 + yf) ; the result will in general be sufficiently near the truth. Or a second approxi- mative value may be found from the formula p" p sin. (z f + //) ; and the value of p" being substituted in the formula sin. p =. sin. p sin. (z f + p") will give a still more correct value of the parallax in altitude. The following series for the parallax in altitude in terms of the horizontal parallax and the true zenith distance is given in several treatises on astronomy : ,. -. N sm. p . , sin.^ P . p (in seconds) =- ^sm. z + "sin. 1" sm. &c. sin. 3 P sm. sin. 3z' and the three first terms are sufficient. 156. If it be assumed that the earth is of a spheroidal figure, and such as would be produced by the revolution of an ellipse about its minor axis, it must follow that the hori- zontal parallaxes, which are the angles formed at a celestial body by a semi-diameter passing through the place of the observer, will be the greatest at the equator and the least at the poles ; and that they will vary with the distance of the spectator from the centre of the earth. The value of the geocentric horizontal parallax at any station whose latitude is given may be investigated in the following manner. Let PSQ be one quarter of the terrestrial meridian passing through any station s; also let P be the pole, and Q the point in which the meridian cuts the equator : again, let M and M' be places of any celestial body, CHAP. V. FOR A SPHEROID. 1.17 as the moon, when in the apparent horizon of an observer at the equator and at any other point S on the earth's surface. Then, since CM may be supposed to be equal to CM' (and, on account of the smallness of the ellipticity of the earth, cs may be considered as perpendicular to s M', as Q M is perpendicular to c Q, so that C Q and C s may represent the sines of the horizontal parallaxes at Q and s) we have, by proportion (p, -sr, and p being the horizontal parallaxes at the equator, the pole, and at any place S respectively) CQ CP : CQ cs :: sin. p sin. -cr : sin. p sin. p. But by conic sections, in an ellipse, the semi-transverse axis CQ being equal to unity, the semi-conjugate axis = c, the excentricity = e, and SCQ the geocentric latitude (art. 151.) of s = Z; we have, as in art. 149., e being very small (0.08 nearly when CQ =: 1), CS = 1 \ ce 1 sin.' 2 /, and CQ cs = | ce 1 sin. 2 /; or the decrements of CQ from the equator towards the pole vary with sin. 2 /. 157. Hence, rad. 2 : sin. 7 /:: CQ CP : CQ cs, or, 1 : sin. 2 I :: sin. p sin. -or : sin. p sin. JP. Let the ratio of CQ to CP be as 305 to 304 as in art. 152.; then 304 305 : 304 :: sin. p : sin. p (= sin. r, the sine of the oUO polar horizontal parallax) ; , . 304 . 1 . . ,., and sin. p sin. p, or ^-^ sin. p expresses the difference oUO oUO between the sines of the equatorial and polar horizontal parallaxes. Consequently, and this last term expresses the difference between the sines of the horizontal parallaxes at the equator and at any place s ; also sin. p sin. p sin. 2 I is equivalent to the sine of the geocentric horizontal parallax at such place ; let it be repre- sented by sin. p'. The following table, which depends on the equatorial hori- zontal parallax of the moon and upon the latitudes of stations, will show by inspection the value of the second term in the I 3 118 DIUKNAL PAKALLAX CHAP. V. C O C P expression ; the ellipticity of the meridian, that is - > CP being = . The terms are to be subtracted from the equa- oUo torial horizontal parallax of the moon (in the Nautical Almanac) in order to reduce it to the geocentric horizontal parallax at any station whose latitude is given. Hor. par. 54 56 58 60 62 Latitudes. 10 20 30 40 50 60 70 80 90 0.9 0.9 1.0 1.0 1.0 1.3 1.3 1.4 1.4 1.5 2.7 2.8 2.9 2.9 3.0 4.5 4.6 4.8 5.0 5.1 6.3 6.5 6.6 6.9 7.0 8.1 8.4 8.7 9.0 9.3 9.5 9.8 10.2 10.5 10.9 10.4 10.8 11.1 11.5 11.9 10.8 11.0 11.6 12.0 12.4 158. If the celestial body be at any point M" elevated above the horizon of the spectator at s, the perpendiculars ca and cb let fall from the centre of the earth upon M'S, M"S produced may represent the sines of the horizontal paral- lax and of the parallax in altitude ; and the hypotenuse C S of the right-angled triangles sc, s&c being common, we have C : cb :: sin. cs : sin. cs; that is, the sines of the parallaxes are to one another as the sines of the distances of the celestial body from the geo- centric zenith. But sin. c s a may be considered as equal to radius (=1) without sensible error; therefore, if the ap- parent geocentric zenith distance ZSM" be represented by z 9 and the horizontal parallax by P', we have rad. : sin. z : : sin. p' : sin. z sin p', and the last term expresses the sine of the geocentric parallax in altitude. 159. When the moon and the sun, the sun and a planet, or the moon and a planet, are very near one another, as in eclipses and occupations, instead of computing separately the parallaxes of the two luminaries in altitude, it is more convenient to substitute in the last expression the difference between their geocentric horizontal parallaxes at the station of the observer : the result will be the difference between the parallaxes of the luminaries in altitude. Thus, if at any place, as s, P' be the geocentric horizontal parallax of the moon, and p' that of the sun (the luminaries being near one another), the difference between their parallaxes in altitude will be expressed by sin. z sin. (P' //)*? where P x p' is CHAP. V. FOUND BY OBSERVATION. 119 called the moon's relative horizontal parallax. The like is to be understood of the difference between the parallax of the sun and a planet, and of the moon and a planet. 160. The horizontal parallax of the moon or of a planet may be found from the difference between the zenith dis- tances of the moon or planet and of a fixed star, observed at two places on or near the same terrestrial meridian. Thus M being the place of the body whose parallax is to be found, E the centre of the earth, and s the place of any fixed star, let A and B be the stations of two observers, which for sim- plicity may be supposed to be on the same meridian, and let them be so situated that the moon or planet, when on the / meridian, may be southward of / one observer and northward of the other : also let it be supposed that the latitudes of the stations A and B are known. Then, by previous agreement between the observers, the zenith distances ZAS and ZAm, Z'BS and Z'BW' being taken when the star, and the moon, or a planet, respectively culminate or come on the meridian ; the arc Sm or the angle SAm, and the arc sm' or the angle SBW' will represent the differences between the zenith distances at the respective stations. But the fixed star s being incal- culably remote, BS may be considered as parallel to AS; hence the angle SA'm may be taken for SAm, and the dif- ference between SA'm or SAm and SB w', will be equal to the angle m'Mm or A MB, which is therefore found. Now, joining the points E and M, the angles A ME, BME, are the parallaxes of the body at M, in altitude, for the ob- servers at A and B respectively, and we have rad. : sin. ZAm :: sin. horizontal parallax : sin. A ME, also, rad. : sin. z' B m! : : sin. horizontal parallax : sin. B M E : therefore, by proportion rad. : sin. ZAW + sin. z'B m! : : sin. hor. par. : sin. AME + sin. BME. If the arc which measures the horizontal parallax, and those which measure the angles AME, BME, be substituted for their sines, then AME + BME being equal to the angle A MB, which was obtained from the observation, we have from the last proportion, i 4 120 DIUKNAL PARALLAX CHAP. V. sin. ZAM + sin.z'BM : rad. :: AMB (in arc, or in seconds) : hor. par. (in arc or seconds) : and thus the horizontal parallax of the moon or planet at the time of observation is found. It should be observed that this method is not applicable to planets beyond the orbit of Mars ; and if the observers be not situated precisely on the same terrestrial meridian, it would be necessary to correct the observed zenith distance of the moon or planet, at one of the stations, on account of the variation in its decimation during the time in which it is passing from one meridian to the other. Since the horizontal parallax of a celestial body may be represented by the angle SMC (arts. 154. 156.), in which the angle at S is, or may be, considered as a right angle, and that S c = M S tan. SMC; it follows that, for different celestial bodies, the tangents of the horizontal parallaxes vary inversely with the distances of the bodies from the earth. 161. The parallax of a celestial body in altitude being obtained, the deviation of the apparent from the true place of the body in any other direction (as far as it depends upon the place of the observer) may be readily found by Plane Trigo- nometry, when the angle at the celestial body between a vertical circle passing through it and a circle of the sphere, also passing through it, in the direction for which the devia- tion is required, is known ; since that deviation may be con- sidered as one side of a right-angled triangle, of which the hypotenuse is the parallax in altitude : but this is not always the most convenient method of determining the deviation in a direction oblique to the vertical circle ; and the following are investigations of formula, by which the parallaxes of a celestial body in right ascension and declination (that is, in the directions which are particularly required for the solution of problems relating to practical aslronomy), are generally obtained. Let THQ (fig. to art. 147.) be part of the celestial equator, stereographically projected, P its pole, and let z be the geocentric zenith of the spectator's station. Let s' be the true place of the celestial body, suppose the moon, and the equinoctial point, so that at a given time H' is the moon's true right ascension (in arc) and PS' her true north polar distance. At the same instant let s be the apparent place of the moon, so that H is the moon's apparent right-ascension (in arc), and PS her apparent polar distance. Then PQ being the meridian of the station, QPS' is the moon's true, and Q p S her apparent, horary angle at the given time. Again, zs' is the true, and zs is the apparent, zenith distance of the CHAP. V. IN RIGHT ASCENSION. 12 1 moon, and ss' is the moon's parallax in altitude; the angle SPS' is her parallax in right ascension, and the difference between PS and PS', is her parallax in declination : also ZP is equal to the colatitude of the station. By Spherical Trigonometry (art. 61.) we have, in the tri- angle SPS', sin. p s : sin. s s' : : sin. s' : sin. SPS'; sin. SS'sin. s' whence, sin. SPS' = ^ f& - ; and in the triangle z p s', sin. z s' : sin. p z : : sin. z p s' : sin. s' : . . sin. P z sin. z p s' whence, sm. s'= - . ^- and by substitution, sin. s S x sin. P z sin. z P s' sin. SPS ; r . sin. PS sin. zs' Let I represent the geocentric latitude of the station. A the true right ascension of the celestial body. D the true declination. z the apparent geocentric zenith distance. p' the geocentric horizontal parallax. a the parallax in right ascension. S the parallax in declination. r the true horary angle. T + a the apparent horary angle. and let the sine of the geocentric parallax in altitude (=rsin. ss') be expressed by sin. p'sin. z. (Art. 158.) Then, by substitution in the last equation, sin. P' sin. z cos. I sin. (T 4- a) sin. a ; a 1 = cos. D sin. z sin. p' cos. I sin. (T + a) , > cos. D. or developing sin. (r + a) in order to eliminate a, sin. p' cos. / . . sin. a (sm. TCOS. a cos. D v and dividing both members by cos. a, tan. a or transposing sin. P x cos. / . . . > sin. a (sm. TCOS. &+ sin. a cos. r) ; cos. D v sin. P cos. I f . ^ tan. a = (sin. r + tan. a cos. T) ; cos. D v } sin. p r cos* I . sm. T cos. D , N tan. a - . - - .... (n), . sm. P r cos. / 1 COS. T COS. D 122 DIUKNAL PAKALLAX CHAP. V. and, approximative^, the second term in the denominator being very small, p' cos. / sin. r , v a~- - .... (in). cos. D Thus is found the parallax in right ascension ; and, on putting any of the formulae in numbers, logarithms with five decimals will suffice. If for sin. r in the last formula there be put the sine of the equatorial interval between any wire and the mean wire in a transit telescope, the resulting value of a will be the effect of parallax in right ascension on the moon's limb at the time of an observed transit at the former wire ; and being divided by 15, it will give, in time, the increase in the time of coming up to the mean wire, or the diminution in the time of passing away from the mean wire, according as the time of observa- tion preceded or followed the transit at the latter wire. This is the correction alluded to in art. 93. ; in the former case it is additive, and in the latter subtract ive. 162. In the spherical triangles PZS, PZS', from either of the formulas (), (5), or (c), art. 60., we have cos. P s cos. PZ cos. zs , cos. z - -. - ; - and sin. P z sin. z s cos. PS' cos. PZ cos. zs' COS.Zrr - ; -- -. - - . sm. PZ sin. zs' Equating these values of cos. z, and cancelling sin. p z which is common, cos. PS cos. P z cos. z s _ cos. PS' cos. P z cos. z s' sin. zs sin. zs' hence, cos. P s sin. z s' cos. P z cos. z s sin. z s' = cos. p s' . sin. z s cos PZ cos. zs' sin. zs ; and by transposition, cos. PS sin. zs' cos. PZ (cos. zs sin. zs' cos. zs'sin. zs)~ cos. P s' sin. z s ; or again, cos. PS sin. zs' cos. PZ sin. (zs' zs) = cos. PS' sin. zs. But sin. (zs' zs), or sin. ss', is equal to sin. P' sin. zs' (art. 155.); therefore, cos. PS sin zs' cos. PZ sin. p' sin. zs':=cos. PS' sin. zs, ^1T1 7 ^ and, (cos. PS cos. PZ sin. p') -^ =cos. PS'. 'sin. zs CHAP. V. IN DECLINATION. 123 Now, in order to eliminate zs and zs', we have (art. 61.), in the spherical triangles, ZPS, ZPS', / sin. PSsin.ZPSx sin. zs : sm. PS :: sm. ZPS : sin. z = ^- , V sm. zs / / sin. PS' sin. ZPS' \ and sm. z s' : sin. PS.: sm. ZPS. sm. z ( = : , ) * \ sin. z s / Equating these values of sin. z, we obtain sin. z s' sin. p s' sin. z P S' B sin. zs sin. PS sin. ZPS* therefore, by substitution, /x sin. PS' sin. ZPS' (cos. PS cos. PZ sm. P') . -. - = cos. p s' ; ' sm. pssm. ZPS (cos. PZ sin. p' sin. ZPS' cotan PS : - \ - = cotan. PS'; sm. PS / sm. ZPS that is, / sin. / sin. p' \ sin. (r + a) , ^ ( tan. D - v '- = tan. (D 8). \ cos. D I sm. T Thus, D being known, we might obtain S, the parallax in declination ; but, as in the process, logarithms with seven decimals must be employed, it would be advantageous to have a formula for & alone. The equation for cotan. P s' may be put in the form , . cos. P z sin. P' sin. ZPS' cotan. P s sm. ZPS' cotan. P s' sm. ZPS = ^ - ; sm. PS and dividing the first member by cotan. PS cotan. PS', it may be put in the form r. , cot. PS' (sin. ZPS' sin. ZPS)) , A \ sm. ZPS' + , (cot. PS cot. P s') = c cotan. PS cotan. p s' 3 v cos. PZ sin. P' sin. ZPS' , N . . ( A ). sm. PS But (PI. Trigon., art. 41.) sin. ZPS' sin. ZPS =2 sin. J (ZPS' ZPS) COS.^(ZPS' + ZPS), or =2 sin. -| a cos. (T + -J a), sin. (PS' PS) and cotan. PS cotan. p s' = ^ r ~ ; sm. P S sm. p s therefore, by substitution, the equation (A) becomes QI n ( T* ^f -L --- n T* ^ i T - : -^ -. { sin. ZP s' + cotan. P s' 2 sin. a cos. (T + A a) = sm. P s sm. P s' cos. P z sin. P' sin. z p s' sin. PS 124 MOON'S AUGMENTATION CHAP. V. . sin.PSsin.PS' or, transposing and subsequently multiplying by -- : 7-, sin. z I* S sin. (p s' P s) =3 cos. P z sin. P' sin. P s' . . . , , x sin. PS - cos. PS' 2 sin. I a cos. (r + Ja) that is, sin. S = sin. I sin. P 7 cos. (D 8) 2 sin. i a cos. (r + J a) / &N - / -- n ^ -cos. D sm. (D 8). sm. (T + a) But (PL Trigon., art, 35.) 2 sin. la- . Bin ' i g ; COS* ~Q CL and in this fraction substituting for sin. a, its equivalent, sin. P' cos. I sin. (r + a) , ^ , , -, (i) above, we have cos. D . sin. p' cos. / sin. (r + a) . f . . 2 sm. ^ a= - - : therefore sin o =2 cos. D cos. I a . 7 . x t >. sin. P' cos. Zcos. (r + ia) . , ^ / \ sm /sm. p'cos/ D o) -- - r - - - ^sm.(D o) . . ( iv ). cos. \ a But again, cos. (D S) m cos. D cos. 8 + sin. D sin 8, and sin. (D 8) = sin. D cos. 8 cos. D sin. 8 ; therefore, substituting these values, and dividing by cos 8, we have sin. P' sin. Zcos. D - sin ' P/ cos ' l cos ' < T + * a > sin. D cos. a tan. S = - .. r . , . . . sin. p' cos. I cos. (T + i o) , 1 ] sin. P sm. Z. sin. D + - - - 7 cos.D f c cos. j a 3 and, approximative^, 8 P' (sin. / cos. D cos. / sin D cos (T + \ a)} ..... (vi). In finding the value of tan. 8, or the tangent of the paral- lax in declination, logarithms with five decimals will suffice. It is easy to perceive that, when a celestial body is on the eastern side of the meridian of a place, the parallax in right ascension increases the true right ascension of a body ; and, when the body is on the western side of the meridian, the parallax diminishes the right ascension : the parallax in decli- nation increases the polar distance of the celestial body in both situations. 163. When the altitude of the upper or lower limb of the sun, the moon, or a planet, is obtained from an observation, the altitude of the centre of the celestial body is found by adding to it, or subtracting from it the angular measure of its semidiameter : this element is given in the Nautical Almanac ; arid when the celestial body is the moon, it is necessary CHAP. V. IN ALTITUDE. 125 that there should be applied to it a correction depending on her altitude above the horizon. A corresponding correction for other celestial bodies is scarcely necessary. In order to investigate it for the moon, let c (fig. to art. 154.) be the centre of the earth, s the place of a spectator on its surface, and let Z'M'M be part of a vertical circle passing through the moon : let also M be the place of the moon when in the horizon of the spectator, and M' her place when elevated above it. Let fall c A perpendicularly on M' s produced ; then if MS, M'A, z'c be each considered as equal to the distance of the moon from the earth's centre, so and SA may represent the diminutions of the moon's distance from the spectator at s when she is at z' and at M' respectively, compared with her distance when at M. But in the triangle SCA, right angled at A, sc : SA :: rad. : sin. SCA(= sin. M'SM, or the sine of the moon's altitude). Therefore the diminution of her distance from the spectator varies with the sine of her altitude. But the angle subtended by any small object increases as the distance from the spectator diminishes, and the augmentation of the angle varies with the diminution of the distance : hence, since the angle subtended by the moon's semidiameter is greater when she is in the zenith than when she is in the horizon, by a certain number of seconds, which may be represented by a ; it follows that Rad. : sin. moon's altitude : : a : a sin. moon's altitude, and the last term expresses the augmentation of the subtended angle, when the moon is between the horizon and the zenith. The value of the angle subtended by the moon's semi- diameter, when in the horizon, depends upon her distance from the earth, which is variable ; and consequently the aug- mentation of the semidiameter when in the zenith experiences variations. The second line in the following table shows the values of those augmentations between the limits within which the moon's horizontal semidiameter may vary ; and the fourth exhibits the augmentation in the zenith corresponding to different values of the moon's horizontal parallax. Moon's hor. "1 semidi. J 14' 30" 14' 50" 15' 10" 15' 30" 15' 50" 16' TO" 16' 30" 16' 45" Augment. ~\ in Zenith. J 13". 5 14". 2 14". 9 15". 5 56' 16".2 16". 9 17". 6 18". 1 Moon's hor.T parallax. J 53' 54' 55' 57' 58' 59' 60' Augment. ~\ in Zenith. J 13". 4 14" 14".6 15". 2 15".8 16".3 16".8 17". 4 126 DEPRESSION CHAP. V. 164. When the altitude of a celestial body s, above the visible horizon of a spectator is taken, it is obvious that, since the eye of the observer cannot be on the surface of the earth or water, but must be elevated above it as at A; and since the edge of the sea horizon is the circumference of a circle on which a cone, having A for its vertex, would be a tangent to the surface, the observed altitude SAH (the line AKH being a tangent to the earth's surface in a vertical plane passing through the observer and the celestial body, and the effects of refraction being neg- lected), will exceed the altitude by the angle h AH, called the dip of the horizon, or by its equal ACK. Hence a correction must be made for this dip or depression, and it may be found from the triangle ACK, in which AB (the height of the spec- tator above the surface) being given, as well as the semidia- meter of the earth, we have AC, CK, and the right angle at K ; to find the angle ACK, or its equal HA h. 165. But the dip of the horizon is evidently affected by the refraction of the rays of light in passing from the edge of the sea to the eye of the spectator, in consequence of which, the sea line generally appears to be higher than it is in reality : and a like source of error exists in the observed angular ele- vations and depressions of objects on land. In order to de- termine the amount of refraction in these cases, the reciprocal elevations or depressions of two stations at the distance of several miles from each other are observed at the same instant (by signals or otherwise), and then a computation is made by means of a formula which is thus investigated. Let A and B be the true, a and b the apparent places of two stations; let AH, BH' be tangents to spherical surfaces passing through A and B, and let the depres- sions H'B, HA be observed; then, c being the centre of the earth or of the spherical surfaces just mentioned, the angles CAH, CBH' are right angles; the angle AOB is the supplement of C which is subtended by the terrestrial arc between A and B, and is there- fore supposed to be given. The sum of the angles OAB, OB A is consequently equal to the known angle at C ; then, if from c, or from the sum of those angles, there be taken the sum of the two depressions H'B, HA&, CHAP. V. OF THE HORIZON. 127 the remainder will evidently be the sum of the two refractions, viz. ftAB 4- AB; and assuming that they are equal to each other, half that sum will be the refraction at each station A or B. If one of the apparent places, as ', had been above the line BH', drawn through the other, the sum of the refractions at the two stations would have been expressed by c + H'B ' HA ft, and half this quantity would have been the refraction at either station. Again, if one of the stations, as A, had been above the horizontal line BH' drawn through B, the line HA being produced would have met BH' in some point, and would have made with it an angle equal to C ; and in this case also, the sum of the refractions would have been ex- pressed by c -h H'B' HA ft. 166. From numerous observations made during the pro- gress of the trigonometrical survey of Great Britain, it has been determined, that the value of the refraction at each of two stations is equal to about one-twelfth of the angle ACB at the centre of the earth, between the radii drawn through the stations. Therefore, if D represent the dip of the horizon, or the value of the angle ACB (corresponding to AC K in the preceding figure), assuming that, by the effect of refraction, the elevation of the sea-line above K, or the angle KAL is equal to ~ D, we have D y 1 ^ D, or T J D for the apparent dip or depression AAL of the horizon ; and this formula is used in the construction of the best tables of the dip, in treatises of navigation. In order to obtain the value of D unaffected by refraction, let r represent the semidiameter of the earth, and h the height AB (in the preceding figure) of the station A above the sur- face of the sea : then, in the right-angled triangle A C K, r -f h : h :: rad. (= 1) : cos. ACK, or cos. AAK, or cos. D ; therefore cos. D = -77- : hence by trigonometry, ^ \v - / TT\*7 =sm. D. (r +A)V And developing the fraction, rejecting powers of A, above the first, we get sin. D - ( J *; and - ^ (-- ) * will express \ 7* s Sill* -L \ / s the value of D in seconds. Assuming that the path of a refracted ray from one station to another, or from the edge of the sea to the eye of the ob- server, is an arc of a circle, Mr. Atkinson determined (Mem. 128 DIP OF THE HORIZON. CHAP. V. Astron. Soc., vol. iv. part 2.), that the apparent dip or the angle &AL is expressed by the formula -; ( ) > sm. 1" \ r n J where - is the value of the refraction in terms of the angle n ACK (in general n = l2 as above stated). If n were equal to, or less than 2, it would follow from the formula that the dip then becomes zero or imaginary : and in these cases the surface of the sea is visible as far as objects on it can be distinguished, or the water and sky appear to be blended together so that no sea-line appears to separate them. CHAP. VI. DETERMINATION, ETC. 129 CHAP. VI. DETERMINATION OF THE EQUINOCTIAL POINTS AND THE OBLIQUITY OF THE ECLIPTIC BY OBSERVATION. 167. WHEN the observer has succeeded in obtaining the latitude of his station, he is prepared with an astronomical circle, to ascertain the apparent declinations of fixed stars, of the sun, the moon, and the planets. For, supposing the ob- server to be in the northern hemisphere, it will be evident from an inspection of the figure in art. 150., in which HZO represents half the meridian, z the zenith, p the pole, and Q the place where the equator cuts the meridian, that if any celestial body culminate south of the zenith, as at M, its ob- served distance ZM from the zenith at the time of culmination being subtracted from ZQ, the geocentric latitude of the station will give MQ the required declination ; and this will be either north or south of the equator, according as the re- mainder is positive or negative. If it culminate north of the zenith and above the pole, as at M', the declination will be equal to the sum of the observed zenith distance and the latitude of the station ; and lastly, if it culminate north of the zenith and below the pole, as at M", the declination will be equal to the supplement of such sum. 168. The declinations of those which are called fixed stars are not always the same ; for independent of certain proper motions in the stars themselves, the plane of the equator, from which the declinations are reckoned, changes its position in the celestial sphere by the effects of planetary attraction on the terrestrial equator ; but the declinations of the sun, moon, and planets are subject to considerable variations. 169. If the celestial body be the sun, the observations of the declinations may be considered as the first step in the determination of the elements of its apparent motions ; for let it be imagined that, by means of the zenith distances ob- served during a whole year, beginning, for example, at mid- winter, and continuing to the next succeeding midwinter, the declinations of the sun are obtained every time that the lu- minary arrives at the meridian of the station ; the corrections on account of the errors of the instrument, and the effects of refraction and parallax being applied, on comparing such de- clinations one with another it will be found, that at midwinter K 130 THE EQUINOCTIAL POINTS CHAP. VI. the observer being in the northern hemisphere, they have their greatest value on the southern side of the equator, that they afterwards gradually diminish, at first slowly, then more rapidly till the 21st of March, when the declination is exactly or nearly zero ; and that they afterwards increase, the sun appearing to be on the northern side of the equator till June 21. From that time till the next midwinter, the changes of de- clination indicate that the sun descends towards the south, the differences of declination following the same law, nearly, as they followed while the sun was ascending from south to north. 170. The greatest observed northern and southern decli- nations of the sun may be considered as constituting ap- proximate values of the angle at which the plane of the ecliptic, or of the earth's orbit, and the plane of the equator intersect each other ; and the two times at which the decli- nations are nearly zero, are the approximate times when the sun, in ascending and descending, crosses the plane of the earth's equator produced ; but as the observations are only made at the moments of apparent noon at the station, it is scarcely possible that the maxima or the zeros of declination should take place precisely at such moments for any one station; and therefore computation must be made by the rules of trigonometry, in order to obtain those elements with sufficient correctness. Before such computations can be made it will be necessary to explain the method of obtaining by ob- servation the daily movements of the sun in right ascension, or from west to east. 171. The transit telescope and the sidereal clock are used for this purpose. And, supposing the former to be duly ad- justed so as to move in the plane of the meridian, also the clock to be regulated, so that the hour hand will perform exactly one revolution (twenty-four hours) in the time of the earth's rotation on its axis ; the times of the transits of the sun's centre must be obtained, from the daily observed times of the transits of the limbs : the daily transit of some re- markable fixed star must moreover be observed, the telescope being supposed capable of rendering the star visible should it pass the meridian during the daylight. The times of these transits may be observed on the sidereal clock, though the latter may not indicate the absolute value of sidereal time, which, in fact, cannot be known from observation till the precise moment of the sun's arrival in the plane of the equator has been determined. The differences between the daily times of the sun's transit are evidently equal to the apparent daily movements of the CHAP. VI. FOUND BY OBSERVATION. 131 sun in right ascension, or those by which the luminary appears to advance from west to east ; and the difference, on any day, between the times of the transit of the sun and of the fixed star, will be the difference between the true right ascensions of the sun and star at the instant of the sun's transit. 172. Now, for several days before and after the 21st of March or the 23rd of September, let the daily transits and declinations of the sun be observed, and let the daily differ- ences of each be taken. Let ss' be part of a great circle representing the sun's path in the ecliptic for one day, and mm' be part of the equator in the celestial sphere ; also let s?7z, s'm' be portions of de- clination circles, or perpendiculars let fall from s and s' on the equator: then, Sm being the declination at the noon preceding that in which the sun ascends above the equator, and s'm f the decli- nation at the following noon, also mm' expressed in sidereal time, being the difference between the sun's right ascensions at the two consecutive noons ; if these arcs be considered as straight lines, on account of their smallness, and s'n be drawn parallel to mm', sn ( Sm + s'ra') will be the difference be- tween the declinations at the consecutive noons, and the right- angled triangles sras' and Swz, considered as plane triangles, will be similar to one another : therefore we shall have Sn : s'n mm' :: Sm : Thus ra 5 in time, being added to the time expressed by the clock at the preceding noon, will give the time which would be shown by the clock when the sun is precisely in the plane of the equator. If a like computation be made with the dif- ferences of right ascension and declination for the several days before and after the noon of March 21st, and a mean of all the times at which the sun is in the equator be taken, that time may be considered as correctly ascertained. Then the difference between the right ascensions of the sun and of the fixed star being found by observation, the distance of the star, in right ascension, from the equinoctial point becomes known ; and if the clock be set to indicate that distance, at the moment when the star next comes to the meridian of the station, the clock will indicate the zero of time (twenty-four lours) when the equinoctial point comes to the meridian : from that time, if well regulated, it will show the absolute right ascension of any celestial body at the moment of cul- mination. 173. The place of the equinoctial point may, however, be K 2 132 PRECESSION CHAP. VI obtained more correctly by the following process : in the right-angled spherical triangles T^S, vm'S' we have (art. 62. 00) sin. T m = cotan. V tan. m s, or tan. nr sin. V m = tan. m S ; , tan. m$ whence, tan. v = , . ... and in like manner, tan. nr = sin. tan. w's' Equating these values of tan. v we obtain sin. T m tan. m' s' = sin. T m! tan. m s ; whence, sin. T m : sin. T m' : : tan. m s : tan. m! s', and by proportion, sin. V ?rc + sin. T m' : sin. m v w') -, sin. (m S m' S 7 ) ^ tan. J ( wz -f T wz') sin. (m S + m' s') ' therefore, since ms, w's' and T^Z + m' are known from the observations, the value of m f is found, and conse- quently T m and f m f are separately determined. It follows from what has been said that the sidereal clock will express in time, at any instant, the angle contained be- tween the plane of the meridian of a station and a plane passing through the axis of the earth and the equinoctial point T : that is, the time indicated by the sidereal clock at any instant expresses the right ascension of the mid-heaven or of the meridian. 174. Computations similar to those which are supposed to have been made in March may be made in September when the sun is again near the place at which it crosses the equator; and thus the situation of the other equinoctial point may be determined. If the sidereal clock were correct it would be found to indicate 12 hours when the sun is in this second point ; from which it may be inferred that the two equinoctial points are distant from one another 180 degrees, that is, half the circumference of the heavens, or 12 hours in right ascen- sion, or that the intersections of the planes of the celestial CHAP. VI. OF THE EQUINOXES. 133 equator and ecliptic are diametrically opposite to one another in the heavens, in a line passing through the earth. On comparing together, with respect to a fixed star, the positions of the equinoctial points determined in the manner above described, in different years, it is found that the ap- parent places of the points are not fixed in the heavens ; and the phenomena indicate either that the points have a con- tinual motion from east to west about the earth, or that all the stars called fixed have a general movement in a contrary direction. The movement, in either sense, is called the pre- cession of the stars, or of the equinoctial points ; and physical considerations prove that, in reality, those points are in move- ment on the celestial equator. From the comparisons of ob- servations made at intervals of several years it has been found that the retrogradation, or movement from east to west, takes place at a mean rate of about 50". 2 annually; and M. Bessel expresses the mean precession for one year by the formula 50".21129 + 0".0002443 t, t denoting the number of years which have elapsed since 1750. The interval between the times at which the sun arrives at the point T is 365.242217 days, which is the length of a tropical year ; also the interval between the times at which the sun has the same distance in right ascension from a fixed star is 365.256336 days, and this is the length of a sidereal year. 175. The absolute right ascension and declination of the sun being obtained for the noon of any day, the rules of sphe- rical trigonometry enable the astronomer to compute the inclination of the plane of the ecliptic to that of the equator, or the obliquity of the ecliptic; for s'(fig. to art. 172.) being the place of the sun at the noon of any day, T m' is its right ascension and m'S' its declination; then (art. 62. (V)) in the spherical triangle S' m! T , right angled at w', and the last factor is the required obliquity. In order to obtain from observation the obliquity of the ecliptic, or the sun's declination at the time when that element is the greatest, that is, at the time when the sun is in either of the solstitial points (June 21. or December 21.), it is con- venient to observe the declination at the noon of each day for several days before or after such epoch ; and the obliquity of the ecliptic being known approximative^, let the correspond- ing longitudes of the sun be computed. Then if s be the place of the sun at the noon preceding (for example) the day of the solstice, or that on which the declination is the greatest, s r the observed declination and T the equinoctial point, we 134 OBLIQUITY OF THE ECLIPTIC. CHAP. VI. shall have, in the spherical triangle s r V right angled at r, representing the angle at v by q (art. 60. (e^) 9 r sin. sr = sin. g x sin. < 5. The arc ^ s is the sun's longitude at the time of the obser- vation ; and if s" be the solstitial point, so that T s" is a quad- rant, and S"R the solstitial declination, v 5 and 5 r will differ but little from ; then (PI. Trigo., art. 56.) EA COS. AEF = EF, Or X COS. = EF, and MA sin. AMD = AD = FA', my sin. FA'; therefore x cos. + y sin. =r EA' x'. Again EA sin. AEF = AF, or x sin. = AF, and MA cos. AMD MD, or y cos. < = MD; therefore y cos. < x sin. = MA' = y'. 179. Imagine E to be the centre of a sphere of which EP is a radius, and let this be considered as unity ; also imagine the arcs of great circles XY, PX, PY, PX', PY', PZ to be drawn, and let ZP be produced to meet the arc XY in m. Then, re- presenting the angle AEM by #, PEM by b, in the spherical triangles PXm, PYW, we have (art. 60. (rf)), cos. PX = cos. Pm cos. xwz, or x cos. d cos. , and cos. PY = cos. PTTZ cos. Ym, or, XY being a quadrant, y cos. d sin. a. Also in the spherical triangles PX'ra, PY'TW, we have cos. PX' = cos. PTTZ cos. x'ra, or x' = cos. d cos. (a <), and cos. PY X =. cos Pm cos. Y r m, or, x x Y r being a quadrant, y' = cos. d sin. (a ). CHAP. VII. OF CO-OHDINATES. 137 Again, in the spherical triangle PXm (art. 62. (/')) cos. pxrn = cotan. PX tan. xm; where the spherical angle PXTTZ, or its equal PAM, is the in- clination of the plane PEX to the plane XEY. 180. Let the co-ordinate planes be turned upon EX as an axis, and let them take the position XEY', XEZ', ZEY or Z'EY'. As in the preceding articles, let P M be the common section of two planes passing through p, one perpendicular to EX and the other to EY, and cutting the plane XEY in AM and MB. Let p N be the common section of the plane passing through P and AM (which plane, since EX is the common section of the planes XEY, XEY', is perpendicular to XEY') and another plane also perpendicular to XEY' and to the line E Y ; these planes will cut XEY' in AN and NB' ; and both PN and PM are in a plane passing through A per- pendicularly to XEY and XEY 7 . Then, since PM and PN are perpendicular to these planes respectively, it follows that the angle MPN is equal to the inclination of the planes to one another, that is, to the angle NAM or to the spherical angle YXY'. Now the co-ordinates of p with respect to the first system of axes are EA (= ar), EB (== y), PM or EC (= z), and the co-ordinates of P with respect to the second system are EA (= ar), EB' (= y'\ PN or EC' (= z). As before, let P be a point on the surface of a sphere whose centre is at E ; let E p be unity, and XY, XY 7 , Z'ZY arcs of great circles of the sphere in the co-ordinate planes : imagine also ZPW, z'pn to be arcs of great circles passing through z and z' which are the poles of XY and XY'. Produce AM till it meets, as in Q, a perpen- dicular let fall upon it from N : let x/w or the angle XEm be represented by , Pm or PEra by d ; also let xra be repre- sented by L and p n by X ; and let the spherical angle at x, or the inclination of the plane XEY 7 to XEY, be represented by 6. Then (PL Trigon., art. 56.) AN sin. NAQ = NQ, and, drawing n O parallel to MQ, PN COS. MPN =: PO =. PM NQ ; whence PN COS. MPN -f AN sin. NAQ = PM, 138 TRANSFORMATIONS CHAP. VII. or z cos. 9 + y' sin. 6 = z. In like manner AN cos. NAQ PN sin. MPN rr AM, or y f cos. z f sin. y. Multiplying the equation for z by cos. 6, and the last equation by sin. 6, and subtracting the last result from the other, we get z z cos. 6 y sin. 6. Again, multiplying the equation for z by sin. 6, and the equation for y by cos. 9, and adding the results together, we get y' z sin. 6 + y cos. #. 181. Let fall the arc pp perpendicularly on the circle Z'ZY; then in the spherical triangle Z'ZP (art. 60. 3 Cor.) cos. Z'P cos. zp =r cos. ZP cos. z'p i cos. z'p or cos. z'p = cos. ZP - : cos. zp but z'p = z'z -f zj9 = + z/> ; therefore (PL Trigonom., art. 32.) cos. z'p cos. cos. zp sin. sin. zp, Substituting this value of cos. z 'p in the equation for cos. z'p, the latter becomes cos. Z'P, or sin. X ( = z') = sin. d (cos. 6 sin. 6 tan. zp) : but again, in the spherical triangle PZp (art. 62. (/')) cos. PZ rr cotan. ZP tan. z, sn. . or sin. a = tan. d tan. zp ; whence - > tan. z ; there- tan. d fore the equation for cos. z'p becomes sin. X =: sin. d cos. 6 sin. 6 sin. a cos. d . . . . (A). In like manner, from the equation cos. z'p cos. zp = cos. ZP cos. zp cos. z', we have cos. ZP = cos. z'p -- /-, m which substi- cos. zp tuting for cos. zp or cos. (z'p 0) its equivalent (PI. Trigon. art. 32.), we get cos. ZP = cos. z'p (cos. 6 + sin. 6 tan. Tip). But in the spherical triangle PZ'/> we have (art. 62. (/')) cos. PZ'/> = cotan. pz' tan. z'p, or sin. L = tan. A, tan. z'p : whence tan. z'p =. - tan. X Substituting this value of tan. z'p in the last equation for cos. ZP, we obtain sin d ( = cos. ZP) = sin X cos. + sin. sin. L cos. X ... (A). CHAP. VII. OF CO-ORDINATES. 139 Again, in the spherical triangle ZY'P (art. 60. 3 Cor.) cos. Y'P cos. ip =: cos. ZP cos. Y'p, , cos. Y' p Or COS. Y P rrCOS. ZP - I COS. Zp but Yp = ZY' zp = ^ zp; therefore (PI. Trigon., art. 32.) cos. Y'P =r cos. ( j cos. Zp + sin. ( 6) sin. Zp, or cos. Y p = sin. 6 cos. z/? + cos. sin. zp. Substituting this value of cos. Y'p in the equation for cos. Y / P we have, after reduction as before, cos. Y'P ( y) = sin. d sin. 6 + cos. sin. a cos. ef. Next, in the spherical triangle XPY', we have (art. 60. 3 Cor.) cos. PX cos. TZY' zr cos. PY' cos. rax, or, cos. PX being equal to cos. a cos. d (art. 179.), and XY' being a quadrant, cos. a cos. d sin. L (sin. d sin. 6 + cos. sin. a cos. ef) cos. L ; sin. d sin. 6 + cos. sin. a cos. d , whence tan. L = cos a cos. d tan. d sin. 6 + cos. sin. a .... (B). cos. a By similar processes the value of tan. a may be found : thus, in the spherical triangle Z X YP, we have (art. 60. 3 Cor.) cos. YP cos. z'p =. cos. Z'P cos. YJO ; whence, cos. Y cos. YP = cos. Z X P - -A COS. 7/p But Yp = + - z r p; therefore (PL Trigon., art. 32.) cos. YJO = cos. (fr 4*' 4) cos. z'p + sin. (|- + ^) sin. Tip sin. 6 cos. z x j0 + cos. sin. z jo. This value being substituted in the equation for cos. YP, the latter becomes cos YP z= cos. z'p ( sin. 9 + cos. 6 tan. 7/p) ; and again, substituting the above value of tan. z'p, also for cos. z'p putting its equivalent sin. X, we have cos. YP =r cos. 6 cos. \ sin. L sin. sin. X. Now, in the spherical triangle XPY, we get (art. 60. 3 Cor.) cos. PX cos. WY cos. YP cos. xw ; 140 TRANSFORMATIONS CHAP. VII. but XY is a quadrant, and in the right-angled triangle (art. 60. (d)) cos. PX cos. X cos. L; substituting this value of cos. PX, and the above value of cos. YP in the last equation, the latter becomes cos. X cos. L sin. a = (cos. 6 cos. X sin. L sin. X sin. 6) cos. a ; cos. 6 cos. X sin. L sin. X sin. 6 whence tan. a cos. L cos. X cos. 6 sin. L tan. X sin. 6 . . . (B'). COS. L Lastly, in the spherical triangle XPW, cos. PX cos. XTZ cos. PTZ, or cos. a cos. d cos. L cos. X; cos. a cos. d therefore cos. L = cos. X 182. If the plane oblique to XEY, instead of cutting the latter plane in EX, were to cut it in some other line as EX'; the investigation of the co-ordinates of P with respect to the new position of the system of planes would be conducted in a manner similar to that which has been above explained. Let X'EY" be one of the planes in the new position, and z" its pole : then, on describing the great circles through z", as in the figure, we should have zz" equal to the inclination of X'Y" to X'Y, which represent by &>. ZP, as before, is equal to d; and, XEX' being represented by sin. o> cos. d sin. (a the obliquity of the ecliptic, being given, in the right-angled spherical triangle T L R we have y L ( 1) and the angle at y ( = 0) j to find y R ( = ) Now (art. 62. (/'))> cos. = cotan. / tan. a, or putting - for cotan. Z, cos. 6 tan. I tan. , the required right ascension ; or ^ tan. I ; by which the longitude may be found. I. 2 148 REDUCTION FROM THE ECLIPTIC CHAP. VII. If the declination LR (= d) were required, we should have (art. 60. (ej), the longitude being given, sin. Z sin. 6 = sin. d, by which the declination is obtained. Or, the right ascension being given (art. 62. (^)), sin. a zz cotan. tan. d, or sin. a tan. 6 = tan. d. Again, in the triangle T LR we have (art. 60. (dj) cos. I =: cos. cos. e?; from which equation the longitude may be found when the right ascension and declination are given. 190. But, for the computation of ephemerides, it is found convenient to have in tables the values of the differences be- tween the longitudes and right ascensions of the sun, in order that the reduction from one element to the other may be effected by simple additions or subtractions, such differences being obtained by inspection : and the formula for the tabular values of the differences is investigated in the following manner : By " Plane Trigonometry " (art. 38.) we have /7 N tan. I tan. a tan. ( i a ) ~ ; ' 1 + tan. I tan. a ' and substituting the above value of tan. , the second member i /-i /i\ tan. I becomes (1 cos. 0) w - ^ : ' 1 + cos. 6 tan. 2 / but again (PL Trigon., art. 39.) tan. 2 / = I- cos. 27 ^ 1 ~p COS. 2i i multiplying both terms of the second member by 1 4- cos. 2 /, we have 1 cos. 2 2 I sin. 2 2 / tan. 2 / =: r =-5 or = 7- -^ ; whence tan. / = (1 + cos. 2 1) 2 (1 + cos. 2 Z) 2 sin. 2 / F+cos. 21' therefore, after reduction, /T \ /-, /i\ sin. 2 / . tan. (I a) = (1 cos. 0) ^ ^ ^ , , ; 1 + cos. + (1 cos. 0) cos. 2 I ' 1 cos. sin. 2 / or 1 + cos. 1 cos. cos. 2 1 + cos. let this be represented by . * S1IK 2 l . *T+I cos. 2 1 Now d(l- -**><* putting 1 ( = rad. 2 ) for cos.' 2 2 I + sin. 2 2 /, t 1 sin. 2 27 1 + 2 cos. 2 7 + * 2 also sec. 2 (/_,)= 1+ 7^3^ = (i + , C os. 2 7) 2 ; consequently, by substitution, N 2 # (cos. But (PL Trigon., art. 47.) 2 cos. rc 7 = e W " + - where s is the base of the hyperbolic logarithms, and n may be any whole number. Now when n = 2, let the second member be represented by P + may be represented by - - - ; or, since the last de- I + tp + - + P nominator is equal to (1 + tp) (1 +-), the fraction may be t B- A. T) resolved into - + *-r, which being brought to a com- L+t P mon denominator, and the co-efficients of 1 and of - separately equated, gives A B = - -^ Therefore the fraction is equivalent to - P and the fractions between the parentheses being developed separately by division, the value of - becomes L 3 150 REDUCTION PROM THE ECLIPTIC CHAP. VII. Then, substituting the equivalents of p + -, p 1 + -^ &c. viz. 2 cos. 27, 2 cos. 4Z, &c., the value of d (I a) becomes *lt&^+JL { i _ 2 , cos . 2 l + 2 t * cos. 4 Z 2 * a cos. 6 I + &c.}dZ. Multiplying t cos. 2 Z + 2 into the terms between the braces, the second member becomes cos. 2 7 - 2 * 2 cos. 2 2 Z 4- 2 3 cos. 2 Z cos. 4 Z + * 2 -2 t 3 cos. 27 2 t 4 cos. 2 Z cos. 6 / + &C. 1 ) , 7 + 2 * 4 cos. 4 Z - &cJ then, from the general formula 2 cos. A cos. B cos. (A + B) H- cos. (A B) (PI. Trigo., art. 32.) we have 2 t 2 cos/ 2 2 / = t 2 cos. 4 Z + # 2 , 2 t 3 cos. 2 Z cos. 4:1 t Q cos. 6 Z + t 3 cos. 2 Z, &c. Substituting the second members of these equations for their equivalents in the above expression, and dividing by 1 t' 2 we obtain d (I a} = 2{t cos. 2 Z - t 2 cos. 4 Z + * 3 cos. 6 Z - &c.}^Z: the two members of this equation being integrated give (/ ) = * sin. 21 ^t* sin. 4 Z + t 3 sin. 6 Z &c. But t, or ~ COS> ^ is (PI. Trigo., art. 39.) equal to tan. 2 6> ; I ~r~ COS. i/ therefore Z a tan. 2 i- sin. 2 Z \ tan. 4 sin. 4 Z + tan. 6 l sin. 6 Z - &c. This second member is expressed in terms of radius ( = 1 ) ; and, to obtain it in seconds, every term must be divided by sin. 1" ; or, since 2 sin. V is nearly equal to sin. 2", &c., we have finally 7 / A c \ tan - 2 i sin. 2 Z tan. 4 sin. 4 Z / a ( in seconds ot arc ) - ^ ___ 2 ___ - __ i- sin. 1" sin. 2" tan. 6 I sin. 6 Z ~sinT3^~ ~ &C ' The values of these terms are arranged in tables, and the ex- pression (of which three terms will suffice) being subtracted CHAP. VII. TO THE EQUATOE. 151 from the sun's longitude, there remains the value of his right ascension. 191. It is evident that by spherical trigonometry, or, at once, from the formulae in art. 181., there may be obtained the latitude and longitude of the moon, a planet, or a star, when the right ascension and declination have been obtained from observation, and the obliquity of the ecliptic is known. Let T Q (% to art. 187.) be part of the equator, T c part of the ecliptic, and let the angle C T Q> the obliquity of the ecliptic, be represented by 6. Let s be the celestial body, T R its right ascension ( = a) and R s its declination ( = d) ; also let T L, the longitude, be represented by L, and L s, the latitude, by X ; then by substitution in (A) and (B) (art. 181.), we have the values of sin. X and tan. L. Conversely, the longitude and latitude being taken from tables, there may be found from the formulae for sin. d and tan. a, the values of the declination and right ascension. L 4 352 THE EARTH'S ORBIT. CHAP.VIII. CHAP. VIII. THE ORBIT OF THE EARTH. ITS FIGURE SHOWN TO BE ELLIPTICAL. SITUATION AND MOVE- MENT OF THE PERIHELION POINT. THE MEAN, TRUE, AND EX- CENTRIC ANOMALIES. EQUATION OF THE CENTRE. 192. FOR the sake of a more ready comprehension of the manner in which the figure of the earth's orbit may be de- termined, it will be convenient, for a moment, to imagine that the earth is at rest, and that the sun describes about it the periphery of a curve similar to that which the earth describes about the sun ; it is evident that, as the means employed to determine the figure involve only the mutual distances of the earth and sun, and the angular movement of either, the form of the required curve will be the same whether the former or the latter be supposed to move about the other. 193. From the observed declinations and right ascensions of the sun obtained daily during a whole year, let the lon- gitudes of that luminary be computed (art. 189.), and let the daily differences of the longitudes be found by subtraction. These daily differences, which may be considered as the ve- locities in longitude, are not equal to one another ; and a com- parison of them will show that, in the present age, they are the greatest soon after mid-winter, and the least soon after mid-summer ; on the first day of January they are about 61' 11".5, and on the first day of July, about 57' 12".5, the mean of which is 59' 12". The angular distance between the places of the sun on the days of the greatest and least velocity is 180 degrees or half the circuit of the celestial sphere, so that the two places appear to be at the extremities of a line drawn through the earth and produced each way to the heavens. Again, if the angle subtended by the diameter of the sun be accurately measured by means of a micrometer daily, or at intervals of a few days, it will be found that this element is variable ; on the first day of July it is the least, being then equal to 31' 30", and on the first of January it is the greatest, being then equal to 32' 35". Now, by the laws of optics, the distance from the observer, of any object which subtends a small angle, is inversely proportional to its apparent magnitude ; it must be inferred therefore that the sun is at a greater distance from the earth in summer than in winter ; CHAP. VIII. ITS ELLIPTICITY. 153 the ratio of the two distances being as 32' 35" to 31/ 30", or as 1.0169 to 0.9831. The variations of the longitudes and of the angular magnitudes follow the same law on both sides of the line of greatest and least distances ; and it follows that the curve apparently described by the sun about the earth in one year, or that which is described by the earth about the sun in the same time, is symmetrical on the two sides of that line. 194. An approximation to the figure of the earth's orbit may be conceived to be obtained from a graphic construction in the following manner. From any point s representing the sun draw lines making the angles A SB, ASD, ASE, &c., equal to the sun's increase of longitude for one day, two days, three days, &c., and make the lengths of SA, SB, SD, &c. inversely propor- tional to the apparent angular measure of the sun's diameter ; then if a line be drawn through the points A, B, D, &c., it will represent the figure of the orbit, and will be found to be nearly the periphery of an ellipse of which the sun occupies one of the foci. Or if, with the given values of the angles A SB, ASD, &c., assuming AC or the half of A p to be unity, the lengths of the lines s B, s D, &c. be computed by the formula for the distances of B, D, &c. from the focus of an ellipse, the values of those lines will be found to agree very nearly with the values obtained by supposing the same lines to be inversely proportional to the apparent angular measures of the sun's diameter ; and thus the ellip- ticity of the orbit may with more certainty be proved. On computing the areas comprehended between the radii vectores SA and SB, SB and SD, &c., they will be found to be equal to one another when the times of describing the arcs A B, BD, &c. are equal to one another, that is, the sectoral areas imagined to be described by the radii vectores about s vary with the times of the description ; or, if t be the time in which the earth moves from B to D for example, and T be the time in which it describes the periphery of the ellipse, we have area BSD: area of the ellipse : : t : T. The ellipticity of the earth's orbit was first discovered by Kepler, and the above relation between the areas is designated one of Kepler's laws. 1 95. Though it is now known that the orbit is not, strictly speaking, an ellipse, yet the latter being the regular curve which is next in simplicity to a circle, astronomers for conve- nience consider it as the figure of the orbit described by the sun about the earth, or of the earth about the sun. The point 154 THE EARTH'S ORBIT. CHAP.VIII. A or P at which the sun or the earth is when the two bodies are at the greatest or the least distance from one another, is called an apsis, and a line joining the points of greatest and least distance is called the line of the apsides. If it be assumed that the earth revolves about the sun the same points are respectively called the aphelion and the perihelion points, the sun being supposed to be in one of the two foci of the ellipse. 196. If BSD be one of the triangles described by a radius vector r in a unit of time (one second, one minute, or one hour), and if v represent the angular velocity of the sun or earth (a circular arc intercepted between SB and SD, and sup- posed to be described about S as a centre with a radius equal to the unit of length) ; then (by similarity of sectors), r v may be considered as equal to the line Dm (a perpendicular let fall from D on s B), and by mensuration, J r 2 v may be considered , , , area BSD , as equal to the area BSD, or v oc ^ nearly ; but the area B s D is constant when the times are equal ; therefore the angular velocity is inversely proportional to the square of the radius vector (nearly). 197. The earth's orbit being symmetrical on each side of the line of apsides, it will follow that the instants when the earth is in the aphelion and perihelion points successively, must differ in time by half the period of a complete revolution of the earth about the sun, as well as that the longitudes of the points must differ from one another by 180 degrees; for it is evident that if any other line as XY be imagined to be drawn through the sun it will cut the orbit in two points which will differ in longitude by 180 degrees, while the times in which the earth moves* from one to the other on opposite sides of the line will be unequal, because the move- ment is more rapid about the perihelion than about the aphe- lion point. It should be observed that, among the registered observations which may have been made during a year, there may not be two which give by computation longitudes differing by exactly 180 degrees ; but if there be found two longitudes which differ by nearly that quantity, then from these, with the known velocity of the sun, the times when the longitudes so differ may be computed by proportions simply. 198. If the longitudes of the sun be computed for two times distant by many years, the daily differences of longitude being at both times the least or the greatest, or being equal to one another and nearly equal to the least or greatest, the longitudes will be those of the sun at, or nearly at, the instants when the earth is in the aphelion or perihelion point, CHAP. VIII. MOTION OF THE PERIHELION. 155 or at equal distances from either ; these longitudes will be found to differ from one another, and the difference will evidently express the quantity by which the perihelion point has moved in longitude in the interval. This movement which takes place in the " order of the signs," is called the progression of the perigee, or of the perihelion point, and its mean value for one year, if reckoned by the different angles which, at given times, SP or SA makes with a line drawn through the sun and the first point of Aries, is found to be 61".9. 199. The time in which the earth revolves once about the sun from the perihelion point to the same is called the anom- alistic year : it is evidently equal to the length of the tropical year, together with the time in which the sun moves through 61 ".9 in longitude. But the equinoctial point (the first point of Aries) retrograding 50". 2 (art. 174.) annually by the general precession, the progression of the perigee, if measured by the different angles which at given times SP or SA makes with a line drawn through s and a fixed point (supposed to be the place of a star) in the heavens is only equal to II".? an- nually. The anomalistic year, therefore, exceeds the length of a sidereal year by the time in which the sun moves through 11 ".7 in longitude. 200. If from a register of the computed longitudes of the sun there be selected two which differ from one another by exactly 180 degrees, and which correspond to times when (the daily differences of longitude being nearly the least and the greatest) it may be considered that the earth was nearly in the aphelion and perihelion points ; the instant at which the earth was in the latter point, and also the longitude of that point, may be found in the following manner. Let AP (fig. to art. 194.) be the line of the apsides, s the sun, x and T the two places of the earth when near A and P, and when in the direction of a right line through s ; also let the required time in which the sun will move from Y to P be represented by t: then, by Kepler's law (art. 194.) sector YSP : sector ASX :: t : t' (t r denoting the time of moving from x to A). But the sectors being supposed to be similar to one another, SP 2 : SA 2 :: sector YSP : sector ASX, or as t : t' ; and the angular velocities at P and A, being inversely as the squares of the distances from s, if those velocities be repre- sented respectively by v and v f (which are known, being equal to the increments of longitude in equal times, at or near the perihelion and aphelion points), we have 156 THE EAKTH'S OKBIT. CHAP.VIII. v v' : v :: t : t' ; whence t' t.. v' But t' t, or the difference between the times of describing the half periphery of the ellipse, and of describing the arc YPX is known, it being the difference between half the anom- alistic year and the given interval, in time, between the ob- servations which were made when the sun was at Y and at X : let this difference be represented by D ; then tv whence t may be found. Hence the time of being at Y, and the longitude of Y, as well as the velocity at that place being known, the time of arriving at p, and also the longitude of P may be found. 201. The excentricity of the earth's orbit, considered as an ellipse, may be conceived to be known approximatively from the relative values of the aphelion and perihelion distances, for these being represented by a and p respectively, a + p will represent the major axis of the orbit, and ^ (jo) will be the distance of the centre from the focus of the ellipse ; also P will be an expression for that distance when the semitransverse axis of the ellipse is supposed to be unity : this excentricity is usually represented by e ; and a more accurate method of determining it will be presently explained (art. 207.). 202. Having determined the periodical time of a tropical revolution of the earth about the sun, astronomers divide 360 degrees by the number of days or hours in the length of the year, and consider the quotient as the mean daily or hourly motion of the sun ; then, dividing the longitude of the peri- helion point, found as above, for a given time by the mean daily motion of the sun, the result will be the number of days from that time since the sun's mean longitude was zero, or since the sun was in the mean equinoctial point. The mean longitude of the sun for any given time may then be found on multiplying the number of days, hours, &c., which have elapsed since the sun had no mean longitude by his daily motion in longitude ; and the difference between this mean longitude of the sun and the longitude of the perihelion point for the given time constitutes what is called the mean anomaly at that time. The true anomaly is the difference between the true longitude of the earth and the longitude of the perihelion point. A third anomaly, which is called excentric, is used when it is required to find the relation between the true and mean anomalies. CHAP. VIII. MEAN, ETC. ANOMALIES. 157 203. In order to investigate this relation, let A DP be half the periphery of the elliptical orbit, C its centre, S one of the foci, or the place of the sun, p the perihelion point, and let E be the place of the earth. With c p as a radius, describe the semicircumference AQP, and draw the radius vector SE ; also through E draw QR perpendicular to CP, join S and Q, c and Q, and let fall SM perpendicularly on CQ. Then the angle P s E is the true, and P c Q the excentric anomaly. By the natures of the circle and ellipse, KQ : RE :: segment QPR : seg. EPR, and (Euc. 1. 6.) RQ : RE :: triangle QSR : triangle ESR; therefore, since by conic sections RQ : RE :: area of circle : area of ellipse, by equality of ratios and composition, sector SQP : sector SEP :: area of circle : area of ellipse, or ellipse : sector SEP : circle : sector SQP. But the areas described by the radii vectores being propor- tional to the times of describing them (art. 194.) if T represent the time in which E describes the periphery of the ellipse, and t the time in which it describes the arc P E, Tit:: area of ellipse : sector SEP: therefore, by equality of ratios, T : t : : area of circle : sector SQP; and if CP zz 1, so that the area of the circle is represented by ?r(=:3.1416) the sector SQP is equal to t. To this sector adding the area of the triangle SCQ or \ CQ.SM which (if c Q or cp = l, cs=e, and the angle SCQ = &) is equal to \ e sin. u, we have the area of the sector CQP equal to - t + \ e sin. u. Now the same sector is equal to ^CP.PQ, or \ u ; therefore, 7T . , 27T - t + \ e sin. u \ u ; whence t = u e sin. u. Q_ But again, expresses the mean angular motion of E about s, being equal to the quotient arising from the division of the circumference of a circle whose radius is unity by the time of 158 THE EARTH'S ORBIT. CHAP.VIII. a complete revolution in that circumference. Let this be represented by n ; then n t =r u e sin. u ..... (A). Since t is reckoned from the instant that E was at the peri- helion point P, n t expresses the mean anomaly ; therefore, from this equation we have the mean, in terms of the true anomaly. 204. Next, by conic sections, SE =r CP CS.CR, which, since CP or CQ = 1, CS = e, and the angle QCP = u, also, representing s E by r, becomes r = 1 e cos. u . . . . (B) ; but, by conic sections, representing the angle PSE by 0, we have ~ l+e cos. 0* Equating these values of r, we get, 1 - 1 e cos. u = l+e cos. ' whence successively, l+e cos. e cos. u e* cos. u cos. = 1 e*, l+e cos. e cos. u (1 + e cos. 0) 1 , or the difference between the mean and true angular movements of the earth, that is, the equation of the centre, is evidently equal to zero when the earth is at the perihelion, and also when it is at the aphelion point of the orbit ; and it follows, from what has been said, that it attains its maximum value on either side of the line AP at the instant when the elliptical movement becomes exactly equal to the mean movement. Hence, if the longitudes of the sun be taken from a table of such as have been computed from the observed right ascensions and declinations, for the two times when the daily differences of longitude are equal to the mean daily difference, that is, when the elliptical velocity of the sun or earth is equal to the mean velocity, the difference between those longitudes being represented by the angle p SJP', or twice ASp, while the mean angular movement of the earth or sun, during the time elapsed between those observations, may be represented by E s E' or twice A s E ; the difference between A sp and A SE, or between p s E and P Sp will be the maximum equation of the centre, which may therefore be so determined. It may not be possible to obtain from the observations two longitudes of the sun at the precise moments when the true velocities of the sun or earth were equal to the mean velocity ; but, since the equation of the centre varies very slowly, it is evident that if two longitudes of the sun were taken when the true velocities are equal to one another, and nearly equal to the mean velocity, the error in the computed value of ESp will be very small, and it may afterwards be corrected by other means. It should be remarked that the longitudes obtained from the observations need not be corrected, on account of the movement of the perihelion in the interval of time between the instants at which the earth is in p and in //, since this movement affects equally the true and the mean longitudes. 207. It has been shown above that the mean angular velocity is expressed by , which, if T be given in seconds, will be the value of a circular arc (whose radius is unity) described about S by uniform motion in one second of time. Now, if v represent the angular velocity described by the CHAP. VIII. EQUATION OF THE CENTRE. 161 earth in the periphery of the ellipse at any time, when the radius vector is represented by r ; that is, if it represent an arc whose radius is unity, which is described by a point in SE in one second of time; then (art. 196.) %r~v is the sectoral area described by SE in one second of time. But, by conic sections, CP being supposed to be unity, 7Tv/(l 2 ) expresses the area of the ellipse; therefore, by the pro- portionality of the times to the areas (art. 194.), T : V :: TTA/(I e 2 ) : |r 2 v; whence v = . r 2 T Equating the true and mean angular velocities we have 27T and the last expression is the value of the radius* vector at the time when the equation of the centre is the greatest. Now, in general (art. 204.), r = (1 *e cos. u) and r = I _ e i - - 2i; and, substituting the above value of r in each of these equations, we have (1 e a )*=l ecos. u also (I-* 2 ! = ^ 3-, l+e cos. 6 or (1 e 2 ) =1 e cos. u, and \-\-e cos. 6 (1 e 2 ). Developing by the binomial theorem, and neglecting powers of e above the square, we have 1 ^e 2 = 1 e cos. M, also 1 + e cos. 0= 1 f ^ ; whence cos. u = \e and cos. 6 je. But at the time when the equation of the centre is the greatest, u and are very nearly equal to right angles, u being less, and being greater ; therefore, for u putting u', and for 6 putting + 0', we get J > cos. u = sin. u', and cos. 6 = sin. X ; also, sin. w' rr |e, and sin. & -=.\e\ or, since w r and X are very small, u'-^e, and 0'= |*. Then, since the equation of the centre is (art. 206.) equal to - nt, or (art. 203. (A)) equal to u + e sin. w ; M 162 NATUKE OF CHAP. VIII. substituting for and u their values at the time when the equation of the centre is the greatest, and representing that greatest equation by E (expressed by a circular arc whose radius is unity), we have or, developing the cosine, or E or again, E = 2e y^* 3 - Whence e = ^E nearly. From the equations (A) (B) and (c) (arts. 203, 204.) there may be obtained by the formulas for the developments of func- tions in infinite series, values of 0, of r, and of 6 nt, in ascending powers of e, with sines and cosines of arcs which are multiples of n t, the mean anomaly. 208. Astronomers have computed the mean movement of the sun, or the angular space which the luminary would ap- pear to describe about the earth in a day, an hour, &c. if its movement were quite uniform, or such as it might be supposed to have if it revolved in a circular orbit undisturbed by any attractions exercised upon it by the other bodies of the system ; and they have added the mean place of the sun in the ecliptic, that is, its distance in longitude from the mean position of the equinoctial point, at a certain epoch, as the commencement of a particular year or century, together with its mean place in longitude at the commencement of the preceding and following years, and supplementary tables by which the use of the table may be extended to 10,000 years before and after the nineteenth century. They have also computed the mean longitude of the perigeum for the epoch, with the mean position of that point at the commencement of each year and the mean daily movement of the same point, in longitude ; and, from the tables of such positions and move- ments, it is evident that the mean longitude of the sun and of the perigeum, or the distance of either from the mean place of the equinoctial point, can be computed for any given time by a simple addition of the numbers. Again, the value of the equation of the centre corresponding to any given mean anomaly (mean dkfance of the earth from the perihelion point of its orbit) is also given in astronomical tables ; and these values being applied to the mean longitude CHAP. VIII. THE SOLAR TABLES. 163 above mentioned, there results the longitude of the sun for a perfectly elliptical orbit : the effects of the perturbations produced by the different planets are also computed, in order that, being applied to the elliptical place of the sun, the true longitude for the given instant may be obtained. Lastly, the values of the radius vector of the earth's orbit, computed on the supposition that the orbit is an ellipse and that the semi- axis major is equal to unity, are given, and to these are added tables of the variations produced in that element by planetary perturbations. M 2 164 THE MOON'S ORBIT. CHAP. IX. CHAP. IX. THE ORBIT OF THE MOON. THE FIGURE OF THE MOON'S ORBIT PERIODICAL TIMES OF HER REVOLUTIONS. THE PRINCIPAL INEQUALITIES OF HER MOTION. HER DISTANCE FROM THE EARTH EMPLOYED TO FIND AP- PROXIMATIVELY THE DISTANCE OF THE EARTH FROM THE SUN. 209. The processes employed to determine the apparent path of the moon in the celestial sphere correspond to those which have been explained in speaking of the sun. The right ascension and declination of the moon are to be observed every day when she comes to the meridian ; and the obliquity of the ecliptic being given, from these observations her longi- tudes and latitudes, as indicated in art. 191., may be found by computation from the formulae (A) and (B) in art. 181. In the series of latitudes so computed there may be found two, which are equal to zero, and the moon in those points is said to be in the nodes of her orbit, for the plane of any orbit will cut the ecliptic in a line passing through the points of no latitude. The longitudes of the moon being computed for the times when the latitude is zero, it follows that the longitudes of the nodes are thereby determined : and since it is found that the nodes of the moon differ in lon- gitude about 180 degrees, it is evident that the line joining them passes through the earth. In the series of the moon's computed latitudes there may also be two which are equal to the maximum value of the element (about 5 9'), one on the north, and the other on the south side of the ecliptic ; and this greatest latitude expresses the obliquity of the moon's orbit to that plane. But as it is scarcely probable that, from any one of the ob- served right ascensions and declinations of the moon when on the meridian of the observer, the computation should give for the latitude exactly zero, methods similar to those which are put in practice for finding the place of the equinoctial point (arts. 172, 173.) may be used to determine the places of the moon's nodes. Thus let two observations be chosen, from one of which the moon's place M' is found to be south of the ecliptic, and from the other the place M is to the north : let E'E be part of the ecliptic in the heavens, M'M part of the CHAP. IX. INCLINATION AND NODES. 165 moon's apparent path ; then N will be the apparent place of the ascending node : also if M'E', ME be parts of great circles passing through the moon's centre perpendicularly to the ecliptic, those arcs will be the computed latitudes, and E'E the difference of the moon's longitudes between the two times of observation : then E'N NE being found by the method explained (art. 173.) for the equinoctial point, since E'N + NE is known, we have E'N and NE separately. Consequently, the longitude of the moon when at M' being known, we have the longitude of the node ; and from one of the right-angled triangles, as NEM, the angle at N may be computed : this angle will be the inclination of the moon's orbit to the ecliptic. 210. Or, the inclination of the moon's orbit to the ecliptic, and the position of the line of nodes, or intersection, may be determined by the methods described in arts. 185, 186, from the longitudes and latitudes of the moon ; these being de- duced by computation from the formula (A) and (B) in art. 181., and the right ascensions and declinations being obtained from observations made at two different times. Thus, let s (fig. to art. 184.) be the centre of the earth, and P a place of the moon at one of the times of observation : let XEY be the plane of the ecliptic, and sx', parallel to EX, the line of the equinoxes ; also let x, y, z be rectangular co- ordinates, SA", SB", MP of P. The angle X'SM will express the longitude of the moon at P, which may be represented by L, and PSM the corresponding latitude, which may be re- presented by X. Let S P be represented by r ; then imagining a line as SN to be drawn through s perpendicularly to the plane of the moon's orbit, we shall have, p on this line being zero since the orbit passes through s, x cos. NSX' -|- y cos. NSY' + z cos. NSZ' = o (art. 183.) But (art. 184.) x r cos. A cos. L. y =n r cos. A sin. L. z = r sin. A ; substituting these values of x, y, z in the equation, the latter becomes cos. A cos. L cos. NSX' + cos. A sin. L cos. NSY' + sin. \ cos. N s z' o : M 3 166 THE MOON'S ORBIT. CHAP. IX. whence cos. NSX' . cos. NSY' cos. A cos. L - ? + cos. A sm. L - , + sin. A = o. COS. NSZ COS. NSZ' In like manner, for another place P' of the moon, we should have . cos. NSX' . . .cos. NSY' cos - A 08 - L + cos ' V sm " L +sm ' v = 0: from these two equations there may be obtained the values of cos. NSX' cos. NSY' , , cos. 2 NSX' + cos. 2 NSY' , , and also of - 5 - ; COS. NSZ COS. NSZ COS. 2 NSZ' the last fraction expresses (art. 185.) the square of the tangent of the angle Z'SN, and this is equal to the required inclination of the orbit to the ecliptic. The position of the line of nodes is found as in art. 186., a and |8 being the co-ordinates of any point in that line, which, as above observed, is to pass through s : for we have a COS. NSX' 4- /3 COS. NSY' = O : a COS. NSY' T ,, whence -= =: --- -, = tan. x s M , and B s m, the com- plement of X'SM', is the angle which the line of nodes makes with sx', that is, with the line of the equinoxes. 211. The longitude of the moon's nodes as well as the obliquity of her orbit to the plane of the ecliptic are found to vary with time, the movement of the nodes taking place in retrograde order. The amount of the retrogradation can be determined approximately by computing the longitude of the moon when she is in one of the nodes, and again when, in her revolutions about the earth, she is next in the same node, and taking the difference between the longitudes (the interval in time being also known) ; but it is evident that the move- ment will be determined more accurately if the computation be made for times very distant from each other. In this manner it may be ascertained that the retrogradation of either of the nodes to the amount of 360 degrees, or the time in which either of the nodes performs a revolution about the earth, from a point in the ecliptic at any distance from the place of the vernal equinox to a point at an equal distance from that place, is accomplished in about 18f years (rr6798 days, 12 hours, 57 minutes, 52 seconds). 212. The place of the node being determined for any given instant, and the latitude ME (fig, to art. 209.) being found for the same time ; the arc MN, or the angular distance of the moon from the node, on her orbit, may be found by trigono- CHAP. IX. ITS FORM ELLIPTICAL. 167 metry in the right angled spherical triangle MEN. If a series of such distances be computed from the observed right ascensions and declinations of the moon when on the meridian, and the intervals, in time, between the transits be also ob- served ; it is plain that the daily angular motion of the moon in her orbit about the earth may be determined. It should be remarked that the longitude of the moon may be converted into the distance on her orbit, from the point y , and the con- verse, by the rules of spherical trigonometry, as in art. 189., or by the formula for / a in art. 1 90. : in either case, for these purposes, I must be made to represent the distance f M on the orbit, a the longitude y E, and 6 the angle M T E. 213. The relative distances of the moon from the earth may be ascertained approximately from the observed angular magnitudes of the moon's diameter ; or the absolute distances may be ascertained by means of her horizontal parallaxes, the latter being determined by the method explained in art. 160. Thus, let it be supposed that the earth is a sphere, and that M, the centre of the moon, is in the horizon of an observer at s (fig. to art. 154.); the angle CSM will be a right angle, and the angle SMC the horizontal parallax; therefore (PI. Trigo., art. 57.) sin. SMC : radius:: sc : CM; hence SC the semidiameter of the earth being known, CM the required distance will be found, (about 237,000 miles). Then, by means of the moon's daily angular motions in her orbit, and her daily distances from the earth, the figure of the moon's orbit may be determined by a graphical construction, as that of the earth was supposed to be determined (art. 194.). It would thus be found that the orbit is nearly elliptical, having the earth in one of its foci ; and from the properties of the ellipse, the excentricity of the orbit may then be ap- proximatively ascertained. The places of the perigeum and apogeum, and the equation of the centre, sometimes called the first inequality, may evidently be determined by methods similar to those which have been described in the account of the earth's orbit. 214. A series of the daily longitudes of the sun and moon being obtained from observations, there may be found by inspection, and by proportions founded on the observed daily motions of the two luminaries in longitude, two successive times at which the sun and moon had the same longitude, or were in conjunction in longitude (times of new moon), or two successive times at which the longitudes of the sun and moon differed in longitude by 180 degrees, that is, two successive 168 THE MOON'S ORBIT. CHAP. IX. oppositions (times of full moon) ; and the interval between the two times, in either case, constitutes the period of a synodical revolution of the moon. Let this synodical period, O ^J/~vO in days, be represented by s ; then - will be the excess of s the moon's mean daily motion in longitude above that of the sun or earth. But the moon's motion being subject to great irregularities, the mean movement deduced from a single synodical period can only be considered as a first approxima- tion to that element. In order to obtain it with greater cor- rectness, the recorded time of an ancient eclipse of the moon, at which time the sun and moon were nearly in opposition, should be compared with the time of a modern eclipse, the moon being nearly in the same part of her orbit with respect to the points of apogee or perigee : the difference between these times divided by the number of synodical revolutions which have taken place in the interval, the number being found from the approximate time of one revolution deter- mined as above, will give the accurate mean time of a synodical revolution. 215. Ptolemy has stated in the Almagest the occurrence of three eclipses of the moon, which were observed by the Chaldeans in the years 721, 720, and 719 B. C. ; and, from the time elapsed between those eclipses and one which hap- pened in the year 1771 of our era, Lalande determined the time in which a synodical revolution was accomplished. The time of such revolution has also been found by a comparison of the Chaldean eclipses with three of those which were ob- served at Cairo by Ibn Junis between the years 997 and 1004, and also by comparisons of eclipses which have been observed within the two last centuries ; and from a combination of the results, not only has the time of a revolution been determined, but certain differences have been found in the durations of the revolutions from whence it is ascertained that the mean motion of the moon has for many ages experienced an accele- ration which varies with time. La Place has determined that, in the, present age, the synodical revolution is performed in 29 days, 12 hours, 44 minutes, 2*8 seconds. 216. The sidereal period of the moon, or the time of her revolution from one fixed star to the same, may be found thus. The number of days between the times of two observed eclipses of the sun or moon, at an interval of many years, is known, and from the approximate length of a synodical revolution, the number of such revolutions in the same in- terval is also known. Then, since the angular space described by the moon about the earth between two successive con- CHAP. IX. HEK HE VOLUTIONS. 169 junctions or oppositions of the sun and moon exceeds 360 degrees by the angular motion of the sun during the same revolution ; it follows that if n be the number of synodical revolutions, and m be the number of degrees described by the sun with his mean sidereal movement in the whole time be- tween the observed eclipses, we shall have n 360 + m for the number of degrees described by the moon in that time : let N be the number of days in the time, then 360::N : N' ; where N' is the number of days during which the moon would describe the circuit of the heavens from any fixed star to the same: thus the time of a sidereal revolution of the moon about the earth is found to be 27.321661 da., or 27 da. 7 ho. 43m. 11 -5 sec. The period of a tropical revolution of the moon may from thence be immediately deduced: for let p ( = 3*75 sec.) be the amount of precession during one sidereal revolution of the moon ( 50". 2 annually) ; then 360 : 360 p:: 27.321661 da. : 27.321582 da., or 27 da. 7 ho. 43 m. 4-68 sec. This is the periodical lunar month, or the time of her revolution from one equinox to the same. 217. The places of the moon's nodes, found as above men- tioned by the determinations of her place when in the ecliptic, or when her latitude is zero, are observed to vary with time ; and this movement, which takes place in retrograde order, can be found by computing the longitudes of the nodes at times very distant from each other. By comparing together many times when the moon was in, or at equal distances from the nodes, it has been found that a tropical revolution of the nodes (a revolution from one equinox to the same) takes place in 6788.54019 da., and a sidereal revolution, in 6793 da. .42118. 218. The duration of an anomalistic revolution of the moon may be found by taking the interval between two times when she is equally distant from, and very near the point of apogeum or perigeum; and from thence the place of her perigeum may be found, as the perihelion point of the earth's orbit was obtained. On comparing the computed places of the moon's perigeum at different times, it will be found that the major axis of the orbit has, during one sidereal revolution of the moon about the earth, a movement in direct order equal to 3 2 / 31 // *6 ; and hence the period of an anomalistic revo- lution is equal to 27.5546 da. It may be remarked that these 170 THE MOON'S ORBIT. CHAP. IX. periods are seldom now obtained from observation ; astro- nomers confining themselves chiefly to the determination of a correct value of the mean tropical motion of the moon by means of her longitudes observed at great intervals of time. The duration of a mean revolution being already known very nearly, the number (N) of complete circumferences described by the moon in any interval between the times of two ob- served longitudes will be known : consequently N. 360 added to the difference between the observed longitudes will give the whole number of degrees described by the moon during the interval ; and this sum, divided by the number of days in that interval, the years being considered as Julian years (=365.25 days each), such being the nature of the years in astronomical tables, will give the mean daily tropical motion : in the present age this is found to be equal to 13 10' 34".896. A formula has been investigated by astronomers for deter- mining the correction which must be applied to the mean motions obtained from the lunar tables in order to reduce them to their value at a given time : the correction thus found and applied, produces results which accord very nearly with the results of observation within 1000 or 1200 years before or since the epoch for which the tables are computed. The variation of the moon's mean motion affects the periodical times of the tropical, sidereal, and other revolutions of the moon, and renders it necessary to apply continually to those times certain small corrections in order to obtain their true values. 219. In seeking the longitude of the moon for any given time, the mean longitude is first obtained from the tables as if the moon revolved uniformly about the earth in a circular orbit coincident with the plane of the ecliptic ; then the application of the equation of the centre, and the correc- tions for the motions of the perigeum and nodes, give the place of the moon as if the projection of her orbit on that plane were a perfect ellipse. But the moon is subject to many inequalities of motion from the perturbations produced by the mutual attractions of the bodies composing the solar system ; and physical astronomy both assigns the several causes of these inequalities and determines the effects due to each. Some of these inequalities are, however, of sufficient magnitude to be capable of being detected by comparisons of observations made in particular positions of the moon; and accordingly they were discovered and their values computed before the theory of gravitation was employed to account for the phenomena of the solar system. 220. The greatest of these inequalities is that which is CHAP. IX. HER PRINCIPAL INEQUALITIES. 171 called the moon's evection. Astronomers before the time of Ptolemy being accustomed to observe the longitudes of the moon only at the times of the eclipses, that is, when the sun and moon are in conjunction or opposition; on comparing the longitudes of the moon deduced from the mean motion as above mentioned, with the longitude obtained from the observation of the eclipse, they considered the difference to be (for the distance of the moon at that instant from apogee or perigee) the value of what was called the first inequality, or the equation of the centre. But, on obtaining the longi- tude of the moon from observations made when she was in quadrature, or at either extremity of a diameter of her orbit at right angles to that which joins the points of conjunction and opposition, it was discovered that, in order to make the longitudes given by the tables agree with those which were observed in such situations of the moon, the first inequality ought to be augmented. The variable quantity by which the equation of the centre should be increased was subse- quently called the evection : it was found, after many trials, to be a maximum when the excess of the mean longitude of the moon over that of the sun is 90 degrees, and when at the same time the mean anomaly of the moon is also 90 degrees. It was found to vanish when the mean longitude of the sun is equal to that of the moon, and also when the mean anomaly of the moon is 180 degrees or zero. Now, when any deviation from a law of the angular motion which one body may have about another is observed to be periodical ; that is, to be zero in a certain position, to attain a maximum when at 90 degrees, and again to vanish when at 180 degrees from thence; that deviation may evidently be considered as depending upon the sine of the angular distance of the body from the first position. There- fore, since the sine of an angle goes through all its variations, while the angle increases from zero to 180 degrees, and twice while it increases from zero to 360 degrees; the formula expressing the amount of the deviation or inequality may be represented by P sin. Q, where p is some constant number to be determined by observation and Q is a variable angle : thus, M being the mean longitude of the moon, s that of the sun, A the moon's mean anomaly, and E the evection, the equation may be represented by E = P sin. [2(M s) A}. From the latest observations the value of P is found to be 1 20' 29 // .54, and the period in which the evection goes through all its variations is 31.811939 da. 17-2 THE MOON'S OEBIT. CHAP. TX 221. The longitudes of the moon computed from tables which include the evection, on being compared with the longitudes obtained from observations at the times of syzygy and quadrature, were found to agree very nearly ; but Tycho Brahe discovered that there was a third inequality which attained its maximum value when the moon was in the octants, or when the difference between the mean longitude of the moon and that of the sun was 45 degrees, and vanished both in the quadratures and syzygies; this inequality is therefore represented by p' sin. 2 (M s). The value of p' is found from observations to be equal to 35 / 41 // .64, and the period in which the inequality goes through all its variations is evidently half a synodical revolution of the moon about the earth. This inequality is what is called the moon's variation. 222. A fourth inequality, which is called the moon's annual equation, was discovered on comparing the observed with the computed longitude of the moon when the earth was at its mean distance from the sun. For, after correcting all the above deviations, it was found that then the excess of the computed above the mean longitude was greater than that of the observed longitude above the latter : the excess was found to diminish gradually, and to vanish when the earth is in the aphelion and perihelion points. From these points the increase of the observed above the mean motion recommences, and it again becomes the greatest at the mean distances of the earth from the sun. The form of the equation is P" sin. A', where A? is the sun's mean anomaly, and P" is equal to 11' 11".976. It is evident that the period in which the inequality compensates itself is half an anoma- listic year. Besides these, there is a small inequality called secular, of the moon's motion in longitude, which with' several others have been detected by the aid of physical astronomy. 223. The latitude of the moon is, in like manner, affected by periodical variations, of which the most important was discovered by Tycho Brahe. When its value is the greatest it changes the inclination of the moon's orbit to the ecliptic by 8' 47". 15, and it depends on the cosine of twice the dis- tance of the sun from the moon's ascending node. There is also an inequality depending on the sine of the sun's distance from the node ; and others have been obtained from theory. The radius vector of the moon's orbit, and consequently the equatorial parallax of the moon, which depends on the radius vector, is also subject to consider- able variations : a mean equatorial parallax whose value is 57' 4". 165, has been determined both by theory and by ob- servation, and this is called the constant of the parallax. CHAP. IX. DISTANCE OF THE SUN. 173 224. The distance of the moon from the earth being ascertained, it is possible by its aid to find that of the earth from the sun by plane trigonometry : it must be admitted, however, that the result so obtained is inaccurate; yet, as it will be advantageous that an approximative knowledge of that distance should be given previously to an explanation of the phenomena by which, in the present state of astronomy, it is determined, the method may without impropriety be here stated. Let E be the centre of the earth, M that of the moon, and s that of the sun ; all these points being supposed to be in one plane, which is that of the ecliptic ; then a plane passing through a~M.b perpendicular to the ecliptic and to the line SM will separate the enlightened from the dark hemisphere of the moon. Let a plane pass through M n per- pendicular to the ecliptic and to the line E M ; then, the semidiameter M n of the moon being very small compared with her distance from the earth, the line E&m may be considered as perpendicular to M n, and m n which represents the visible breadth of the enlightened part of the moon's disk to an observer on the earth will have to M?Z the ratio that the versed sine of the angle bMn has to radius. Now, by instruments, the angles subtended at the earth by M n and m n can be measured : and by trigo- nometry, i/Ln : mn :: 1 : 1 cos. , mn , MTZ mn whence = 1 cos. OMW, or cos. bw.n = - : Mn Mra therefore the angle bMn may be found. Then, since SM is a right angle, the angle SMTZ is the complement of &MTZ, and EM rc being a right angle, the angle EMS is found. Conse- quently, if at the time of measuring the breadth mn, the angular distance MES, of the moon from the sun be observed (both sun and moon being above the horizon), there will be given, in the plane triangle EMS, the side EM and all the angles; to find ES, which is the required distance of the earth from the sun, (about 95,000,000 miles). The difficulty of the problem consists in obtaining the measurement of M n and m n by the micrometer with sufficient accuracy ; and, if the angular distance of the sun from the moon be found immediately by observation, in rightly determining the effects of parallax. 174 APPARENT DISPLACEMENTS. CHAP. X. CHAP. X. APPARENT DISPLACEMENTS OF THE CELESTIAL BODIES ARISING FROM THE FIGURE AND MOTION OF THE EARTH. THE EFFECTS OF PRECESSION, ABERRATION, AND NUTATION. 225. THE observations of the ancients on the places of the fixed stars were not sufficiently precise to allow the law of their apparent motions to be accurately determined by com- parisons of those places with such as have been obtained in a later age. In endeavouring to ascertain the value and law of the motions, it has therefore been found proper to compare the observations made in recent times with those of a date not more remote than the middle of the eighteenth century, when celestial observations first acquired the correctness necessary to render a comparison advantageous. On comparing the longitudes and latitudes computed from the right ascensions and declinations of many hundred stars observed at different epochs since the year 1750, it has been found that the longitude of each has increased at a mean rate of 50". 2 annually, while the latitudes undergo in a year changes not exceeding half a second: and, as it is scarcely probable that all the stars have equal angular movements about the axis of the ecliptic, the circumstance suggests the hypothesis that the equinoctial point 7% from which the longitudes are reckoned, has a retrograde movement on that great circle of the sphere at the mean rate above mentioned, as stated in describing the solar orbit (art. 174.). This hypo- thesis being adopted, it should follow that the annual variations in right ascension and declination, when computed on that hypothesis, by the rules of trigonometry, agree with those which are found by observation to take place. 226. In order to find the laws of the variations in declina- tion and right ascension : let if C L be part of a great circle of the sphere representing the trace of the ecliptic, which may be considered as fixed, and let p be its pole. Let T q be part of the equator, and P its pole : so that the angle Cf<7, the obliquity of the ecliptic, is measured by pp. Let S be the place of a star, and draw the great circle p ST ; then T T will be the star's longitude, and TS its latitude. Draw also the declination circle P S m ; then, at a given epoch, CHAP. X. PRECESSION IN DECLINATION. 175 suppose at the commencement of any year ( y being place of the equinoctial point), y m will be the right a anrl vn A tViA rJf>r>lir>ntinn nf flip. Ktnr then the ascension, and m s the declination of the star. Now, at the commencement of the next year (for example) let the equinoctial point be at E, so that yE=z50".2; and let the position of the equator be E/, having p' for its pole. Draw the declination circle p' S n ; then, at the second epoch, Era will be the right ascension, and sn the declination of the star. Again, p~p' or its equal the angle CE cos. y p' s ; or (0 being the obliquity, and y P'S being considered as the given right ascension ( = a) of the star), p' (in seconds) 50". 2 sin. cos. . But p' may be considered as the difference between the polar distances P s and P'S, or the variation of the declination between the two epochs. Thus the annual precession in declination is equal to 50". 2 sin. cos. a: but 50". 2 sin. is nearly equal to 20". 06 ; therefore the precession in declination may be represented by 20".06 cos. a. This variation is negative, or the effect of precession on a star whose right ascension is less than 6 hours, is to diminish the polar distance, and when the right ascension is zero, the variation has the greatest negative value, being then about 176 PRECESSION IN RIGHT ASCENSION. CHAP. X. equal to 20".06 : when the right ascension of the star becomes 6 hours, cos. #, and consequently the precession in north polar distance, vanishes. The variations afterwards increase till the right ascension is 12 hours, when they attain their greatest positive value, being then about + 20". 06 : they afterwards diminish till the right ascension is 1 8 hours ; and they again increase till the right ascension is 24 hours. 227. The annual precession in right ascension is the dif- ference between y m and E n : in order to find it let fall T e perpendicularly on E q f ; then in the right angled triangle Eye, which may be considered as plane, we have ET 50". 2, the angle at E equal to JOP ( 0) and the right angle at e ; therefore (PI. Trigo., art. 56.) E f cos = E e (in seconds). Now, because of the smallness of the angle subtended at the pole P or P' by the interval in right ascension between m and n, that part of the precession in right ascension may be represented by nri '; and this may be found from the two spherical triangles P S t and n S n f , which have an equal angle at S, and in which the angles at t' and n may be considered as right angles. PS and STZ are known, the former being the given polar distance, and the other being supposed to be equal to its complement, that is, to the given decimation : also in the triangle PP', p (= pp' sin. r P's) =: 50".2 sin. 6 sin. a. Therefore, in the spherical triangles P s t and n s n' (art. 60. (*?)), r sin. P sin. S sin. PS and r sin. nn' sin. s sin. S n. Consequently sin. PS : sin. Pt:: sin. sn : sin. nn' \ or substituting the small arcs or angles for their sines, sin. PS : P t : : sin. s n : nn' \ whence n n' =. -. Pt ; or (d being the decimation) Sill* Jr o nn' i ^ 50".2 sin. sin. a 50".2 sin. sin. a tan. d. cos. d Therefore the whole annual precession in right ascension ( =. E e + n n f ) is equal to 50". 2 (cos. 6 + sin. sin. a tan. d). And these formulae for the annual precession in declination and right ascension are found to express very nearly the differences between the declinations, and between the right ascensions when observed at considerable intervals of time. 228. About the year 1750, Dr. Bradley, while observing certain stars which massed the meridian near the zenith of his CHAP. X. ABERRATION. 177 station, discovered that, after having made the necessary cor- rections in the right ascensions and declinations on account of the effects of general precession, such stars apparently described in the heavens the circumferences of very small circles, or the peripheries of small ellipses each returning to the place in which it was first observed at the end of one year. Thus, at the time of the vernal equinox, the star 7 Draconis was observed on the meridian near the zenith about the time of sun-rise, when, consequently, the plane of the meridian must have passed through a tangent to the earth's orbit ; and the star then appeared to be to the south of that which, according to computation, should have been its true place in declination, while its apparent right ascension was unchanged. At the summer solstice the star was on the me- ridian about midnight, at which time the plane of the meri- dian is nearly at right angles to a line touching the earth's orbit ; and then the star was nearly in its true place with respect to declination, but its transit took place at a time later than the computed time, as if its right ascension had increased, or the star had moved eastward. Again, at the time of the autumnal equinox, 7 Draconis was on the meri- dian about sun-set, when the plane of the meridian must have passed through a tangent to the orbit ; and the star appeared to the north of its computed place, while the right ascension became the same as at first. Lastly, at the winter solstice, the star passed the meridian above the pole at noon, at which time the declination was the same as at first, but the time of the transit took place earlier than the computed time, as if the right ascension had diminished, or the star had moved westward. The like phenomena were exhibited by every other star on which observations were made. These variations of position were found to correspond to such as would result from the movement of the light transmitted by the star, and that of the earth in its orbit ; and this cause of the phenomena being admitted, it is evident that similar variations must result from the movement of light combined with that of the earth on its axis, but these are found to be very small. 229. The apparent deviation of a star on account of the combined movements of light and of the earth, is called the aberration of the star ; and the manner in which the pheno- mena take place may be thus explained. Let s be the place of a star, and ME the position of the axis of a telescope when the eye is at E, and when a par- ticle of light in a ray proceeding from the star S is at M ; then, while the particle moves from M to E', the eye of the spectator- is carried from E to E' by the motion of the earth ; so that N 178 ABERRATION CHAP. X. E'S' becomes the position of the axis of the telescope when the star is visible, and s' becomes the apparent place of the star. Now the ratio of ME' to EE' is equal to that of the velocity of light to the velo- city of the earth in its orbit ; and (PI. Trigon., art. 57.) ME' : EE' :: sin. MEE' : sin EME'. T E/ E But the angle MEE' may be considered as equal to SE'T, and EME' is equal to SES'; therefore, substituting the small angle SB'S' for its sine, the velocity of light is to that of the earth as sin. SE'T is to SB'S'. Again, the ratio of the velocity of light to that of the earth being constant, it follows that the aberration in the plane SE'T varies with the sine of the angle SE /r r. Here, from the smallness of the earth compared with the celestial sphere, E and E' may be considered as either at the surface or at the centre of the earth, and E T as a tangent to the earth's orbit : therefore the angle SB'S', or the aberration, is always in a plane passing through the star, the centre of the earth and a tangent to the orbit. Hence, on the hypo- thesis that the earth revolves annually about the sun, it is evident that the effect of the combined motions of light and the earth may be represented by supposing each star to de- scribe annually in the heavens a curve similar and parallel to the earth's orbit ; the star being always in advance of the point corresponding to the place of the earth's centre by the value of the angle SB's'. But, in the region of the fixed stars, the magnitude of the earth's orbit scarcely subtends any sensible angle ; and therefore the star appears to describe, about the place which it would seem to occupy if light passed instantaneously to the earth, a circle or ellipse at a distance equal to the measure of the angle SB's': the visible form of this curve will evidently be different for different stars ac- cording to the position of the plane in which such curve ap- pears to be projected on the celestial sphere. 230. Since the angle of aberration SB'S' varies with sin. SE'T, it will evidently be a maximum when SE'T is a right angle, in which case the star would be in the direction of a line drawn from the earth perpendicularly to the plane of its orbit, that is, on account of the great distance of the stars, in the pole of the ecliptic. Now the angle SE'T being known for some particular star, and the angle SB'S' being found by observation, the amount of aberration for a star in the pole of the ecliptic may consequently be obtained from the proportion sin. SE'T : radius :: SB'S' : A. CHAP. X. IN LONGITUDE. 179 This last term is generally called the constant of aberration, and, from the best observations, it is found to be equal to 20 // .36. Therefore the formula 20".36 sin. SE'T gives, for any star, the value of the aberration in a plane passing through the star, the earth and a tangent to the earth's orbit. The position of the last-mentioned plane for any star at a given time being found, the effects of aberration in longitude and latitude, in right ascension and declination, may be computed by the rules of spherical trigonometry. 231. Let the primitive circle ETA, of which a part only is drawn, represent the ecliptic in the heavens, let p be its pole, and s the place of a star: let T be the equinoctial point, and for the given time, let T O be the longitude of the sun ; then E, at the extremity of a diameter passing through o, will be the place of the earth when projected in the heavens by a line drawn from the sun. Since the earth's orbit may be considered as very small, the place both of the earth and of the sun may be conceived to be at the projection p of the pole of the ecliptic ; and pT, at right angles to OE, may be considered as a tangent to the earth's orbit. Imagine a great circle to be drawn through s and T, and also a circle of longitude p A through s ; then the plane pST is that which passes through the star, the earth and a tangent to the earth's orbit, and is consequently the plane of aberration, or that which above is designated SET. The arc GET (reckoning from o in the order T GET, that is of the signs) is equal to three quadrants, TO (= L) is the longitude of the sun, and T A ( = /) is the longitude of the star; therefore AOET is equal to L + 270 /. Now, in the plane s/?T, the aberration is equal to 20".36 sin. ST: let this be represented by SS', and imagine pa to pass through s'; also let s'<7 be drawn through s' perpendicular to pa: then s'g, when reduced to the ecliptic, on dividing it by cos. s'a (art. 70.) becomes A a, the aberration in longitude, and sq is the aberration in latitude. 232. To find the aberration in longitude. N 2 180 ABERKATION IN LATITUDE. CHAP. X. In the triangle SAT, right-angled at A, we have (art. 60. (e)) sin. AT (= sin. AOET) z=sin. AST sin. ST : therefore sm. (L + ) _ g .^ AST ^ or _ cog> ss /^ sin. s T But a'q = SS' cos. SS'? = 20".36 sin. ST - L g ^ gT = 20 // .36 sin. (L + 270 -7); Or (PL Trigon., art. 32.) s'2=20".36 {sin. (L + 270) cos. Z cos. (L -f 270) sin. Z} ; and supposing, as in the above figure, that Z*is less than 90, also L + 270 less than 360, we have sin (L + 270) = cos. L, and cos. (L + 270) = sin. L; therefore a'q = 20 // .36 (cos. L cos. I + sin. L sin. Z), = 20".36 cos. (L /); which being reduced to the ecliptic, we have (art. 70.) X re- presenting the latitude A s of the star, A (the aberration in longitude) = 20".36 cos - ( L ~ l \ COS. A 233. To find the aberration in latitude. Imagine a great circle s'l (of which s'^ may be considered as a part) to pass through s' perpendicularly to pA. and to cut the ecliptic in I ; the point I will be the pole of p A and the angle at I will be measured by SA, the star's latitude. Then, in the triangle SIT (art. 61.), sin. ST : sin. i :: sin. IT : sin. IS'T. But sin. I = sin. \ ; and since IA =r TO, each of them being equal to a quadrant, we have I T =: T O T A = L Z ; also S'T may be considered as equal to ST; ,, p / / \ sm - A sm - ( L therefore sm. is r T ( = sin. ss^) = ; * *. sm. ST and s^ (the aberration in latitude) = ss r sin. ss'y = 20 /r .36 sin. ST sin. ss^ = 20 r/ .36 sin. A sin. (L Z). 234. To find the aberration in right ascension. Let w T M be the projection of the celestial equator, and P its pole : draw the declination circles through P s and P s x , cutting the equator in M and N, and the former cutting the ecliptic in A' ; also let s r t be drawn from s r perpendicular to PS: then MN will represent the required aberration in right ascension. CHAP. X. ABERRATION IN RIGHT ASCENSION. 181 In the triangle SA'T we have (art. 61.) sin. ST : sin. A' :: sin A'T : sin. A'ST; , sin. A? sin. A'T whence r - sm. A'ST rr cos. ss'f. sin. ST And (reckoning in the order A'OET, that is of the signs) sin. A'T rr sin. ( T T r A') = sin. y T cos. T A' cos. if T sin. f A' ; (PL Trigon., art. 32.) therefore sin. A' sin. A'T = sin. A' cos. T A' sin. if T sin. A' sin. T A' cos. if T .... (A). Now in the triangle PA'B, right angled at B (art. 62. (e')\ r sin. A'B z= cotan. A? tan. PB But if B =270, according to the order of signs ; there- fore A'B = 270 if A', and sin. A'B = cos. if A' : substi- tuting this term for its equivalent, and multiplying both members by sin. A?, the last equation becomes, cos. A' being negative, r sin. A' cos. if A' cos. A' tan. PB. Also, in the same triangle we have (art. 60. (/)), r cos. A' = sin A'PB cos. PB ; therefore the preceding equation becomes sin. A' cos. Y A' = sin. A'PB sin. PB. But A'PB = 90 + f M; therefore sin. A'PB = cos. if M = cos. a (a representing the star's right ascension) : also P B = 90 + pp ; therefore sin. P B = cos. P/? = cos. (6 representing the obliquity of the ecliptic). Consequently sin. A' cos. if A' = cos. a cos. 6. Again, in the triangle if A r M, right angled at M (art. 60. (e)), sin. A' sin. if A' = sin. T M = sin. a. Lastly, if T (according to the order of signs) = L + 270 ; therefore sin. f T ~ cos. L, and cos. if T =r sin. L. Then, substituting all the values just found in the equation (A) we get sin. A' sin. A X T m (cos. L cos. a cos. 6 + sin. L sin. a). Now s r (= ss' cos. ss'f) = 20".36 sin. ST cos. SS'T, or = 20 7/ .36 sin. A' sin. A'T : Hence it follows that s7 20".36 (cos. L cos. a cos. 6 + sin. L sin. a), which being divided by cos. s M, or cos. d (d representing the star's declination) is reduced to the equator, and becomes the value of MN, the aberration in right ascension. N 3 182 ABERRATION IN DECLINATION. CHAP. X. 235. To find s the aberration in decimation. Imagine the great circle s'l', of which s't may be con- sidered as a part, to be drawn through s' perpendicularly to P s, cutting the equator in w, and the ecliptic in i' : then w will be the pole of PSM, and, st being very small, the angle at w may be considered as measured by MS, the star's declination. Now in the triangle s'i'T (art. 61.) sin. S'T : sin. i' :: sin. I'T : sin. I'S'T ( = sin. ss'); ,, , , sin. i' sin. I' T therefore, putting ST for S'T, sin. I's T ; - . sin. s T But, reckoning according to the order of the signs, I'T = T T r i' ; therefore (PI. Trigon., art. 32.) sin. I'T = sin. r T cos. Y i' cos. T T sin. T i' a and sin. i' sin. I'T = sin. I' cos. f i' sin. f T sin. i sin f i' cos. y T ..... (B). Now in the triangle I'wy (art. 61.) sin. if : sin. w f : : sin. w : sin. i x f ; therefore sin. i' sin. i' T = sin. w sin. w f : but (reckoning according to the order of the signs) M w =270; therefore T w = T M 4- 270 = a + 270 ; and sin. r w = cos. , also cos. fw = sin. a. Again, multiplying both members of the last equation by cos i' *v . ' r or by cotan. i' T , that equation becomes sin. I *]p sin. i 7 cos. i x if = sin. w sin. w *y cotan. i' y : but (art. 62., 2 Cor.) sin. WT cotan. r'f =cos. WT cos.T + also, as above, cos. f T =: sin. L, and sin. f T =: cos. L ; therefore substituting the values of the terms in the above equation (B) we have sin. i' sin. I'T^: cos. L sin. d sin. a cos. 6 + cos. L cos. d sin. 9 sin. L sin. d cos. a ; which being divided by sin. ST in order to have the value of sin. i r s' T, or s s r t, and then multiplied by s s', or by its equi- valent, 20".36 sin. ST, the result wiU be the value of at, the required aberration in declination. 236. The value of the angle EME' (fig. to art. 229.) for any star, being supposed to be determined by observation, and that of the angle SE r T being obtained by computation; it will follow, by trigonometry, that the ratio of ME' to EE' is CHAP. X. DIUKNAL ABERRATION. 183 nearly as 10,110 to 1 ; and since it is not convenient, in this place, to introduce an account of the celestial observations by which the velocity of light was first determined, it will be proper at present to admit the motion of light as an hypo- thesis, and to consider the ratio above mentioned as a result of the observations by which the aberration was ascertained. It will then follow, the distance of the earth from the sun and the mean daily motion of the former in its orbit being supposed to be known, that the earth must describe in the orbit an arc subtending at the sun an angle of 20". 3 6 in the time (8' 13".5, art. 282.) that a particle of light passes from the sun to the earth ; and hence it will be evident that when the sun's centre appears in the axis of a telescope, that centre is apparently 20". 3 6 in advance of the place which it would occupy if no time elapsed in that passage. The effect of aberration on the place of the sun is, therefore, the same as that which takes place on a star situated in the pole of the ecliptic. Aberration necessarily affects also the ap- parent places of the planets, and its value for each depends upon the relative motions of the light, the earth and the planet. The formula by which its effects are computed are investigated in several treatises of astronomy. See De- lambre, " Astronomic," torn. iii. ch. 29. 237. The diurnal aberration of light is a phenomenon re- sulting from the movement of light combined with the ro- tation of the earth on its axis : and it differs from the annual aberration above described, in consequence merely of the difference between the velocity of the earth on its axis and in its orbit : now the annual movement of the earth in the time in which light passes from the sun to the earth is, as above mentioned, 20". 3 6 ; but if r be the semidiameter of the earth, and r' that of the earth's orbit supposed to be circular, we have r' - (p being the sun's equatorial sin. p ^ J parallax, or 8". 6) ; therefore the movement of the earth in its orbit in that time is -^ 20".36 sin. 1", or putting 8".6 sm. p 20" 36 r sin. V for sin. p, the movement in the orbit is * or 8 .o 2.355 r. The angular motion of the earth on its axis in the same time is equal to 2 3' 18", or in linear measure, 7398" sin. 1". r; and the annual aberration of a fixed star when a maximum is 20". 3 6 : therefore 2.355 r : 7398" sin. 1":: 20".36 : 0".3085, or in sidereal time, 0" 0206. This is the diurnal aberration N 4 184 NUTATION, CHAP. X. in right ascension for a star in the equator at the time that it is on the meridian of a station, the latter being on the ter- restrial equator; but for a station whose latitude is L, the diurnal rotation being less on the parallel of latitude than at the equator in the ratio of radius to cos. L, the aberration of a star on the meridian of the station is, in sidereal time, equal to 0".0206 cos. L ; if the star be not in the equator, this ex- pression denotes only the aberration on the parallel of the star's declination, and in order to reduce it to the celestial equator, or to the value of the aberration in right ascension, it must be divided by the cosine of the star's declination. The diurnal aberration in declination is too small to require notice. 238. The displacement of the pole of the world in con- sequence of the general retrogradation of the equinoctial point is such that, while the latter point would appear to describe the circumference of the ecliptic in the heavens, the pole of the equator would describe a circle about the pole of the ecliptic with an equal angular motion (50". 2 annually), and at a distance equal to the mean obliquity of the two circles. But Dr. Bradley remarked that the observed declinations of stars differed from those which resulted, by computation, from the hypothesis of a uniform precession by small quantities which varied very slowly between the extreme limits (about + 9" and 9"), and he discovered that the star returned to the same position in declination in about eighteen or nineteen years ; in which time either of the nodes of the moon's orbit revolves about the earth. It may be observed that this ap- parent displacement of the stars was a discovery made sub- sequently to that of aberration ; and that its effects in the course of one year are much smaller than those which result from the movement of light and of the earth. 239. In order to explain the nature of the displacement let QN be the mean place of the celestial equator and p its pole at a time when the moon's ascending node K is at the equinoctial point Q. At such time it was observed by Bradley that the polar distance of a star s whose right ascension was six hours, instead of being represented by PS, was less than the value of that arc by about 9" ; while the polar distance of a star s' whose right ascension was eighteen hours exceeded PS' by the same quantity ; at the same time a star at s" or s'", whose right ascension was or twelve hours, appeared to suffer no change of declination ; and therefore the phenomena were CHAP. X. LUNAR AND SOLAR. 185 such as would take place if the pole of the equator were at A, at a distance from p equal to about 9" of a degree. Again, let Q N represent the mean place of the equator and p its pole at a'time when, by its retrogradation, the moon's node is at N, its longitude being then 270 degrees. At such time (about four and a half years from the former time) the polar distance of a star s'" whose right ascension is zero, instead of being re- presented by P s w was less than the value of that arc by about 7", while that of a star s" whose right ascension was twelve hours exceeded p s" by the same quantity : at that time a star at s or s' appeared to suffer no change of declination ; there- fore the phenomena were such as would take place if the pole of the equator were at B, at a distance from p equal to about seven seconds. When the moon's node was at L (about nine years from the commencement of the observations) the ob- served declinations of stars were such as would take place if the pole of the equator were at c in the solsticial colure NPN', at a distance from p equal to AP ; and during the following nine years the variations of declination were observed to take place as in the preceding period but in a reverse order. These observations were made between the years 1727 and 1746, and Dr. Bradley's conclusion was that the phenomena might be represented by supposing an oscillation of the plane of the equator upon the line of the equinoxes, combined with the general movement of that line on the centre of the ecliptic ; so that the pole of the world, or the extremity of the axis of the equator, which axis is always at right angles to the plane of that circle, seems to describe a small elliptical figure ABCD about the point P, the mean place of the pole. 240. It is easy to conceive, since P is supposed to be con- stantly moving in the circumference of a circle at the mean rate of 50". 2 annually, while the revolving pole describes the peri- . phery AB c D in nineteen years, that, if PQ represent the part of the peri- phery described in that time by the mean pole P about the pole of the ecliptic, the curve line P'Q', a species of epicycloid, will represent the corresponding path of the revolving pole in the same time. This oscillatory motion of the pole about its mean place is designated nutation ; and the effects of such movement in changing the apparent right as- censions and declinations of stars are called nutation in right ascension and nutation in declination. That which has been described above depends on the movement of the moon's as- cending node ; and it is therefore designated the lunar nuta- 186 NUTATION" CHAP. X. tion. Corresponding variations are found to depend on the place of the sun, and these constitute the solar nutation. When the position of the pole P' in the ellipse of nutation is determined for any given time, the effects of nutation on the right ascension and polar distance of any particular star may be determined by spherical trigonometry in a manner exactly similar to that which has been described in finding the effects of the general precession. 241. Thus, let p be the pole of the ecliptic, and p the mean place of the pole of the equator at a given epoch : also, agreeably to the hypothesis of Bradley, let it be supposed that ABCD in which the true pole P' is situated is an ellipse, and imagine the circle Abed to be described about it. The pole p' being at A when the moon's ascending node is at Q (fig. to art. 239.), and at B when the node is at N, and so on; let the angle APE be equal to the distance of the node from Q in a direction contrary to the order of the signs (that is, let APE be equal to 360 minus the longitude of the node) : then drawing E F perpendicular to A c, the place of P' may be con- ceived to be the intersection of this line with the periphery of the ellipse. Now, by conic sections, AP : BP :: EF : P'F, and by trigonometry, EF : P'F :: tan. EPA : tan. P'PA. But, from the best observations, AP, called the constant of nu- tation, = 9".239 and BP = 7". 18 (Mem. Astr. Soc., vol. xii.); and putting N for the longitude of the moon's node, tan. EPA = tan. N, also cos. EPA =. cos. N ; nrr TO therefore tan P' p A = ' tan. y Again, by trigonometry, PE : PP':: sec. EPA : sec. P'PA, or :: cos. P'PA : cos. EPA; ,, n PE cos. EPA cos. N therefore PP' = - - - = 9".239 cos. P'PA cos. P'PA Then, in the spherical triangle ppp' (art. 61.), sin. APP' : sin. P'/>P :: sin. p'p : sin. PP' ; or, putting pp (equal to the obliquity of the ecliptic) for p'p and representing it by 6 ; also putting small arcs or angles for their sines, sin. APP' : p'jt?p::sin. 6 : PP'; CHAP. X. IN DECLINATION. 187 PP' sin. APP' therefore P pP = - . Q . sin. u Substituting the above value of PP', we have 9".239 cos. N sin. APP' 9".239 cos. N A P'P =: - ; 5 . = . >, tan. APP'. sin. 6 cos. APP' sin. 6 And again, substituting the above value of tan. A P p', we have 9".239 cos. N 7". 18 P > P : -ST0 ' 9*339 ^ *' which, after reduction, becomes 18". 03 sin. N. This is called the lunar equation of precession, or the lunar equation of the equinoctial points in longitude. From the value of P P' above, we have PF (= PP' cos. P'PA) = 9".239 cos. N. This is called the lunar equation of obliquity. 242. By processes exactly similar to those which were employed for finding the precession in right ascension and declination, the effects of nutation in those directions may be found. For let f C represent part of the trace of the ecliptic, of which p is the pole ; f R the trace of part of the mean equator whose pole is P, and E R' the trace of part of the true equator at the given time ; and, for this time, let N represent the longitude of the moon's node, let P' be the pole of the equator, and let s be the place of a star. The figure being completed as before (art. 226.), the arc TE is equal to the angle J?p P', and repre- sents the lunar equation of the equinoctial points in longitude: then, the angle at E being repre- sented by 0, the obliquity ; and y being let fall perpendicularly on EH', we have (PI. Trigo., art. 56.) E T cos. 6 = E t ; or, since E f ?'p p, 18".03 sin. N. cos. = E. This is called the equation of the equinoctial points in right ascension, in seconds of a degree; or when divided by 15, it expresses that equation in time. 243. Let fall p'r perpendicularly on p s ; then p x pr ( p r pn RPS) = P X PR /!" a\ (a representing T m the right ascension of the star) : or P 7 P r p' p R -f a ? 188 NUTATION. CHAP. X. Also Pr = PP' cos. p'Pr = PP' sin. (P'PR + a) = PP' (sin. P'PR cos. a + cos. P'PR sin. a) ; then, for PP' sin. P'PR substituting P'^P sin. 6 (art. 241), or 18".03 sin. N sin. 0; and for PP 7 cos. P'PR substituting 9".239 cos. N, we get Pr = 9".239 cos. N sin. a 18".03 sin. N sin. 6 cos. a ; and this is the nutation in north polar distance. 244. In finding the nutation in right ascension, n n' may be considered as the measure of the angle subtended at either pole P or P' by the interval between n and m ; also s n may be considered as equal to the declination d of the star, and SP' as its complement: then, as in the case of general preces- sion (art. 227.), sin. P'S : p'r :: sin. $n : nn f , that is cos. d : p'r : : sin. d : n n ; consequently nn' = p'r tan. d. But p'r = PP' sin. p'pr = PP' cos. (P'PR + a) PP' (cos. P'PR cos. a sin. P'PR sin. a) = 9 r/ .239 cos. NCOS, a + 18".03 sin. N. sin. 6 sin. a; therefore nn' (9".239 cos. N cos. a + 18."03 sin. N. sin. sin. a) tan. d. This value of nn' being added to the above value of E gives the required lunar nutation in right ascension. 245. The stars are known to suffer an apparent displace- ment depending on the position of the sun with respect to the point of the vernal equinox, and similar to that which de- pends on the moon's node ; but, on account of its smallness, it is incapable of being detected in a separate form by actual observations, and is only known as a result of theory. Its effects may be represented by a motion of the pole of the world in an ellipse about its place in that which depends on the moon : the major axis of such ellipse is 0".435 and the minor axis is 0".399. But, since the inequality is compen- sated twice in each year, being the greatest at the times of the solstices (June and December) and vanishing at the times of the equinoxes (March and September) the angle corre- sponding to APE (art. 241.), instead of being equal to the excess of 360 above the longitude of the moon's node, is equal to twice the sun's longitude. CHAP. XL DISTANCES OF PLANETS. 189 CHAP. XL THE ORBITS OF PLANETS AND THEIR SATELLITES. THE ELEMENTS OP PLANETARY ORBITS. VARIABILITY OF THE ELEMENTS. KEPLER'S THREE LAWS. PROCESSES FOR FINDING THE TIME OF A PLANET'S REVOLUTION ABOUT THE SUN. FOR- MULAE FOR THE MEAN MOTIONS, ANGULAR VELOCITIES AND TIMES OF DESCRIBING SECTORAL AREAS IN ELLIPTICAL ORBITS. MODI- FICATIONS OF KEPLER'S LAWS FOR BODIES MOVING IN PARA- BOLICAL ORBITS. MOTIONS OF THE SATELLITES OF JUPITER, SATURN, AND URANUS. IMMERSIONS ETC. OF JUPITER'S SATEL- LITES. THE ORBITS OF SATELLITES ARE INCLINED TO THE ECLIPTIC. SATURN'S RING. 246. THE distance of a planet from the sun may be deter- mined approximative!^ in terms of the earth's distance from that luminary by various processes. For example, if S re- present the sun, E the earth, and v one of the inferior planets, Mercury or Yenus, supposed to revolve in a circular orbit about s, in the plane of the ecliptic; and if the angle SEV be observed with an instrument at the time when the planet appears to be at its greatest elonga- tion from the sun, or if SEV be taken to repre- sent the difference between the longitude of the sun and the geocentric longitude of the planet at that time ; then the visual ray v E being a tangent to the orbit of v, and consequently at right angles to sv, we have (PL Trigon., art. 56.) SV SE sin. SEV. Thus sv, the required distance, may be found. With respect to a superior planet, its distance from the sun, in terms of the distance of the latter from the earth, might be computed approximatively by means of observations made at times when the planet appears to be stationary, be- fore and after its apparent movement is in retrograde order. Thus, let s be the sun, p a superior planet, supposed to be in the plane of the ecliptic ; and let E be the place of the earth in its orbit when a tangent to the latter would pass through p : at this time p appears to an observer at E to be stationary in the heavens, the apparent movement being subsequently retrograde or from p towards p. At the same time let the arc 190 PLANETAEY ORBITS. CHAP. XL sp, the apparent distance of P from a fixed star s in the plane of the ecliptic, be found by an angular instrument. Now if, for the present, the motion of P towards P 7 , in its orbit, be neglected; when the earth, having described the arc EAE', arrives at E' where a tangent to its orbit would also pass through P, the planet is again sta- tionary; and in this situation of the earth let the arc sp f , or the apparent distance of P from 5, be measured. Then EP or SP being very small compared with the distance of s from E or s, the arc p p' ( = sp sp) may be considered as the measure of the angle p?p or EPE', and half of this arc or angle may be considered as equal to EPS: therefore, in the plane triangle PES, right angled at E, the distance SE being given, the line SP or the distance of the planet p from the sun may be computed. But while the earth is moving from E to E 7 , the planet will have moved from p to P'; in which .case the apparent angular distance of P' from s will be sp", and the angular movement of P about the sun, that is the angle PSP',' which maybe considered as equal to PE'P', must be added to the arc pp" obtained from the observations, in order to have the arc pp\ or the angle EPE'. The distance of the planet Mars from the earth may be obtained by means of its diurnal parallax; the latter being deduced from corresponding observations at two points on the earth's surface as described in art. 160., and such observa- tions are made by astronomers when opportunities present themselves. 247. The observed right ascensions and declinations of a planet enable an astronomer to obtain a series of points in the celestial sphere, which being supposed to be joined, there will be formed a trace of the line of the planet's motion in space, as it would appear to an observer on the earth's surface ; and neg- lecting the diurnal parallax, it may be considered as the trace of the geocentric path. But if it be assumed that the planets revolve about the sun as the centre, or a focus of their orbits ; then, preparatory to determining the forms of the orbits, the places which a planet seems to occupy in the heavens, when viewed from the earth, must be reduced to the places which it would appear to occupy if seen from the sun : with respect, however, to the longitude of a planet, no such reduction is necessary when the planet is in conjunction with, or in opposition to the sun ; the geocentric longitude being then the same as the helio- CHAP. XL PLACES OF THE NODES. 191 centric longitude, or differing from it by exactly 180 degrees. The two inferior planets, Mercury and Venus, frequently come in conjunction with the sun ; and every year the superior planets are in one or both positions. 248. The elements to be determined for the purpose of ob- taining a knowledge of a planet's orbit are : the longitude of the ascending node, the inclination of the plane of the orbit to the ecliptic, the time in which a sidereal revolution of the planet about the sun is performed, and the excentricity. In order to ascertain the place of a planet in its orbit at a given time, there must previously be known the longitude of the planet at a certain epoch, and also the longitude of the perihelion point at the same epoch. 249. Now, as indicated in art. 191., the geocentric lon- gitudes and latitudes can at all times be computed by the formulae (A) and (B) in art. 181., from the observed right ascensions and declinations, the obliquity of the ecliptic being given ; and, the sun being supposed to be always in the plane of the earth's path, it follows that when the geocentric latitude becomes zero, the heliocentric latitudes will also be zero, or the planet will be in the plane of the ecliptic, that is in one of the nodes of the orbit : hence the first step in the determination of a planet's orbit about the sun should be the discovery of the places of the nodes. For this purpose it is merely necessary to find in the registers of the computed geocentric longitudes of the planet, two such longitudes at times in which they are nearly the same as those of the sun, or differ from them by nearly 180 degrees, and in which the latitudes are nearly zero, the planet being at the first time southward, and at the other northward of the ecliptic ; for between the times of these observations, the planet must have been in the ascending node. Then, by a simple proportion founded on the apparent angular movements of the planet in longitude and latitude at the time, which may be obtained by taking differences between the computed daily longitudes and latitudes, the longitude of the node and the time of the planet being in it can be found : thus there is obtained an approxi- mation to the heliocentric longitude of the ascending node. If when the latitudes were nearly zero, the first latitude had been northward, and the other southward, the longitude determined in like manner would have been that of the descending node : and as it will be found that the difference between the longitudes of the two nodes is nearly equal to 180 degrees, it is evident that, at such times, the line of nodes passes accurately or nearly through the sun and earth. The earth having but a momentary position in this line, it follows 192 PLANETAKY ORBITS. CHAP. XL that the latter passes constantly through the sun only ; and therefore the sun must be considered as the centre or focus of the orbit of each planet. 250. When the geocentric place of a planet's ascending or descending node has been found, it is easy to determine the inclination of its orbit to the 'ecliptic ; and the manner of finding this element may be thus explained. Among the geocentric longitudes and latitudes of the planet, whether superior or inferior, which may be computed from the ob- served right ascensions and declinations, let one such longi- tude and latitude be taken for a time when the longitude of the sun is equal to that of the node ; when consequently both the earth and sun will be in the direction of the line of nodes. Then, let NPN' represent the planet's orbit, and N/?N' its orthographical projection on the plane of the ecliptic; let also NN' be the line of nodes, in which let S be the sun and E the earth. Again, let P be the planet, and imagine Pp to be let fall per- pendicularly on NpN' : draw p A per- pendicular to NN', also join p and A, p and E, and p and E. The angle PAp (zrPNj?) is the inclination of the planet's orbit to the ecliptic ; NEJO is the difference be- tween the longitude of the sun, or node, and the geocentric longitude of the planet, and PEJO is the planet's geocentric latitude. In the triangles PE/?, EAJP, right angled at p and A, we have (PI. Trigon., art. 56.) pp^p tan. PE/?, and Ap=Ep sin. therefore ^ ( =tan. PAD) = tan ' ^ Ap sin. and thus the angle PAp or the required inclination is found. It must be remarked that, in a register of observations, it will be scarcely probable that any should be found for the time at which the longitude of the sun is precisely equal to that of the node ; but as the daily angular motions of the sun and planet in longitude are supposed to be known approxi- matively, the longitude of the planet may, by proportion, be found for the required moment of the conjunction of the sun and node. 251. The longitudes of a planet's nodes being found for two times distant by a considerable interval, and allowance being made for the precession of the equinoxes during the CHAP. XL THE ELEMENTS FOUND. 193 time, it will be discovered that the places of the nodes are not fixed in the heavens : the inclinations of the orbits experience also some variations ; but these, as well as the changes in the excentricities, and the positions of the perihelia, can at present be determined correctly by theory alone. If the times of two successive arrivals of the planet at the same node, with the longitudes of the node, be obtained from observations, the interval will be the periodical time of the planet's tropical revolution about the sun ; and hence its mean movement about the sun may be determined : or, if instead of the lon- gitudes of the node, there be found the distances of the node from a fixed star, the time of a sidereal revolution will be obtained. If the planet have made many revolutions about the sun between the observed times of its arrival at the same node, and allowance be made for the movement of the node in the interval, the mean movement may be very correctly obtained. 252. The longitude of the ascending node of a planet's orbit, and also the inclination of the orbit to the plane of the ecliptic being found, the planet's radius vector, and its distance from the node may be determined for any given time by means of its observed right ascension and declination : and the method of rectangular co-ordinates is that which is now generally employed for the purpose. Thus (fig. to art. 184.), let E be the centre of the earth, s that of the sun, and p the place of a planet : let the plane XE Y be the ecliptic, and let a plane passing through P and s cutting the ecliptic in A" SB be the orbit of the planet, the line A" SB, which for sim- plicity is supposed to be parallel to EX, being the line of the nodes. Imagine also EQ, SQ' to be drawn in the plane EXT to the place of the equinoctial point in the heavens: these lines, on account of the great distance of that point, may be considered as parallel to one another. Then EA, EB, the sun's co-ordinates with respect to E, being repre- sented by x and Y, E s by R, the angle Q' s x' or the longitude of the node by n, and Q E s the sun's longitude by L, we have, as in art. 184., x = R cos. (L TZ), Y = R sin. (L n). Next EA', EB', PM, the geocentric co-ordinates of the planet P being represented by x, y, z\ QEM, the planet's geocentric longitude by Z, so that x E M = I n, and p E M, its geocentric latitude by A ; also E p being represented by r, we have, as in the same article, x = r cos. A cos. (I n)~} fx' = r cos. A cos. (Z n) R cos. (L n) y = r cos. A sin. (I n) I ; whence -I y' = r cos. A sin. (/ ) R sin. (L n) z = r sin. A [_z = r sin. A and S P 2 ( = r' 2 ) =R 2 -f r 2 2 R r cos. A cos. (/ L). o 194 PLANETARY ORBITS. CHAP. XI, The inclination PA"M of the plane of the orbit to the ecliptic being known and represented by 0, we have also r sin. A = {r cos. A sin. (I n) R sin. (L n)} tan, 6 : from these equations there may be obtained the values of r, r', and subsequently those of x', /, and z. Then in the plane triangles, SA"P, SA"M, right angled at A", we have SP : SA" :: rad. : cos. PSA", SA" : A"M :: rad. : tan. MSA"; and in the triangle SMP, right angled at M, SM ( = (SA" 2 + A"M 2 )*) : PM :: rad. : tan. PS M. From the first of these proportions we obtain PSA", which is the planet's heliocentric angular distance from its node, 011 the orbit ; from the second, the planet's heliocentric distance in longitude from the node ; and from the third, its heliocentric latitude. 253. If the geocentric longitude and latitude of the planet had been found from observations for the time when the planet P was in conjunction with the sun (the earth being still at E), and consequently the heliocentric as well as the geocentric longitude of the planet being the same as the longitude of the sun, the com- ' putation for determining the planet's heliocentric latitude and radius vector would have been more simple. For, the perpendicular let fall from P upon the plane of the ecliptic being supposed to meet the latter in N, and A"N being drawn perpendicularly to SA" or to EX, P N PN TJ- tan. PA"N = tan. 6, and -- = tan. PSN; A"N SN therefore A"N tan. 6 =. SN tan. PSN, or SN : A"N :: tan. 6 : tan. PSN. But in the right angled triangle SA"N SN : A"N :: rad. : sin. A"SN [=rsin. (LW)]; therefore rad. : sin. (L n) :: tan. 6 : tan. PSN. Thus PSN, the planet's heliocentric latitude, is found. Now the angle PEN, the planet's geocentric latitude, is known from the observation, therefore the angle SPE, the planet's parallax CHAP. xr. KEPLER'S LAWS. 1 95 in latitude ( = PSN PEN), is also known ; and in the triangle PSE, sm. SPE : sin. PES :: SE : SP. Thus s P, the radius vector, is obtained in terms of the earth's distance from the sun. By similar processes the values of the radius vector and heliocentric latitude might be found from observations made when the planet is in opposition to the sun. 254. If the values of SP be computed from the geocentric longitudes and latitudes obtained from observations made when the planet is at different distances from either node, those values will serve to determine the figure of the orbit ; and from such values it has been found that the orbits of the planets are nearly elliptical. In each orbit, also, the areas described by the radii vectores s P, in different times, are found to be nearly proportional to the times ; and for all the planets, the squares of the periodical times of their revolutions about the sun are found to be nearly proportional to the cubes of their mean distances from that luminary. These are called Kepler's three laws, having been discovered by that astro- nomer ; and Newton has proved that they are necessary con- sequences of the law of general attraction which prevails among the bodies of the universe. It must be understood, however, with respect to the third law, that it holds good only on the supposition that the revolving planet is a ma- terial point : in fact, if P be the periodical time of a planet's revolution about the sun, M the sum of the masses of the sun and planet, and a the semi-axis major of the orbit, we should have, from Physical Astronomy, (TT =. 3.1416) P 2 = a\ or P 2 would vary with . 255. The truth of Kepler's laws being admitted, those laws may be, and are employed for the purpose of obtaining the elements of a planet's orbit by the help of data obtained from observation. If, for example, the periodical time P of a planet's sidereal revolution about the sun have been ascer- tained, the semi- axis major of the orbit, supposed to be an ellipse, may be readily determined : thus, the earth being considered as a planet, and the period of its sidereal revolution being 365.256384 da., also the semi-axis of its orbit being sup- posed to be unity ; we have, conformably to Kepler's third law, 365.256384 days : P (in days) :: I 1 : a' 1 ; where a is the semi-axis required. o 2 196 PLANETARY ORBITS. CHAP. XL 256. On the other hand, having, from observations made at short intervals of time, two or any number of the geo- centric longitudes of a planet with the longitudes of the sun at the several times of observation, it is possible by successive approximations, without a knowledge of the time in which an entire revolution about the sun is performed, to find that time and also a radius vector of the orbit. Thus (fig. to art. 253.), let s represent the sun, E the earth, and P' the planet ; also let N' be the projection of P' on the plane of the ecliptic, and let E Q, or s Q' drawn parallel to E Q, be the line of the equinoxes : then the angle SEN'( = QEN' QES) will represent the difference between the geocentric longitude of p'and the longitude of the sun. Again, let ES (=: unity) be the earth's mean distance from the sun. Now, the planet being (for example) supposed to be beyond the earth with respect to the sun, and being conceived to revolve about the sun in a circular orbit, with uniform motion, let SP' or SN', supposed also to be equal to one another, be represented by any assumed number r greater than unity ; then, in the plane triangle SEN' we shall have r : 1 :: sin. SEN 7 : sin. EN'S The angle EN'S may be called the planet's annual parallax, and being added to SEN', the sum (in the figure) is equal to NSN' : this last angle being added to NSQ', or its equal QES, gives the value of Q'SN', which is the heliocentric longitude of the planet. Compute in like manner, with an equal radius /*, the heliocentric longitude QSN" corresponding to a second observation, and let t, t f , be the two times of observation ; then Q'SN" Q'SN' will be the difference between the helio- centric longitudes corresponding to the difference f t be- tween the times : and, the motion in the orbit being considered as uniform, we have Q'SN" Q'SN' : t't:: 360 degrees : P; where P represents the time in which the planet would make a revolution about the sun. Now, comparing the revolutions of the earth and planet about the sun, by Kepler's third law, we have, I 1 : r 1 :: 365.256384 da. : p. The like computations may be made with other observed longitudes and times : and as, probably, none of the periodical times determined by the first proportion will agree with that which is determined by the second, the assumed value of r must be changed till such agreement is obtained in the results. CHAP. XL THE ELEMENTS FOUND. 197 257. If there be obtained from observations, by means such as have been already stated, three radii vectores, as s P, with the three corresponding heliocentric angular distances of the planet from the node ; the angular distance of the perihelion point of the orbit from the same node, the semi-axis major and the excentricity may be determined in the following manner. Let r, r', r" be the three radii vectores ; /, Z', I" the three observed angular distances from the node ; p the required angular distance of the perihelion point from the node ; a the required transverse axis of the ellipse, and a e the required excentricity. a (\ __ e *\ Then, since, from the nature of the ellipse, r = = - - ^ ^, l+ecos. (lp) we have from the three observations, 1 + e cos. (lp) = a(l-e*)- .... (i.) l+*cos. (l'-p)=a(l-e^ .... (ii.) 1+ecos. (*" 70 = 0(1**)^. . (in.) Subtracting (i.) from (n.), and (i.) from (in.), we have, re- spectively, the second o 3 198 PLANETARY DEBITS. CHAP. XL member will still be a known quantity ; let it be represented by R : then 5n /-2- ~ sn. or, putting m for (Z' -f Z) and W for J (I" + T), we have sin. (m-p) w hence (PL Trigon., art. 32.) sin. (np) sin. m cos. jo cos. m sin. p =. R (sin. w cos. p cos. w sin. p), and dividing by cos. p, sin. ??2 cos. m tan. jo = R (sin. ?z cos. n tan. p) ; therefore (R cos. w cos. m) tan. jo = R sin. n sin. m : whence tan. p, and consequently p, the required distance of the perihelion point, is found. This value of p being substi- tuted in any two of the equations (i.), (n.), and (m.), there may from thence be obtained the values of a and of ae, the semi-axis major and the excentricity of the orbit. The greatest equation of the centre for the orbit of any planet may be obtained by a method similar to that which has been explained in describing the orbit of the earth. 258. The mean diurnal motion of a planet in its orbit may be found from that of the earth, the truth of Kepler's laws for elliptical orbits being assumed : thus, the squares of the periodical times of revolution being proportional to the cubes of the distances from the sun ; if the semi-axis major of a planet's orbit be expressed in terms of the like semi-axis of the earth's orbit (that is, if the former be represented by a, the unit of which is the mean distance of the earth from the sun), we have, P being the time, in days for example, of the earth's revolution about the sun, and p' the time of the planet's revolution, I 3 : 3 ::p 2 : p' 2 ; a , 360 360 . whence p = Pa 2 , and - or - r is the planet s mean daily angular movement about the sun ; in which - , or 5 9' 8". 19, may be represented by n. Therefore, if t denote any given number of days, the mean angular motion for that time in the orbits of different planets may be expressed by ^ <& 259. To find the angular velocity of a planet in any part of an elliptical orbit, let r represent the radius vector of a CHAP. XL ANGULAR VELOCITY. 199 /I A planet and ^ the angle at the sun between two places of the 7 A planet at an interval of time equal to one day ; then r -y- may represent the orbital arc described by the planet in one day, 7 A and, by mensuration, \ r 2 -=- may represent the sectoral area described by the radius vector in the same time, while by conic sections, a representing the semi-transverse axis, ae the excentricity, and 7r = 3.1416, the area of the ellipse is a 2 (1 e)*7r: but the areas varying as the times, 2 (x _)l v : i r i f*| :: 365.256384 a* : 1 (= 1 day). (A) Therefore -=- =r.0172 k^ expresses the angular move- ment required ; and it follows that the angular velocity in different parts of the same elliptical orbit varies inversely as the square of the radius vector. 260. Previously to finding the time in which the radius vector of a planet revolving in an ellip- tical orbit will describe about the sun, in one of the foci, the sectoral area corre- sponding to a given anomaly, the value of such sectoral area must be determined. For this purpose let vc, the semitrans- verse axis, be represented by a ; cr, the excentricity, by a e ; and the anomaly v F p, reckoned from the perihelion, by : then by conic sections, 1 _ g2 FP, the radius vector, =: a y - -. ; and if d (rad. = 1) represent the angle PFp described by the radius FP in a unit of time, FP.e?0 may be considered as the value of the arc Pp ; and we shall have, by Mensuration, \ a 2 (1 e' 2 ) 2 , (^ J. ~p ^ COS. for the area of the elementary sector PF p. In order to integrate this expression, assume / d _ A sin. / d /(I +e cos. 0) 2 ~ 1 +e cos. + V 1 +e cos. ' then, on differentiating, we get __A cos. (l+e cos. 0) (l+ecos.0) 2 ~ (1+ecos. 0) 2 B (l+e cos. 0) dQ (l+e cos. 0) 2 o 4 200 PLANETARY ORBITS. CHAP. XI. and equating like terms in the numerator, we get 1 1 A TD i o J - > 1 O Next, assume cos. = ^ , an equation equivalent to 1 T- x __(! cos. 0)^ , --^ A fT f7 V then, on differentiating, sin. d6 . FP = ^TTTZ or = QT^ we nave - oT3*T7 for the arc p *l C08. a COB.* dt considered as circular ; hence i - TT^" 7 ma Y represent the cos. J0 area of the parabolical sector P Fp described in a unit of time ; and (art. 264.), f- D 2 being the area FVP' when the perihelion distance is D, |D 2 : -- iTTr TT :: T : * C^ 0116 day). -" -y- = ' TT COS. -g" t7 tt 6 Therefore -y- = ' , or, combining this with the above equation for FP, we have -s-r = . _ _ - r; from which it is at 41.1FP 2 to be inferred that in any one parabola, the angular velocity varies inversely with the square of the radius vector. 269. It sometimes happens that a comet may be seen at night on the meridian of a station, and in such a case the geocentric right ascension and declination of the comet, like those of any other star, can be observed by the transit telescope and mural circle; but frequently a comet, when visible, is too near the sun to be on the meridian when that luminary is below the horizon, and therefore its apparent place must be ascertained by other instruments. With a 206 SATELLITES. CHAP. XI. well-adjusted equatorial, the right ascension and declination of a comet, when any where above the horizon at night, may be obtained immediately by observation ; or an altitude and azimuth instrument may be employed, and then the right ascension and declination must be deduced by calcu- lation. Frequently the comet may be observed in the vicinity of a known fixed star ; when, by means of a micrometer, the difference between its place and that of the star, in right ascension and declination, may be obtained: from these its absolute geocentric right ascension and declination, or its geocentric longitude and latitude, can be deduced. Observa- tions on a comet can however but seldom be made with accuracy on account of its ill- defined disk, and the faintness of its light, which is often such that the comet becomes in- visible when the field of the telescope is sufficiently enlightened to render the micrometer wires perceptible. Three observations of a comet, or rather the mean of several observations on each of three different nights, will in strictness suffice to afford the means of determining the elements of its orbit ; but more are necessary for the sake of greater accuracy, and for the purpose of verifying the con- clusions which have been obtained by the first computations. The geocentric places should be determined for intervals of one, two, or a greater number of days ; and with these data, granting that the orbit is an ellipse or a parabola, also that the sun is in the focus of the latter or in one of the foci of the former, the perihelion distance, the longitude of the perihelion point and all the other elements may be found. The problem is, however, one of great intricacy, and a complete solution of it is best obtained by processes derived from physical astronomy. 270. On directing a telescope to Jupiter it is observed that four small stars accompany that planet, each of them appearing at times on one side of his disk, and at other times on the opposite side ; and hence it is inferred that these stars revolve about the primary planet as the moon revolves about the earth, and in a plane which if produced would pass nearly through the latter. The planets Saturn and Uranus, or the Georgium Sidus, are also accompanied by small stars or satellites, the former by seven, and the latter by six, which exhibit phenomena similar to those presented by the satellites of Jupiter : very powerful telescopes are however required in order to enable the observer to see the satellites of the Georgian planet and most of those which accompany Saturn. 271. The secondary planets, as they are called, of Jupiter, are those only which have been made subservient to the purposes of practical astronomy; and for those purposes the CHAP. XL THEIR ECLIPSES. 207 theory of their motions has been diligently studied. On ex- amining them attentively during many nights it is observed that, in the interval between the time that any one of them ceases to be visible on the eastern side of the planet, and subsequently appears on the western side, a dark spot, which may be conceived to be the shadow of the satellite, is seen to pass across the disk of the planet : the appearance of a satellite on the western side is at first very near the disk ; the satellite from that time gradually recedes from the planet, and having attained a certain angular elongation it returns towards it. At times a satellite disappears at the western limb of the planet, and subsequently reappears at the eastern limb as if it had passed behind the planet; from hence it gradually recedes eastward to a certain distance and subsequently returns towards the disk. Corresponding phenomena are exhibited by the satellites of Saturn ; and hence it may be inferred that they also revolve about their primary planet. With respect to those which accompany Uranus, the move- ments of two only are known, and these appear to revolve about that planet in a plane nearly perpendicular to its orbit. 272. But it happens most frequently that a satellite loses its light at a certain distance from the disk of the planet on one side, and reappears at a certain distance from the disk on the opposite side ; and it may be conceived that such pheno- mena are caused by the entrance of the satellite into a cone of shadow which the primary planet, being an opaque body, casts behind it, or on the side opposite to that which is en- lightened by the sun. The shadow of a satellite on the body of the primary planet produces evidently an eclipse of the sun to the portion of the surface on which the shadow falls ; and during the interval that a satellite is passing through the shadow of the primary it must appear to suffer an eclipse to the inhabitants, if such there be, on the side of the planet which is furthest from the sun. When the sun is on the eastern side of one of the above planets, as Jupiter, the disappearance of a satellite is always observed to take place on the western side of the disk ; and when the sun is on the western side, a reappearance takes place on the eastern side : these circumstances sufficiently indicate that the disappearance is caused by the immersion of the satellite in the shadow cast by the sun behind the planet. When the planet is nearly in opposition to the sun, with respect to the earth, the shadow being then immediately behind the planet, the disappearance and the subsequent reappearance take place very near the disk : and, with respect to the two satellites of Jupiter which are at the greatest 208 SATELLITES. CHAP. XI. distance from his centre, when the axis of the cone of shadow forms a considerable angle with a line drawn from the Earth to Jupiter, the immersion into, and emersion from the shadow take place on the same side of Jupiter's disk. 273. The movement of the shadow of a satellite on the disk of Jupiter or Saturn is always from east to west, and the eclipses take place while the satellite is moving from west to east; it is therefore obvious that the motion of these satellites about their primaries is from west to east, or in the same order as the planets revolve about the sun. Again, as the satellites are not eclipsed every time that they pass from west to east, and as the shadow does not pass across the disk of a planet every time that a satellite passes from east to west, it is evident that the orbits of the satellites have certain inclinations to the orbits of their primaries, as the orbit of the moon has an inclination to that of the earth. 274. The periods in which the satellites revolve about their primaries are ascertained by means of the observed immer- sions into, and emersions from the shadow of the planet ; and the semidiameters of the orbits, in terms of the diameter of the planet, by means of their greatest angular elongations from its centre : the elongations being measured by means of a micrometer. From the vicinity of the first satellite of Jupiter to the body of the planet, it is never possible to see both the immersion and the succeeding emersion, but the other satellites are sufficiently distant from the planet to allow both phenomena to be observed in certain positions of the earth with respect to a line joining the sun and Jupiter ; if therefore the instants of the immersion, and the succeeding emersion of one of those satellites be observed, the middle of the interval of time may be considered as the instant at which the centre of the satellite is in opposition to the sun, or in the direction of a line drawn through the centres of the sun and Jupiter ; and if the middle of the interval between the next im- mersion and emersion of the satellite be found in like manner, the interval between these middle times will evi- s'' dently be the duration of a synodical revolution of the satellite about its primary, or the time in which, S being the sun and J the planet, the satellite setting out from A and revolving about J is carried to the point A' by the motion of the planet from J to J' ; that is the time in which A has described an an^'le about CHAP. XL POSITIONS OF THEIR ORBITS. 209 the moving point J equal to 360H-A'j'a, or 360 + jsj' (as'b being drawn parallel to AS). But JSJ' is the angular movement of Jupiter in its orbit during the same time, and this motion is known ; therefore the time of a sidereal revolution of the satellite, or that in which it would describe merely the periphery of its orbit, can be found by proportion. When the time of one revolution is thus obtained, the mean time of a revolution may be found by determining the instants of opposition from two eclipses separated in time by a long interval, on dividing this interval by the number of revo- lutions. 275. A method of finding the inclination of the orbit of a satellite to that of the primary planet and the places of the nodes may be understood from the following explanation. Let ADM, in the region of the satellite, be a section through the cone of shadow cast by the planet, C its centre, and let AB be half the chord described by the satellite during an eclipse: then AC, the semidiameter of the section, may be found from the known dimensions of the cone of shadow cast by the planet and the distance of the satellite from the latter, and AD may be had from the observed duration of the eclipse with the motion of the satellite in its orbit; therefore letting fall CB perpen- dicularly on AD, AB (=. -J AD) is known, and = sin. ACB; A C thus this angle is found. Next, imagine the triangle BCJ, right angled at C, to be perpendicular to the plane of the section ADM and to meet it in BC ; and let c J be the radius B C of the satellite's orbit : then will express the tangent of the c j inclination of that orbit to the orbit of the planet ; and since BC is known, being evidently equal to AC cos. ACB, that in- clination may be found. When a satellite passes centrally through the cone of shadow, it must be in one of the nodes of its orbit, and the duration of the eclipse is then the greatest : therefore, if from a register of many durations of eclipses, determined as above by the times of the observed immersions and emersions, there be taken those whose durations were the greatest, the times of their occurrence will be the times in which the satellite was in its node ; and the longitudes of the planet will be also the longitudes of the node at the same times. 276. By frequently observing the eclipses of each satellite, astronomers have found that the periods of their sidereal re- 210 SATELLITES. CHAP. XI. volutions are very nearly constant ; and hence, it is inferred that the orbits of the satellites are very nearly circular. Dr. Bradley, however, discovered in 1717, from certain small variations in the time of the revolution of the first satellite of Jupiter, that its orbit has a small excentricity, and "Wargentin, in 1743, discovered an ellipticity in the orbit of the third; that of the fourth satellite is also sensible. In order to de- termine the inequalities which may be suspected to exist in the movements of satellites about their primary planets, as- tronomers observe, with a position micrometer, the differences between the right ascensions and declinations of a satellite, and of the centre of a planet at certain intervals of time, and from thence they compute the arcs which the former describes in its orbit in equal times. From the inequalities of the arcs so described by the satellites of Jupiter, they have been enabled to ascertain the points in which the angular velocities about the planet are .the greatest and the least ; that is, in other words, to determine the points of perijove and apojove, or the extremities of the major axes of the orbits. All the satellites of Saturn, except the sixth and seventh, move in the plane of the planet's equator, and M. Bessel has recently discovered that the orbit of the sixth is elliptical. The satellites of Uranus revolve about that planet in planes nearly perpendicular to the orbit of the planet, which seems to imply that the latter revolves on an axis lying nearly in the plane of the orbit ; and the revolutions of the satellites are supposed to be performed in retrograde order. 277. From the periodical variations in the light of the satellites, Sir William Herschel ascertained that each of them presents always the same face to the planet about which it revolves ; and hence it follows that, like the moon, each of them performs a revolution on its axis in the time of a revo- lution in its orbit. 278. The eclipses of Jupiter's satellites are employed, in a manner which will be hereafter explained, for the determin- ation of terrestrial longitude ; it should be remarked, however, that considerable uncertainty exists respecting the true instant of immersion or emersion, on account of the time during which the whole disk of a satellite is passing into or out of the side of the shadow. This uncertainty, even with the first satellite, whose motion is more rapid than that of the others, may amount to half a minute, with the others it may amount to one, three, and four minutes. The uncertainty is the greatest when Jupiter is near the horizon, or near the sun, and when the immersion or emersion takes place very near the disk of the planet. CHAP. xi. SATUKN'S RING. 211 279. The planet Saturn is distinguished from all the other bodies of the solar system, in being surrounded by a detached ring which, receiving light from the sun on one of its faces, reflects that light to the earth. The plane of the ring coin- cides with that of the planet's equator, and this being oblique to the plane of the ecliptic, the ring, to an observer on the earth, appears of an elliptical form ; its breadth being variable according to its position with respect to such observer. The phenomena which it presents indicate that its true figure is circular, and that the line in which it intersects the plane of the planet's orbit revolves in that plane, as the line of the moon's nodes revolves in the plane of the ecliptic : when that line, which is called the line of the ring's nodes, if produced, would pass through the earth, the ring is invisible, because its edge only is turned towards the spectator, and from this time the apparent breadth of the ring gradually enlarges till the line of section is perpendicular to one drawn from its centre to the earth, after which, as the line of section continues to revolve, the breadth as gradually diminishes, and so on. The ring is also invisible, or appears merely as a luminous thread, when the line of section passes through the sun ; and it is evidently invisible when it presents towards the earth the surface which is not enlightened by the sun. It may be observed that when the enlightened surface of the ring is towards the earth, the part nearest the latter casts a distinct shadow on the body of the planet. 280. On account of the great distance of Saturn from the earth, when compared with the magnitude of the ring, rays of light passing to a spectator on the earth from every part of the circumference of the ring, may be considered as parallel to a line drawn to the spectator from its centre ; and if a plane perpendicular to the last-mentioned line be imagined to cut the cylinder formed by the rays, the section will re- present the visible figure of the ring. Now* if a time be chosen when the nodes of the ring are perpendicular to the axis of such cylinder, that is, when the breadth of the ring appears to be the greatest, to measure with a micrometer the lengths of the greatest and least diameters of the ring, the inclination of the ring to the plane passing through the earth and the ring's nodes may be found. For if E represent the earth, c the centre of the ring and A / planet, CA the semidiameter of the ring (equal to the greatest of the visible semidiameters so mea- sured), and AB drawn perpendi- cular to EC produced, represent its visible projection or the p 2 212 MOTION OF LIGHT. CHAP. XL least semidiameter ; we have ( sin. ACB) for the sine of the required inclination, which from such observations has been found to be between 25 and 30 degrees. Imagine now the semidiameter AC to be produced to meet the ecliptic EF, as in D ; then if to the angle ACB or E c D there be added the angle CED or the geocentric latitude of Saturn, the sum will be equal to CDF, which expresses the inclination of the ring to the plane of the ecliptic. M. Struve makes this angle equal to 28 6'. (Mem. Astr. Soc., vol. ii. part 2.) 281. The positions of the nodes of the ring are determined by observing the instants when the ring disappears in con- sequence of its plane passing through the sun ; for then the heliocentric longitude of the nodes is the same as that of the planet. Those disappearances may be recognised by being such as are observed to take place regularly at intervals equal to half a sidereal revolution of Saturn about the sun ; for the disappearances which are caused by the plane of the ring passing through the earth, occur at intervals which depend on the position of the earth : and, as the situations of the nodes of the ring always correspond to the same points in the orbit of the planet, it follows that the plane of the ring remains constantly parallel to itself. The line joining the nodes of the ring, if transferred parallel to itself on the ecliptic, makes a constant angle equal to about 5930 / with the line joining the nodes of Saturn's orbit. 282. At the latter end of the seventeenth century, M. Homer, on comparing the recurrences of the eclipses of Jupiter's first satellite with the time of its sidereal revolution, ascertained that, between a conjunction of Jupiter with the sun and the next following opposition, the immersions and emersions were continually accelerated, with respect to the times at conjunction ; and that between an opposition and the next following conjunction, they were continually re- tarded, with respect to the times at opposition. Now the difference between the two distances of Jupiter from the earth at the times of conjunction and opposition is equal to a diameter of the earth's orbit ; and, in order to account for the differences between the observed and computed times of the phenomena, it was assumed as an hypothesis that the passage of light from the celestial body to the earth, instead of being instantaneous, as had been till then supposed, takes place in times which depend upon the distances. From the best observations, it is found that the whole acceleration of the eclipses at the time of the opposition, and the retard- ation at the time of the conjunction of Jupiter, is 16 / 27 // '; CHAP. XL MOTION OF LIGHT. 213 in which time, therefore, the particles of light are supposed to move through a diameter of the earth's orbit, and it follows that light should pass from the sun to the earth in half that time, or 8'13".5. What was at first observed with respect to the .first satellite of Jupiter, is now known to take place with all the satellites, and the above hypo- thesis is fully confirmed by the phenomena of aberration; it is, therefore, universally admitted as just. Also, since the latter phenomena can be explained only on the supposition that the motion of light is progressive, and that the earth revolves about the sun, the fact of such revolution is to be considered as established. P 3 214 SOLAR SPOTS. CHAP. XII. CHAP. XII. ROTATION OF THE SUN ON ITS AXIS, AND THE LIBRA- TIONS OF THE MOON. 283. ON the surface of the sun are frequently seen a number of dark spots, which, entering at the eastern limb, appear to move across his disk, and occasionally, having dis- appeared at the western limb, they have been observed, after an interval of time equal to that of their passage over the disk, to re-appear on the eastern limb. It has been consequently concluded that they exist on the surface of the sun ; and, as the time of performing a revolution is the same for all the spots (about 27 days 8 hours), it is further inferred that their apparent motion is caused by the rotation of the sun on an axis, in the same order as the earth and planets revolve about him. The paths which the spots appear to describe are generally of an oval figure, their convexities being towards the lower or upper part of the disk according as the northern extremity of the axis is directed towards, or from the earth ; but twice in each year (in December and June) when the plane of the rotation, if produced, would pass through the earth, they appear to be straight lines. In order to determine the apparent path of a spot it is necessary, with a transit or equatorial instrument, to find the difference between the right ascension of the spot and of the eastern or western limb of the sun, and with a micrometer, the difference between the declinations of the spot and of the upper or lower limb : the right ascension and declination of the sun's centre is known for the same instant ; therefore a series of such observations being made during the time that a particular spot is on the disk, if AB be supposed to repre- sent part of a parallel of de- clination passing through C the sun's centre ; on making c M, CN, &c. equal to the differ- ences of right ascension be- tween the sun's centre and a spot in the situations s and s', and M S, N s', &c., perpendicular to AB, equal to the corresponding differences of declination, a line CHAP. XII. THE SUN'S EQUATOR. 215 joining the points s, s', &c., will be an orthographical pro- jection of the path described by the spot. The true path of the spot is the circumference of a circle whose plane is per- pendicular to the axis of the sun's rotation, or parallel to what may be called the sun's equator ; and therefore, from three observed right ascensions and declinations of a spot it is pos- sible to determine by calculation the position of such circle with respect to the plane of the ecliptic, and the position of the line in which the sun's equator intersects that plane ; but for these purposes it is necessary first to determine the geo- centric longitude and latitude of the spot at the several times of observation. 284. Let DD' represent a part of the ecliptic, and let fall S'R perpendicularly on it. Then, if the difference between the transit of A and s' be observed in sidereal time, and there be subtracted from it the time given in the Nautical Almanac, in which the sun's semidiameter AC passes the meridian, the remainder multiplied by 15 will express an arc of the equator corresponding to C N ; and multiplying this by the cosine of the sun's declination, the product will be the value of C N in seconds of a great circle : s' N is the observed difference between the declinations of C and s', which is obtained by the micrometer in the same denomination ; then in the right-angled triangle CNS', considered as plane, there may be found cs' and the angle S'CN. Now the angle D'CB is equal to the angle of the sun's position between the pole of the ecliptic and that of the equator, or the value of the angle p c P, where p is the pole of the equator and of the parallel AB, and p is the pole of the ecliptic. In the spherical triangle />PC we have/?P, the obliquity of the ecliptic, p C equal to a quadrant, and the angle p P c the supplement (in the figure) of the angle c P Q, which last, since p p Q is the solsticial colure, is equal to the complement of the sun's right ascension : therefore sin. pfC : sin. p c ( = radius) :: sin. pp : sin. p c P, or sin. I/CB. Then, in the right-angled triangle s' c E, considered as plane, we have s'c, the angle S'CR( S'CN D'CB); to find CR and S'R: the former is the difference between the geocentric longitudes of the sun and spot, which may be represented by L I (L being the longitude of the sun's centre, and I the geo- centric longitude of the spot), and the latter is the geocentric latitude (X) of the spot. 285. Next, let E be the centre of the earth, s that of the sun, and through E imagine the rectangular co-ordinate axes EX, E Y, EZ to be drawn, of which EX and E Y are in the plane of the ecliptic, and the former passes through the equinoctial p 4 216 THE SUN'S DOTATION. CHAP. XII. point T . Imagine s' to be the place of a spot on the sun's disk, and s its orthographical projec- tion on the plane of the ecliptic : again, let the co-ordinates EX, EY of the point s be repre- sented by x and Y, and E s by R ; then since the angle T E S, the sun's longitude, is represented by L, we have X = R COS L, Y r= R sin. L. Let now E s' be represented by r ; also let the co-ordinates EX 7 , E Y 7 , s / 's of the spot s' be represented by x,y,z; then T ES, the geocentric longitude of s', being represented by-Z, and S'ES, the geocentric latitude, by X, we have, as in art. 1 84., x r cos. \ cos. /, y zr r cos. \ sin. I, and z r sin. X. Lastly, let the three rectangular co-ordinates of s', with respect to s, be x' (= sx") y' (SY") and z = (s' s), and let the semidiameter s s' of the sun be represented by r' : then, x' =. x x, and y' ~ y Y ; and in these equations substituting the above values, we have for the equivalents of x', y', and z s and subsequently of r', values identical with those in art. 184. From these, by pro- cesses similar to those in art. 185., we may obtain the tangent of the angle which the perpendicular let fall from s on the plane of the circle, described by the spot, makes with the axis S T!' ; that angle is equal to the inclination of the plane of the circle, and consequently of the sun's equator to the plane of the ecliptic. The required inclination is therefore found. The position of the line in which the sun's equator intersects the ecliptic may be found by the method explained in art. 186. ; in which the value of ^ expresses the co-tangent of the angle which the line of section makes with sx", and being negative, the angle (as T'SQin the figure) is greater than one right angle and less than two : this angle represents the longitude of the node of the sun's equator. 286. It must be observed that the time in which the sun revolves on its axis is not precisely that which elapses between the first appearance of a spot on the eastern limb, and its re- appearance at the same limb. For let E be the earth and s s' part of the sun's apparent annual path, so that ABC, A'B'C' may be considered as projections of the upper hemisphere of the sun on the plane of the ecliptic ; and imagine the axis CHAP. XII. THE MOON'S ITERATIONS. 217 of the sun's rotation to be perpendicular to the plane of the ecliptic, which is sufficiently correct for the present purpose : also let A be the place where a spot becomes visible to a spectator on the earth, in a direction of a line drawn from E and touching the disk of the sun. Now while the spot A appears to revolve in the direction ABC, the sun appears to move in its orbit ; therefore let the centre of the sun pass from s to s' while A describes about s the exact circumference of a circle: the spot will then be at a in such a situation that the angle aS'B' is equal to A SB. But the sun must continue to revolve on its axis till the spot arrives at A', making the angle A'S'E equal to ASE before it becomes visible ; that is, it must describe the arc a A? above a complete revolution about s. The arc a A.', or the angle as' A? is equal to B'S'E or to S'ES, which in 27 days 8 hours is equal to 27 7' ; consequently, the exact time of a revolution may now be found by proportion ; thus 387 7' : 360 :: 27 days 8 hours : 25 days 10 hours, and the last term is the time required. 287. It may be seen by the naked eye that the surface of the moon is remarkably diversified with light and shade ; and the telescope shows that it resembles the appearance which the earth would present to a spectator at a great distance from it if the vegetable mould which covers so great a part of the land were removed, and if the beds of the seas were dry. In many places great mountains rise from the general surface, and terminate in points, or form clusters or ridges ; but much of the surface is occupied by deep cavities, which are surrounded by circular margins of elevated ground : there is no appearance of water in the moon, and the ex- istence of an atmosphere about it is doubtful. 288. The spots on the moon's surface always retain the same positions relatively to each other, yet, apparently, they change their places with respect to the circumference of her disk in consequence of certain vibrations of the luminary on its centre, between the east and west, and between the north and south, which are called the librations in longitude and latitude, respectively. The first depends on the in- equalities of the moon's motion in her orbit, combined with a uniform motion on her axis; in consequence of which motions the parts of her surface about the eastern and western 218 THE MOON'S SPOTS. CHAP. XII. limbs are turned alternately towards, and from the earth. The second depends on the axis of her rotation not being perpendicular to the plane of her orbit, and on its keeping parallel to itself during a revolution about the earth ; thus a part of her northern or southern limb, which at one time may be unseen, is at another rendered visible. A third libration, called diurnal, results from a part of the moon's surface about the upper limb becoming visible when the moon rises and sets, the spectator being then above her, and ceasing to be seen when the moon has considerable elevation above the horizon, from the spectator being below her : at this time a part of the surface about the lower limb becomes visible, which in the other positions could not be seen. 289. With a micrometer applied to a telescope mounted equatorially, the difference between the right ascensions of a spot and of the eastern or western limb of the moon, also the difference between the declinations of a spot and of the upper or lower limb, may be found as the corresponding elements of the sun's spots were obtained ; and from these observations the geocentric longitude and latitude of a spot, together with the values of x', y', z f , may be determined. In employing the formula? given in art. 184. there must be put for L the longitude of the moon, and for E its distance from the earth when projected on the plane of the ecliptic by means of the moon's latitude : the distance of a spot from the earth must be represented by r, its geocentric longitude by /, and its geocentric latitude by A. This last is equal to the sum or difference of the latitude of the moon's centre, and the distance of the spot in latitude from the moon's orbit. If the co-ordinates x' , y, z f , are determined on the supposition that s (fig. to art. 285.) is the centre of the moon projected on the plane of the ecliptic, the places of the moon's spots, as they would appear if observed from her centre, may be deter- mined. For if s were the projection of a spot on the plane of the ecliptic, we should have equal to the tangent of its J' selenocentric longitude, and . _ for the tangent of the "> " corresponding latitude. Here z' must be considered as equal to r sin. A K' sin. A', 11' being the distance of the moon from the earth and A' the moon's latitude. By means similar to those which are indicated in art. 185. the position of the moon's equator may be determined : and it is found that the latter forms nearly a constant angle ( = 1 30') with the plane of the ecliptic. The intersection of her equator, when pro- CHAP. XII. HEIGHTS OF HER MOUNTAINS. 219 duced, with the plane of the ecliptic, is found to be constantly parallel to the line of the nodes of her orbit. 290. The determination of the heights of the lunar moun- tains is an object rather of curiosity than of use in practical astronomy ; but as it may be interesting to know in what manner those heights may be found, the following method, which was proposed by Sir William Herschel, is here intro- duced. Let c be the centre and AmN part of a section of the moon made by a plane passing through the earth and sun ; let M be the summit of a mountain, and S m M the direction of a ray of light from the sun, touching the level part of the moon's surface at m, and enlightening the top of M while the space between M and m is in dark- ness ; also let M E, m E' be the directions of rays of light pro- ceeding to the earth, whose distance from the moon is great enough to allow those rays to be considered as parallel to one another. Then mp drawn perpendicularly to EM denotes the breadth of the unenlightened space M m as seen from the earth, and this must be measured in seconds of a degree by means of the micrometer. Now the rays of light from the sun to the earth and moon being, on account of the great distance of the sun, considered as parallel to one another ; if s M be produced to meet CP, which is drawn parallel to ME and is supposed to join the centres of the earth and moon, in the point P, and P T be a line drawn to the equinoctial point in the heavens, the angle CPS or its equal p Mm will be equal to the difference between the longitudes of the sun and moon ( T P s and T P c) at the time of the observation : this angle may be found from the Nautical Almanac ; and in the plane triangle Mp m right angled at p, sin. m ~M.p : rad. : : mp : M m. Thus Mm may be found in seconds of a degree; but the angle subtended at the earth by the semidiameter of the moon, or the value of c m, is known from the Nautical Almanac ; therefore in the triangle CMm right angled at m, CM may be found, and subtracting cm or en from it, we have M/z, the height of the mountain. This is expressed in seconds ; but since the semidiameter of the moon is known in miles, the value of MTZ may, by proportion, be found in miles. From the computations of Sir William Herschel it appears that the heights of the mountains above the level surface of the moon do not exceed 1 miles. 220 PLACES OF FIXED STARS. CHAP. XIII. CHAP. XIII. THE FIXED STARS. REDUCTION OF THE MEAN TO THE APPARENT PLACES. THEIR PROPER MOTIONS. ANNUAL PARALLAXES. 291. AN exact, knowledge of the positions of the stars called fixed is of the highest importance since on those positions depend the determination of the places of the sun, moon, planets, and comets ; and consequently the verification of the results of theory respecting the movements of the bodies composing the solar system. It has been shown that the apparent places of stars are affected by the causes which pro- duce the precession of the equinoxes, the solar and lunar nutation, and what is called the aberration of light; and it may be proper to state here the formulae by which, for any given time, the apparent places with respect to right ascension and declination may be reduced to the mean places, and the converse. In the present Nautical Almanacs (page 435.) these formulae are indicated by the two following series, the values of the several terms being taken from Mr. Baily's paper, " On some new Tables," &c., in the Memoirs of the Astro- nomical Society, vol. ii. part 1. : The apparent right ascension (in ) r i c *? v /= seconds of a degree) J The apparent declination = S + A a' + B b f + c c' + D d r . In these formula a and S are the mean right ascension and mean declination at the commencement of the year, as in the tables, pp. 432 to 434. of the Almanac : the terms indicated byA# + B# constitute the formulae for aberration in right ascension (art. 234.), and those indicated by Aa' + j$b f consti- tute the formulas for aberration in declination (art. 235.). The terms cc + ~Dd comprehend all those which enter into the effects of luni-solar nutation in right ascension, and Cc' -f Dd' all those which enter into the effects of luni-solar nutation in declination. The quantities represented by A, B, c, D, depend on the true longitudes of the sun and moon and the longitude of the moon's node ; and as these vary continually their logarithms are given for every day in the year, in the Nautical Almanacs, page xxii. of each month. A further CHAP. XIII. ORDERS OF MAGNITUDE. 221 correction is required on account of the diurnal aberration of a star in right ascension, (art. 237.) 292. Besides the deviations of the apparent from the mean places of stars, as above indicated, many of the stars have apparent motions which cannot be referred to the causes by which these deviations are produced; and hence they are supposed to be peculiar to the stars themselves, either inde- pendently of the other bodies of the universe or results of some general movement, as yet unknown, of the whole system of stars. In the Memoirs of the Astronomical Society, vol. v., - Mr. Baily has given a list of 3 14 stars which are supposed to have proper motions ; and in a few cases the directions and annual amounts of these motions have been ascertained. They are represented in the formulae of reduc- tion above mentioned by A c for right ascension and by A c r for declination ; and the values of these terms are to be multiplied by t which expresses the fraction of a year elapsed between Jan. 1. and the given instant, or t = ^^-^ : ' The star a Centauri is one of those whose proper motion is the greatest; and the amount of the motion annually is said to be equal to 3"*6 of a degree. 293. The stars called fixed have at all times been classed according to certain orders of apparent magnitude or bright- ness, from the most brilliant, which are said to be of the first order ; the stars of the seventh order being those which are barely visible to the unassisted eye. Stars of the same apparent class are not however equally brilliant ; and Sirius is supposed to emit about three times as much light as an average star of the first order of magnitude. Many of the stars also suffer periodical changes in their degrees of bril- liancy, some appearing constantly to diminish, and others to increase in brightness : a few which are mentioned in the ancient catalogues are not now to be found in the heavens, and some are there at present which do not appear to have been formerly remarked. It may be added that some of the nebulous spots are known to have changed their figure since they were first noticed. In order to form a judgment of the relative magnitudes of stars from the first to the sixth or seventh order, and even to estimate any changes which may take place in the apparent magnitudes of stars within those orders, some astronomers trust to the impression produced by their different brilliancies on the unassisted eye : a greater degree of precision in the estimate may, however, be obtained by comparing a star which may be under notice with one of a known order, 222 FIXED STAES. CHAP. XIII. in the telescope of a sextant, after having by a movement oi the index brought the two stars together in its field. The relative quantities of light received from stars of different magnitudes can be ascertained with considerable precision by diminishing the aperture of a telescope (the object glass if the telescope is achromatic, or the speculum if a reflector) when viewing the brighter of two stars, till the eye is affected in the same manner as it is when the less bright is viewed with the entire aperture ; for the quantities of light emitted from the two stars will evidently be inversely as the areas or as the squares of the diameters of the apertures with which the intensities of light appear to be equal. By such experi- ments Sir John Herschel concludes that the light of an average star of the first magnitude is to that of a star of the sixth as 100 to 1. (Mem. Astr. Soc. vol. iii. p. 182.) In estimating the relative magnitudes of stars below the sixth the same astronomer proceeds on the principle that the designation of magnitude should increase in an arithmetical progression 7, 8, 9, &c., while the light decreases in the geometrical progression ~, J, J, &c. ; so that if two stars apparently of the same magnitude, when brought together so as to appear but as one, affect the eye in the same manner as one of the sixth magnitude, each is to be considered as of the seventh magnitude, and so on. The sensibility of the eye to differences in the degrees of illumination is very great ; and, with telescopes of considerable power, the organ is capable of appreciating magnitudes of the 18th or 20th order. It is customary to express degrees of magnitude between the primary orders by whole numbers with decimals ; thus mag. 3.5 denotes an order equally distant between the third and fourth. 294. Mayer seems to have been the first who observed that several of the large stars are accompanied by smaller ones which revolve about them like satellites: such have since been discovered in almost every part of the heavens ; and Sir John Herschel has given, in the Memoirs of the Astronomical Society, a table of the positions of 735 double stars in the northern hemisphere, to which must now be added those which the same astronomer observed in the southern hemisphere during his residence at the Cape of Good Hope. One of the largest double stars in our hemisphere is a Geminorum (Castor) of which the small one is said to have described about the other, between the years 1760 and 1830, an angle of about 67 degrees, with a variable velocity. The star f Ursre Majoris is double, and the smaller one re- CHAP. XIII. OKBITS OF REVOLVING STARS. 223 volves about the other in a period of about 59 years, which was completed between 1781 and 1839. One of the two stars constituting 77 Coronae is supposed to have completed its period in 45 j years ; and from certain variations in its angular motion (if the observations can be depended on) it may be presumed that its orbit is elliptical. The stars s l and e 2 Lyrae, from a similarity in appearance, and from an equality of angular motion, seem to constitute a binary system, the two stars having a combined rotation, in the same direction, about their common centre of gravity. 295. The angular distances between two stars, one of which revolves about the other, and also their angle of position with respect to the meridian or to a horary circle passing through one of them, when measured with a micro- meter, is liable to much uncertainty from the imperfection of the instrument and the difficulty of adjusting it with suffi- cient precision for the purpose ; so that the path which the revolving star describes about that which is fixed, if deter- mined directly from the measures, would appear to have no regular figure. An ingenious method, however, of obtaining by a graphical construction, from the observations, an ap- proximation to the true values of those distances and angles, and subsequently of ascertaining nearly the figure of the path or orbit, has been given by Sir John Herschel in the fifth volume of the Memoirs of the Astronomical Society. This consists in laying down, by any scale of equal parts, on a line taken at pleasure a series of distances representing the intervals (in years and decimals) between the observations ; and from these points setting up, at right angles to the line, ordinates whose lengths may represent the observed angles of position: then a line being drawn by hand among the extremities of the ordinates so as to present the appearance of a regular curve of gentle convexity or concavity, its ordi- nates may be conceived to represent with tolerable correct- ness the values of the angles of position for the times indicated by the corresponding abscissae on the line first laid down. Next, drawing tangents of any convenient length to this curve, at different points in it, and considering them as the hypotenuses of right-angled triangles whose bases are paral- lel to the line of abscissas, and represent time, while the perpendiculars represent angular motions in the orbit; the quotients of the latter by the former denote the angular velocities of the star in the orbit, at the times indicated on the scale of abscissas, corresponding to the points of contact. Now it being presumed that the path really described by one star about another is a circle or an ellipse, and the ap- 224 FIXED STARS. CHAP. XIII. parent path in the heavens being an orthographical projection of the orbit on a plane perpendicular to the line drawn from the spectator to the star, the latter path will (by conic sec- tions, and art. 27.) also be an ellipse: hence, by the elliptical theory for planets '(art. 259.), the squares of the radii vectores of the revolving star (the other being at the focus) will be inversely proportional to the angular velocities ; therefore, by drawing lines from the star, supposed to be at the focus, making angles with each other equal to the velocities just found, and making the lengths of the lines inversely pro- portional to the square roots of those velocities, the curve joining the extremities, or passing among them, may, as in the method of finding the orbit of a planet, be expected to approach nearly to the form of an ellipse. Let the curve so drawn be an ellipse, as AZBX ; let c be its centre and s be the place of the star about which the other star revolves : then in order to form a scale of seconds of a degree for the diagram, take from the scale by which the projected figure was drawn a mean of all the radii, or lines drawn from s to the periphery, for which the corresponding distances in seconds have been obtained by measurement with the micrometer. From a mean of all these measured distances compared with the mean radius there may be found, by proportion, the number of seconds corresponding to each unit of the scale employed in the projection. 296. Since by the nature of the orthographical projection the centre of the projected ellipse corresponds to the centre of the real ellipse, if a line, as AB, be drawn through s and c, that line will be the projection of the major axis, and so will be that of the excentricity ; also by the nature of the pro- jection, these will bear the same ratio to one another as the Q y~t real major axis and excentricity : therefore we have for the value of e (the excentricity when the semiaxis major is unity). Let a' be the real semi-axis major, b f the real semi- axis minor : then -, = e, and the ratio of a' to b f is known. Next, by trials find the position of a line K L drawn through s so as to make the parts SK,SL equal to one another ; then KL will be the projected latus rectum (for in the real orbit, CHAP. XIII. ORBITS OF STARS. 225 a double ordinate equal to the latus rectum would pass through s and be bisected in that point), and consequently zx drawn through c parallel to KL is the projected minor axis, hence measuring AB and zx by the scale, the ratio ( T) ^ s Z X \ ' known (a and b being the projected semiaxes). Let the line ns passing through the star s be the direction of the meridian, or the hour circle, from which the angles of position in the heavens were measured ; and let it cut zx in R; then with a protractor or otherwise measure the angles n s B, n s K (=ttRz), or the inclinations of the projected axes to that line: let these inclinations be represented by a and ft re- spectively. Again, let A'Z'B'X' be the real orbit, of which the other is presumed to be an orthographical projection formed by lines, as A' A, B'B, C'C, &c. drawn from every part of the real orbit perpendicularly on the plane of the apparent orbit, and let MN be the line of section; then AB, zx being the pro- jected axes, A'B' and z'x' will be the corresponding axes of the real orbit. Let the planes z'zxx', ZMNX be produced till they meet the plane A'X'B' in some point as T, and let the angle n s M be represented by n : then the angle s T c ( = M s K) (3 n, and the exterior angle M s c = a n, or CST=: TT (a n). Now imagine CP to be let fall perpen- dicularly on ST, and C',P to be joined: then (Geom. 2 and 4. Def. Planes) the angle CPC' ( = 7) is the inclination of the projected to the real orbit. Next, imagine c' to be the centre and c c' the semidiameter of a sphere in contact with the plane SCT at the point C; then the plane angles sc'c and TC'C may be represented by two sides of a spherical triangle intersecting one another at c, and SCT as the spherical angle continued between them. Let a c b, in the annexed figure, be such a triangle, and from C let fall the perpendicular C d ; then C d will re- present the angle cc'pin a plane passing through C C' perpendicularly to the base ST of the plane figure, that is, it will represent the complement of the angle CPC' which is the inclination of the plane SC'T to SCT. Now, in the spherical triangle a C d, right angled at d, we have (art. 60. (d)) cos. ac cos. ad cos. c d, and substituting the equivalents, cos. CC'S:=COS.SC'PCOS. CC'P, or sin.csc'=sin. c'SPsin.CPC': Q \ 226 FIXED STARS. CHAP. XIII. sin. csc/ /,x whence sin. C PC' = gS -^ ]p .... (1, In like manner, in the triangle bed, we have sin. CTC' /nN sin. CPC' = - - ; .... (2). sm. C'TP Again, in the triangle acd, we have (art. 62. (V)) sin. dc = cotan. acd tan. d, and substituting the equivalents, sin. PC' c = cotan. SCP tan. SC'P, or cos. CPC' = tan. CSP cotan. C'SP .... (3). In like manner, in the triangle bed, we have cos. CPC' tan. CTP cotan. C'TP . . . . (4). Multiplying together the equations (3) and (4) we get (putting for cotan.) tan. CSP tan. CTP , K > cos. 2 CPC':= - - .... (5). tan. C'SP tan. C'TP The angle SC'T (fig. to art. 295.) being a right angle, the angles c' s P and C'T p are together equal to a right angle, and tan. C'TP = cotan. C'SP; therefore the denominator of the frac- tion is equal to unity ( = rad. 2 ), and cos. 2 CPC' = tan. CSP tan. CTP .... (6) ; that is, cos. 2 7= tan. (OLU) tan. (j3n). Next, since cc' is parallel to BB' and to zz' we have (Euc. 2. 6.) SC : sc' :: CB : C'B', or sc : sc' :: a : a', and CT : C'T :: cz : c'z', or CT : C'T :: b : b' '; ,-,... SC SC' a a' theretore by division, - - : 7- : : -7 : T7 . J CT C / T I ft sc' Now SC'T being a right angle, - 7 --=. tan. STC': C T but in the plane triangles TPC', TPC, right angled at P, PC' zi PT tan. STC' and PC PT tan. STC; whence PC . PC' :: tan. STC : tan. STC'. Again, in the plane triangle PCC', right angled at C, PC = PC' cos. C'PC, or PC = PC' cos. 7; therefore, C'T/ cos. 7 { tan. (a ?z) tan. (/3 71 CHAP. XIII. PARALLAXES OF STARS. 227 Also in the triangle s C T, s C : c T : : sin. s T c : sin. c s T : that is :: sin. (ft n) : sin. (a n): therefore sc = sin. (/3-rc) C T sin. (a n) Whence after squaring all the terms, a 2 . a/*., sin. 2 (13 -n) . tan. 2 (/3-n) b' 2 ' b' 2 " sin.' 2 (a n) ' tan. (* n) tan. (ft 7i) 9 sm ' 2 QQ ft) . sin. (ftn) " sin.' 2 (oL n) ' tan. (a n) cos. (ftn)' or, again, successively, :: tan. (a TZ) cos. (ft n) sin. (ftn) : sin. 2 (a w); :: cos. (/& fl) sin. (ftn) : cos. (a TZ) sin. (a rc); sin. 2 (/3 ft) : sin. 2 (a n)\ : : sin. 2 /? cos. 2 rc cos. 2 /3 sin. 2 ^ : cos. 2 a sin. 2 ?z sin. 2 a cos. 2 n ; : : sin. 2 ft cos. 2 /3 tan. 2 TZ ; cos. 2 a tan. 2 w sin. 2 : Now, multiplying extremes and means, 2 /2 T^(COS. 2 a tan. 2n sin. 2 a) =^ (sin. 2/8 cos. 2 /3 tan. 2ra); whence tan. 2 racos. 2 a + cos. 2 fi\ ^(sin. 2 and finally /2 2 ^72 sin. 2 + psin. 2 a tan. 2 w =-g ^ -- cos. 2 a + cos. 2 Thus ft, or the angle n S M may be found ; and its value being substituted in that of cos. 2 7 above, the inclination of the apparent to the real orbit will be obtained. 297. The distances of the fixed stars from any part of the solar system are so great, that it is even yet doubtful whether, at the nearest of them, the diameter of the earth's orbit subtends a sensible angle ; but, as efforts are now being fre- quently made with a view of ascertaining the existence and amount of such angle, which is designated the annual parallax of a star, it will be proper to notice here the nature of the observations which are required for the purpose, and the manner of determining the parallax from them. The method most generally put in practice hitherto consists in comparing Q 2 228 FIXED STAKS. CHAP. XIII. the observed geocentric right ascensions and declinations of the brightest fixed stars (which are presumed on that account to be the nearest, and which should consequently have the greatest parallax), with their computed heliocentric right ascensions and declinations : and it is evident that, after all the known causes of error in the observations have been removed, if any differences could be found between the elements so compared, the circumstance would indicate the existence, and serve to express the value of the parallax. 298. The right ascension and declination of the star should be observed at the time that the star is in opposition to the sun in longitude ; and, from these, with the known obliquity of the ecliptic, the longitude of the star must be computed by trigonometry ; the right ascension and longitude so ob- tained are geocentric, but at the instant of opposition, the latter is evidently equal to the heliocentric longitude (l r ) of the star. The right ascension and declination of the star should also be observed at a time when the star is 90 from the state of conjunction or opposition ; and from these the star's lati- tude should be computed ; the declination and latitude so ob- tained are geocentric, but at the instant named the latter is equal to the heliocentric latitude (V). The geocentric right ascension (a) and declination (d) of the star are then to be observed as often as possible during the course of a year ; and having corrected them for the effects of precession, nutation, and aberration, the corresponding values of the geocentric longitude (Z) and latitude (X) must be found by computation. Now, it is evident, that if the parallax of the star in longitude or latitude, in right ascension or declination, be insensible, we should.have /' /, X' = X, &c. ; and if the differences between the~ values of I' and Z, X' and X, &c. deduced from the observations should be found to vary, it would follow that the parallax may be appreciated. Let s represent the sun, s a fixed star, E the earth, and E p the orbit which it describes about the sun. Again, imagine sx, SY, to be rectangular co-ordi- nates in the plane of the ecliptic, the former passing through the equinoctial point; let fall EX' EY' perpendicularly on sx and SY, and produce ES to s': also let fall SM perpendicularly on the plane of the ecliptic, and draw MX, MY perpendicular to T SX, SY. CHAP. XIII. PARALLAXES OF STARS. 229 Then, as in art. 184., sx' and SY' being represented by x and Y, SE by r, and the sun's longitude (= TSE -f 180) by L, we have, x = r cos. L, Y = r sin. L. Also, representing sx, SY, SM. by x,y,z, and ss by r', we have as in the article just quoted, putting accents on X and /, x r' cos. \' cos. I' y=r' cos. X' sin. I' z r' sin. X' ; and, joining the points s and M, SM = r'cos. X'. Again, drawing E V parallel to sx, and joining the points E and M, it is evident that T ' E M will express the geocentric longitude of s, and SEM its geocentric latitude: also XSM is the heliocentric longitude, andsSM the heliocentric latitude. Now, by trigonometry, tan. T 'EM ( ==) = rcos.sn. rsm. \ x x/ r cos. X cos. / + r cos. L / SM\ r' sin. X' and sin. sEM(= }== c , V SE/ /r '2 _ r r cos. multiplying each term in the numerator of the value of tan. orm / T ' E M by . ' p which is equivalent to dividing it by tan. sin. I we have, , _ . 7/ . cos. I' r cos. X cos. I +r sin. L =- t tan. T ; EM = tan. I' _ Bm * * . r' cos. X cos. I + r cos. L or dividing by r' cos. X r cos. l' s and putting p sin. 1 " (jp being expressed in seconds of a degree) for - which, when s E is per- pendicular to SE, denotes an arc equivalent to the sine of the very small angle SSE, or of the annual parallax in its maximum state, we get _ sin. L 1 + sin. 1" ^~rr-^ cos.X sin./ tan. T'EMzztan. l'- . COS. L 1 +p sin. 1 cos. \ f cos. I' Then, dividing the numerator of the second member by the denominator as far as two terms only (that is, neglecting powers of p above the first) after bringing the fractions in both to a common denominator, sin. L cos. I' cos. L sin. tan. - %/ . T/ -- ,7 cos. X sin. / cos. /' Q 3 230 FIXED STAKS. CHAP. XIII. or, tan.r' BM -tan.r=tan.r,psm. I", but (PI. Trigo. art. 38.), tan. T 'EM-tan. l'= and since p is extremely small, it is evident from the preceding equation, that tan. T'EM tan. V, and consequently sin. (T'EM I') must be extremely small; therefore, putting (T 'EM 70 Eta- 1" for sin. (T'EM /0; also p^dog^^ cos. / for tan. I ', and considering cos. T 'E M as equal to cos. I', the last equation becomes (in seconds of a degree), T'EM /' (the difference between the geocentric and helio- N p sin. (L I') centric longitudes of the star)=- -- 7 J . By corresponding substitutions the value of sin. SEM above r becomes, on putting, as before, p sin. 1" for - equal to - r - -77- -TT~ "TTvTi ' or > Developing the radical [12/? sin. l"cos.A COS.(L Z')j 3 as far as two terms, we have, sin. s E M = sin. A' [ 1 + p sin. 1 " cos. A' cos. (L 7') } , or again, sin. s EM sin. \' p sin. l"sin. V cos. A' cos. (L I'} : the second member containing 7? as a multiplier is evidently extremely small, therefore the first is also extremely small ; and since (PL Trigo. art. 41.) sin. SE M sin. A' ~ 2 sin. \ (SEM V) cos. J (SEM + V), on putting arcs for their sines, and considering SEM as equal to V, so that cos. X' may be put for cos. \ (SEM + A/)> we have finally in seconds of a degree, SEM A' (the difference between the geocentric and heliocentric latitudes of the star") n sin. Z'cos.(L-/0- On comparing these values with the formula for aberration in longitude and latitude (arts. 232, 233.) a certain corre- spondence will be found to subsist between them, and the latter may be made to agree with the former on substituting p for the constant of aberration (=20" -3 6) and changino- L (the longitude of the sun) into L + 90, in which case cos (L-/0 becomes sin. (L - 1'), and sin. (L - /') becomes cos! (L-/). Therefore making these substitutions in the for- mulae for MN (art. 234.) the result will be the difference between the geocentric and heliocentric right ascensions of the star, or its parallax in right ascension : also making the CHAP. XIII. PARALLAXES OF STARS. 231 substitutions in the second member of the equation (art. 235.) the result will be the difference between the geocentric and heliocentric declinations of the star, or its parallax in de- clination. From numerous observed declinations of a Lyra which were made by the Astronomer Royal in 1836, the computed parallax of that star was found to be in some cases positive, in others negative ; and the greatest value was 0".505. Mr. Henderson, from observations made at the Cape of Good Hope in 1832-3, concludes that the star a Centauri has a parallax equal to about one second ; and the same astronomer found that the parallax of Sirius does not exceed half a second (Mem. Astr. Soc., vol. xi). 299. The efforts to arrive at a satisfactory value of the parallax of a fixed star by means of meridional observations having failed, M. Bessel, in 1837-8, attempted the problem by observations on double stars. For this purpose, by means of a heliometer (art. 121.) he measured the distance of the principal star from a small star designated a in its neighbour- hood (the latter from its smallness being presumed to have no sensible parallax) and also the angle of position between the hour circle passing through the first star and a line supposed to connect it with the star a ; and from these obser- vations he computed, by means of formulas, the investigations of which are given by Mr. Main in the twelfth volume of the Astronomical Society's Memoirs, the effects of parallax on the distance between the same stars and also on their angle of position. 300. In these investigations the first step is to obtain an expression for the parallax of a star, in terms of its maximum value, in the direction of an arc of a great circle supposed to pass through the star, the sun and the earth. Thus, let s be the sun and E the earth ; also let a and s be the places of two stars supposed to be at equal distances from the sun, a being in the pole of the ecliptic or in the line Scr drawn per- pendicularly to S E. Then the angles S cr E and SSE will be the annual parallaxes of the two stars. Produce s E till it meets S m let fall per- pendicularly on it from S; then SE, Sra may be considered as the sines of the parallaxes, or, on account of the smallness of the angles at cr and s, as the parallaxes themselves (in seconds). But SE : sm :: rad. : sin. and the angle s E m may be considered as equal to s s E : there- Q 4 232 FIXED STARS. CHAP. XIII. fore if p represent the angle SO-E, or the maximum, commonly called the constant of parallax, the parallax for any other star, as s, is equal to p sin sSE. 301. Now let T MF be a projection of the celestial equator, T the equinoctial point, and T NE a projection of the ecliptic : let P and p be the poles of these circles, and let s be the place of a star. Imagine a plane to pass through the sun, the earth, and the star s, and to cut the trace of the ecliptic in E ; then by the effect of parallax the star will appear, to a spectator on the earth, to be at some point s' in the great circle ES produced; and from what was said above (art. 300.) we have ss' p sin. SE. Let a be a small star in the neigh- bourhood of s, and let fall s'n perpendicularly on the arc as; then sn will denote the effect of the parallax of s in the direction sa: draw also si perpendicular to s a. In the right-angled triangle ss'n considered as plane, sn^ss'sin. ss'n, ss'sin.isE,or p sin. SE sin. isE ---- (A). But in the spherical triangle ISE (art. 61.), sin. SE : sin. IE :: sin. si E : sin. ISE; therefore sin. SE sin. ISE = sin. IE sin. s IE. Now i E is equal to the difference between the longitude of the earth and that of the point I ; and the longitude of the earth is equal to that of the sun + 180; therefore if L represent the longitude of the sun, we have IE = L -f 180 long, i, and sin. IE = sin. (L long. i) ; therefore sn, or p sin.sE sin. ISE, becomes equal to p sin. (L long, i) sin. si E. Again, in the spherical triangle p s p (art. 61.), sin. p s : sin. s p p : : sin. p p : sin. p s p ( sin. MSN): but PS is the star's north polar distance, the angle spf is the complement of the star's longitude, and ~pp is equal to the obliquity of the ecliptic ; therefore the value of the angle MsN may be found from the proportion. The arc si being at right angles to s a, the angle i s N is equal to the complement of the sum of the angles asM. and MSN : and since Psa is the observed angle of position for the star , its supplement asM. is known; therefore the angle ISN may be found. Lastly, in the triangle ISN, we have SN, the star's latitude, CHAP. XIII. PAKALLAXES OF STARS. 233 the angle ISN, and the right angle at N; therefore (arts. 62. (*0 and 60 (/)) r sin. s N == tan. I N cotan. I s N, and r COS.NIS rrsin. ISNCOS.SN. Thus there are obtained the value of IN and of the angle Nis or its supplement SIE: the arc Ti, or the longitude of I, is equal to IN + TN; that is, to the sum of IN and the star's longitude ; and substituting these values in the formula (A) we have the value of sn, the star's parallax in the direction sa. 302. The double star 61 Cygni was employed by M. Bessel because, consisting of two stars very near together, he was enabled, in using the heliometer (art. 121.), to bring the other star a seen through one half of the divided object glass cor- rectly into the direction of a line joining the two stars com- posing the double star, which was seen through the other half. Numerous measures of the distances between the star a and the middle point between the two which compose the double star were taken during more than a year (1837-8), and these were reduced to the distance for one epoch (Jan.l. 1838) by corrections on account of the annual proper motion of the star and the effects of the aberration of light on the distance between the star a and 61 Cygni. Then, considering the differences between the measured distances and the distance at the epoch as so many values of sn 9 there were obtained as many equations from which the value of p, the constant of parallax, was found. M. Bessel determined the amount of this constant from the stars 61 Cygni and a to be 0".369, and from the first star and another designated b, 0".26l : the difference is ascribed to a sensible parallax in the latter star. 234 TIME. CHAP. XIV. CHAP. XIV. TIME. SIDEREAL, SOLAR, LUNAR, AND PLANETARY DAYS. EQUATION OP TIME. EQUINOCTIAL TIME. REDUCTIONS OF ASTRONOMICAL ELEMENTS TO THEIR VALUES FOR GIVEN TIMES. HOUR ANGLES. 303. SINCE time enters as an element in most of the pro- blems relating to practical astronomy, it will be proper now to explain the kinds which are employed, and the manner of reducing a given interval in time from one denomination to another. The most simple unit of time is the length of the interval in which the earth makes an exact revolution on its axis, or in which a point on its surface describes exactly the circum- ference of a circle by the diurnal rotation : this movement of rotation is known to be uniform, and the interval in which it is performed, invariable. The time of such rotation is fre- quently considered as the length of a sidereal day, but it is not exactly that which is so designated by astronomers, the latter being the interval of time between the instant at which the plane of any meridian of the earth passes through the true place of the vernal equinox, or the first point of Aries, and that at which, in consequence of the diurnal rotation, it passes next through the same point. Now the equinoctial point is subject to a displacement in the heavens in consequence of the general precession and the effects of nutation ; therefore this interval is not precisely the same as that of a revolution of the earth on its axis, being constantly less by about one hundredth part of a second, on account of the general pre- cession merely : it is moreover not invariable, since the effect of nutation on the equinoctial point is to make that point oscillate within certain small limits about the place which it would occupy by the effects of precession alone. This last inequality would not, however, in nineteen years, cause an error exceeding 2 ".3 in the going of a clock ; and therefore the sidereal day, assumed as above, may be considered in- variable. Hence sidereal time at any instant is represented by the angle at that instant, between the plane of the geo- graphical meridian of a place and a plane passing through the CHAP. XIV. SIDEEEAL AND SOLAK DAYS. 235 axis of the earth and the true equinoctial point, or by an arc of the equator between the meridian and that point. 304. The sidereal day is divided into twenty-four hours, and the hour-hand of a clock, regulated according to sidereal 'time, should indicate Oho. Omin. Osec. every time that the equinoctial point is on the meridian. Now if a fixed star were situated precisely in the plane of the hour-circle (the equinoctial colure) passing through the two opposite equi- noctial points, the meridian of the station would pass through the star every time that, by the diurnal rotation, it came in coincidence with that colure : thus the right ascension of such star would be zero or 180 degrees, according to its position with respect to the axis of the diurnal rotation, and the sidereal clock would indicate ho., or 12 ho., when the star is observed on the meridian wire of the transit telescope. There- fore, on any other star being so observed, it is evident that the time indicated by the sidereal clock should express, in time, the apparent right ascension of that star, or the angle between the plane of the meridian and that of the equinoctial colure ; and this sidereal time of the star's transit ought to agree with the apparent right ascension which is obtained by computation from astronomical tables. In the Nautical Almanacs there are now given the computed right ascensions of one hundred principal fixed stars ; therefore if the transit- telescope be well adjusted in the plane of the meridian, and the movements of the clock-hands be perfectly equable, a comparison of the observed time of the transit of one of these stars with the computed right ascension, will immediately show the error of the clock by the difference which may exist between them. Hence the daily rate of the sidereal clock may be found by observing the transit of a star each night, or at intervals not exceeding five or six nights ; in the latter case the difference, if any, being divided by the number of nights in each interval, will give the gain or loss of the clock- time with respect to true sidereal time during twenty-four hours. 305. Besides the time which is regulated by the diurnal rotation of the earth with respect to a fixed star, astronomers also employ, for the sake of simplicity in calculation, a diurnal rotation of the earth with respect to the sun as a measure of time. Now the earth being supposed to revolve about the sun with its true or proper annual motion, each of the inter- vals between the instants at which, successively, by the earth's rotation on its axis, the plane of a terrestrial meridian when produced passes through the centre of the sun, is con- sidered as the length of the apparent solar day; and it is 236 TIME. CHAP. XIV. found that the number of such revolutions, or days, in the mean tropical year is (art. 174.) 365.242217. The obliquity of the terrestrial equator to the plane of the ecliptic, con- sidered as fixed, is here supposed to suffer no variation, and the line of the equinoxes to continue parallel to itself except the uniform angular deviation from parallelism which is due to the effect of general precession (=50". 2 yearly). The ar- rival of the sun at the plane of a terrestrial meridian consti- tutes noon on that meridian, and the shadow of the gnomon indicates on a sun-dial the hour of the day : but the lengths of these solar days are unequal, both because the motion of the earth in its orbit is variable, and because the plane of the orbit is not perpendicular to the axis of the diurnal rotation. 306. Now (= 59'8".33) expresses the mean daily motion of the earth in the ecliptic : and if it be conceived that the centre of the earth is at rest while a fic- titious sun revolves about it in a circular orbit in the plane of the equator with this uniform daily motion, the earth at the same time revolving on its axis as before, each interval between the instants at which successively the meridian of a station on the earth passes through the fictitious sun would constitute a mean solar day. Thus, let QSN be part of the circumference of a circle in the celestial sphere in the plane of the equator B D, of the earth ; s and s', in that circumference, the centres of the fictitious sun in such positions that the angle s P s' is equal to 59'8".33. Also let PA be a projected meridian passing through any point A, which represents a station on the earth : then the time in which that point would, in consequence of the diurnal rotation, describe about P (the projected pole of the equator) the arc AE is the length of the mean solar day, while the time of describing the circumference AEA ex- presses, very nearly, the length of the sidereal day. Thus the length of the mean solar day exceeds that of the sidereal day by the time in which PA would, by the diurnal rotation, move to the position P a ; and this time is expressed by 59'8 /r .33 ^ , or 3 / 56 // .55 in sidereal time. lo The length of the mean solar day may be divided into twenty-four equal parts, or hours, or the half of it into twelve hours ; and a clock whose dial plate is divided into twelve CHAP. XIV. REDUCTION OF TIMES. 237 hours being regulated so that the hour hand may make a complete revolution on the plate in the time of half the mean solar day, as above described, is a perfect indicator of mean solar time. 307. The times at which two consecutive transits of the moon's centre take place at the meridian of any station being observed by a clock regulated according to mean solar time, the interval between those times is designated a lunar day, and is the solar time in which the moon appears to deviate 360 degrees, or twenty-four hours of terrestrial longitude, from the meridian of the station : the length of such lunar day is evidently variable on account of the inequalities of the moon's motion in right ascension; but the period of a synodical revolution being 29 da. 12 ho. 44 min. (art. 215.) its mean excess above the length of a mean solar day is found, on dividing 24 hours by the length of that period, to be 48 / 46 //< 8. In like manner, if the times of two consecutive transits of a planet over the meridian of any station be ob- served by a clock regulated according to mean solar time, the interval between those times may be considered as the length, in solar time, of a planetary day. This length, besides being different for the different planets, is variable for the same planet ; and it exceeds or falls short of the length of a solar day : its variations depending on the order of the planet's movement (direct or retrograde) in its orbit, and upon the inequalities of the motion in right ascension. 308. If it be required to find the hour, minute, &c. of mean solar time corresponding to any given sidereal time ; let QN (fig. to art. 306.) be a portion of the trace of the equator in the celestial sphere, and Q be the true place of the vernal equinox: now there must be a time when the meridian PA of a station lies in the direction PQ, when the sidereal clock at the station A should indicate ho., and if the terrestrial meridian be, at a subsequent instant, in any other position as PE, the angle QPE (in time) w T ill express the sidereal time, at the same instant, on the same meridian ; also if Q P s express the sidereal time, at the mean noon, on subtracting it from QPE, the remainder SPE diminished by SPS", which may represent the increase of the fictitious sun's mean longitude (in time), while the meridian is revolving from PS to PE, will be the angular distance of the mean sun s" from the meridian; that is, mean solar time, at the same instant. In the Nautical Almanac there is given, for every day, the sidereal time at the instant of mean noon (when a clock regulated by mean solar time indicates ho.) at Greenwich. 238 TIME. CHAP. XIV. And if it were required to find the mean solar time at that place, corresponding to any given sidereal time, the following process may be used. From the given sidereal time subtract the sidereal time at mean noon ; the remainder will be the number of sidereal hours which have elapsed since mean noon, and this may be considered as the approximate solar time : but the excess of a mean solar hour above a sidereal hour is equal 3 / 56 / ' 55 to -A* > or to 9". 85 6 5 (sid. ti.) ; therefore the number of sidereal hours in a given interval of time is greater than the number of solar hours in the same interval ; and in order to obtain the required number of solar hours it is necessary to subtract the product of 9 ".85 65 by the above remainder, from that remainder. The quantity 9".8565 (when t expresses any given number of sidereal hours) is called the acceleration of sidereal above mean solar time for that number of hours. A converse operation is performed in the reduction of a given solar time to the corresponding sidereal time. Thus, having found the acceleration for the given solar time, add together the given time, the acceleration and the sidereal time at mean noon ; the result will be the required sidereal time. 309. If the station of the observer be at any distance, in longitude, from Greenwich, the sidereal time at noon, at the station, must be employed instead of the sidereal time at noon at Greenwich. Now the difference between any two con- secutive times in the column of sidereal time at noon in the Nautical Almanac is 3 / 56 // .55, which is, in fact, the acceler- ation of sidereal time for twenty-four hours ; or the variation of sidereal time in the column is 9". 85 65 for one hour of time : but the instant of mean noon at any station distant from Greenwich, in longitude, 15 degrees westward, is later by one hour than the instant of mean noon at Greenwich, and at a station 15 degrees eastward, is earlier by one hour. Therefore, if t, in hours, be the longitude of the station from Greenwich, 9".8565# must be added to the sidereal time at Greenwich mean noon, if the station be west of Greenwich, or the same must be subtracted if the station be east of Greenwich, in order to have the required sidereal time at mean noon. 310. The difference between apparent and mean solar time at any instant is called the equation of time, and in the Nautical Almanac its value is given for the instants both of apparent and mean noon at Greenwich on every day in the year. In order to understand the manner of finding this value, imagine a point to revolve about the earth in the plane CHAP. XIV. EQUATION OF TIME. 239 of the equator with a uniform angular motion equal to 59' 8". 33 in a mean solar day; and let ST, TV, c. subtend each, at P, the pole of the earth's equator, an angle equal to that quantity : then, since this motion is performed in the time that the hour hand of a clock regulated according to mean solar time revolves twice round upon the dial plate, it may be conceived that the hour hand may be so ad- justed as to indicate hrs. when the point arrives at s, T, v, &c. Make the arc SQ, or the angle SPQ, in the plane of the equator, equal to the sun's mean longitude at the given instant, suppose that of mean noon ; then Q is the place of the mean equinox. Make Q Q/ equal to the equation of the equinoxes in right ascension, in arc or in minutes, c. of a degree ; then Q' will be the place of the true equinox. Let Q' s' represent part of the trace of the ecliptic in the heavens ; make that arc equal to the sun's true longitude at the same instant, and let fall s' perpendicularly on QS ; then s' may represent the place of the true sun, and Q't the sun's true right ascension. Now the true sun revolving, apparently, about the earth in the plane of the ecliptic with its proper variable motion ; the time at which the plane of a terrestrial meridian passes, by the diurnal rotation, through s', is the instant of apparent noon, or that at which the shadow of the gnomon on a sun-dial indicates hrs. : but the terrestrial meridians being perpendicular to the equator, that which passes through s' also passes through t\ and the angle PS through which a meridian revolves between the instant of apparent noon and the instant of mean noon is evidently equal to the difference between the sun's mean longitude and his true right ascension. Therefore since the places of s and s' are supposed to be computed for the instant of mean noon, the difference so found is the equation of time for mean noon. If the places of s and s' had been computed for the instant of apparent noon, the difference would have been the equation for that instant. If L, for any given instant, represent the mean longitude of the sun, reckoned from the mean vernal equinox, as it would be found from astronomical tables ; and if + n repre- sent the equation of the equinoctial points in right ascension, in seconds of a degree ; then Q/ s =: L + n. Now the true right ascension of the sun in degrees might be obtained from the Nautical Almanac for the same instant ; but when found from astronomical tables, it consists of the following terms : 240 TIME. CHAP. XIV. the mean longitude (L) of the sun from the mean equinox ; the equation (+E) of the centre; the equation of the equi- noxes (+ N) in longitude ; the effects (+ P) of planetary per- turbation; and lastly, a term (n) which being applied according to its sign, reduces the whole to its projection on the equator : the sum of all these terms is represented by Q' t. Therefore ts, the equation of time at any instant, is expressed by the formula TJ ( 311. The instant of the occurrence of a celestial phe- nomenon is usually expressed in mean, or apparent solar time for the station at which the phenomenon is observed ; and it is consequently necessary to mention also the place of observ- ation in order that the time so expressed may be reduced to the corresponding times for other stations. For the purpose, however, of rendering unnecessary a reference to the par- ticular station, a practice has been of late introduced of giving the time of a phenomenon in mean solar time reckoned from the instant when a mean sun was in the mean equi- noctial point in the year of the Christian era. Time so reckoned is called equinoctial time, and being independent of any station on the earth's surface it serves in some measure to supply the want of a fixed meridian in expressing the time of a phenomenon. The mean sun alluded to is one which is supposed to re- volve about the earth with a uniform motion equal to the mean velocity with which the latter revolves about the sun ; and the mean equinoctial point is a place in the heavens which the intersection of the ecliptic and equator would oc- cupy if its movement were that only which constitutes the general precession. The time in which the mean sun would make one revolution from the mean equinoctial point to the same again, or the length of the equinoctial year, is, from the investigations of M. Bessel, equal to 365.242217 mean solar days, and at the instant of mean noon at Greenwich on March 22d, 1844, the equinoctial time was 1843 years, 0.082875 days, or an equinoctial year terminated at 0.082875 days (or 1 hr. 59' 20" A) before the instant of mean noon at Greenwich. Now in the Nautical Almanac there is given, page XXII. of each month, the number of mean solar days which have elapsed since the mean noon of the day in the preceding March on which the sun was in the mean equinoctial point ; therefore (till March 22d, 1845) the decimal 0.082875 days must be added to the complete number of days elapsed since March 22d, 1844, in order to have the equinoctial time cor- CHAP. XIV. EQUINOCTIAL TIME. 241 responding to the Greenwich mean noon of the given day. The above decimal changes on March 22d, 1845, but a like process must be used during the next equinoctial year ; and so on. If it were required to find the equinoctial time correspond- ing to July 10th, 1844, at 5 P.M., mean time at Greenwich; since from March 22d, to July 10th, there are 110 days, and 5 hours are equal to -/%, or 0.208333 days, that time would be 1843 years 110.208333 + 0.082875 days, or 1843 years 110 2Q1 207 110.291207 days; or again, 1843 3^5 242217 ( = 1843 - 30196 ) years. Should the equinoctial time be required for a given instant expressed in mean solar time for any place distant, in longitude, from Greenwich ; that mean time may be reduced to Greenwich mean time by adding the difference of longitude in time, if the place be west of Greenwich, or by subtracting it if eastward, and then proceeding as before. 312. The right ascension of the meridian or midheaven, at any given instant, is expressed by the time which a sidereal clock indicates at that instant (art. 73.) ; and therefore, the apparent or mean solar time being given, that right ascension may be obtained by finding the corresponding sidereal time, as explained at the end of art. 308. ; such sidereal time is the right ascension required. If a watch or clock regulated according to solar time in- dicate correctly mean time at a station, the moment at which the sun will be on the meridian will be ascertained on apply- ing the equation of time to the hour (12) of mean noon, by addition or subtraction, as directed in the Nautical Almanac (page II. of each month). But the time at which any fixed star will be on the meridian of a station is to be found by the following process : let Q s s' represent a trace of the equator in the heavens ; E the earth, p m a pro- jection of the plane of the meridian, and Q the position of the equinoctial point ; let also Q s' represent the right ascension of the star, which is to be taken from the Nautical Almanac. Then, from the same work let the right ascen- sion of the sun at apparent, or at mean noon, at Greenwich be taken, and let there be added to it, if the station be westward of Greenwich, or subtracted from it, if eastward, the variation of the sun's right ascension for a time corresponding to the distance of the place in longitude from Greenwich; the result (QS), which is the sun's right ascension at apparent or mean noon at the B 242 TIME. CHAP. XIV. station, being subtracted from the star's right ascension, gives S s', the approximate time at which the star culminates, or the time in which the meridian Pm would revolve by the diurnal rotation from PS to PS'. Again, let the variation Ss of the sun's right ascension during that approximate time be found and subtracted from the approximate time ; the remainder will be the angular distance SPS' (in time) of the sun from the meridian when the plane of the latter passes through the star ; that is, very nearly, the apparent, or mean, solar time at which the star will culminate. If the sun's right ascension should exceed that of the star, it would be merely necessary to add twenty-four hours to the latter before the sun's right ascension is subtracted from it ; the remainder will still be the time which is to elapse before the star culminates. 313. It is of importance to have a knowledge of the instant at which a planet will culminate, or be on the meridian of any particular station whose longitude from Greenwich is known exactly or nearly, for the purpose of determining, by means of an observed altitude of the planet, the hour of the observation, or in order to be prepared for observing the me- ridian altitude of the planet and subsequently of determining the latitude of the station. This knowledge may be ob- tained in the following manner. If s s', as above, represent the difference between the right ascension of the sun and the geocentric right ascension of a planet, at mean noon at the station, and consequently the approximate mean time at which the planet will culminate ; let there be next found, for that approximate time, the variation ss of the sun's right as- cension and the variation s' s' of the planet's geocentric right ascension (the latter being set out in the direction sV or s's", according as the right ascension is increasing or de- creasing). Then the arc ss' or ss" ( ss' + sV Ss, or ss' a's" Ss) may be considered as a very near approxi- mation to the mean time at which the planet will culminate at the station. The time at which the moon will culminate at any station may be found in like manner, sV representing the variation of the moon's right ascension in the first approximate time ; and this variation may be obtained from the hourly right as- censions of the moon in the Nautical Almanac. The culmination of a planet at any station may also be de- termined from the times of the daily transits or passages of the planets over the meridian of Greenwich which are given in the pages of the Nautical Almanac containing the geo- centric places. For the day of the month being given, 011 CHAP. XIV. CULMINATIONS OF STAIIS. 243 subtracting from one another the times of transit on that day and the next, there will be found the excess or deficiency of the interval between the times, with respect to a mean solar day : then, supposing that the excess or deficiency takes place uniformly, there may be had by a simple proportion the value of it for an interval of time equal to the given difference of longitude. For example, let it be required to find the time that Venus culminated on March 1st, 1843, at a station whose longitude from Greenwich, in time, is three hours eastward. From the Nautical Almanac it is found that the difference between the times of the meridian passages on the first and second days of March is 0".4 (an increase) ; therefore 24 ho. : 3 ho. :: 0'.4 : 0'.05, which being subtracted from 21 ho. 5 '.6 (the time of the transit at Greenwich on the first day of March) gives 21 ho. 5'.55, or 21 ho. 5' 33" for the time of the transit, in mean time, at the station. The excess is subtracted because the station is eastward of Greenwich, and the transit takes place there earlier than at the latter place : it should have been added if the station had been westward of Greenwich. It must be observed also that, if the times of transit should diminish from one day to the next, the difference corresponding to the difference of lon- gitude must be added when the station is east of Greenwich, and subtracted when it is westward. Thus, let it be required to find the time that Jupiter cul- minated on August 1st, 1843, at a station whose longitude from Greenwich, in time, is 3 hours eastward. From the Nautical Almanac it is found that the difference between the times of the meridian passages on the first and second days of August is 4 '.4 (a diminution) ; therefore 24 ho. : 3 ho. :: 4'.4 : / .55, which being added to 13 ho. 8 '.2 (the time of the transit at Greenwich on the first day of August) gives 13 ho. 8 '.7 5 for the mean time of the transit at the station. If the station had been westward of Greenwich the diminution must have been subtracted. The time of the moon's transit over the meridian of any station whose longitude from Greenwich is known may also be determined with sufficient precision for the purpose of pre- paring to observe the meridian altitude of that luminary. For example ; on the first day of September, 1843, the moon passed the meridian of Greenwich at 6 ho. 15'.2, and on the second day at 7 ho. 12 '.5; the difference between them is R 2 244 TIME. CHAP. XIV. 5 7 '.3 (the excess of the time between two consecutive transits above a solar day) : therefore, if the station be distant from Greenwich in longitude 45 degrees, or 3 hours, eastward, 24 ho. : 3 ho. :: 57'.3 : 7'.16; and .this excess subtracted from 6 ho. 15'.2 leaves 6 ho. 8'.04, or 6 ho. 8' 2" A, for the mean solar time of the transit at the station. The excess obtained from the proportion must have been added if the station had been westward of Greenwich : in the first case the transit takes place at the station before it takes place at Greenwich ; and in the other, it takes place later. 314. If a celestial body having no proper motion were observed at two different instants, the angle at the pole be- tween the hour circles passing through the apparent places of the body at those instants would be expressed in degrees by 15 T, T representing the number of sidereal hours in the interval of time during which a meridian of the earth would, by the diurnal rotation, pass from one place of the celestial body to the other. Also, when the celestial body is the sun, and the interval between the observations is ex- pressed as usual in mean solar time, or that which is given by a clock or watch going correctly, the same interval mul- tiplied by 15 will give the angle between the hour circles in degrees. For, let QSS / (fig. to art. 312.) represent the equator, p its pole, Q the equinoctial point ; and let the plane of the meri- dian of a station pass through p A and P B at the two times of observation : also let Q s be the right ascension of the sun at the first observation ; and, by the change in the sun's right ascension in the interval, let Qs be the right ascension at the second observation. Then, the arc A s in degrees (the angular distance of the meridian from the hour circle passing through the sun at the first observation) being represented by a, the arc AB by b, and Ss by c; the angular distance BS of the meridian from the hour circle passing through the sun at the second observation, is equal to a b + c, and the angular distance between the hour circles passing through the two visible places of the sun is equal to AS BS or b c. In sidereal time that angle is expressed by yj (b c) : but yy c is the acceleration corresponding to the sidereal tune ~j b ; and yj (b c) expresses also the solar time corresponding to y 1 ^ b, or T j (bc) is the given interval. Thus the angle in degrees between the hour circles is equal to the product of 1 5 by the given interval between the observations, in solar time. In strictness, the angle between the two hour circles being CHAP. XIV. HOUR ANGLES. 245 affected by the inequality of the sun's movement in right ascension during the interval, that interval, in mean time, should be reduced to the corresponding interval in apparent solar time, by applying, according to its sign, the variation in the equation of time during the interval before it is multiplied by 15. The variation of the equation does not, however, exceed 30" in 24 hours, and is generally much less. If the interval between the observations on a fixed star at two different instants, were expressed in solar time, it would evidently be necessary, in order to have in degrees the angle at the pole between the hour circles passing through the two visible places of the star, that the interval so expressed should be converted into sidereal time, either by a table of " Time equivalents " (Naut. Aim., pages 554 557.), or by adding the acceleration, before the multiplication by 15 is made. When the interval between two observations on the moon is ex- pressed in solar time, it must in like manner be converted into sidereal time : this last may be represented by yj A B in the figure, that is, by the angular distance (in time) between the places of the meridian at the times of the two observ- ations; therefore, subtracting from it the increase of the moon's right ascension during the interval (represented by T j Ss), and multiplying the remainder by 15, the result will be the required angle at the pole between the hour circles passing through the apparent places of the moon at the observations. In like manner, the interval between two times of observation on a planet being reduced to sidereal time, if from the result there be subtracted the increase, or added the decrease of the planet's geocentric right ascension for the interval ; there will be obtained (in time) the required angle at the pole. 315. It follows from what has been said, that when the hour angle of a celestial body (the angle at the pole between the meridian of a station, and a horary circle passing through the celestial body) is obtained, in degrees, from an observed altitude of the body, the reduction of that angle to solar time, and the converse operation, will be different for the sun, the moon, and for a star. The angle at the pole between the meridian of a station, and a hour circle passing through the sun, being divided by 15, or multiplied by -fa according to the practice of mariners, expresses the solar time of the observ- ation, reckoning from the instant of apparent noon. For a fixed star, the like angle at the pole, in degrees, on being divided by 15, is expressed in sidereal time ; and the acce- leration for the number of hours must be subtracted from the quotient, in order that it may be expressed in solar time : R 3 246 TIME. CHAP. XIV. it will thus denote the solar time of the observation reckoned from the instant at which the star culminates. If a like angle at the pole be obtained in degrees from an observed altitude of the moon, the number of degrees divided by 15, give the sidereal time corresponding to the interval: if to this be added the increase of the moon's right ascension for the same interval the sum will be (nearly) the hour angle in sidereal time, which may be reduced to solar time by sub- tracting the acceleration. The result will be (nearly) the time of the observation, reckoning from the instant at which the moon culminates. A process, precisely similar, must be em- ployed when the hour angle of a planet is obtained from an observation, the variation of the planet's right ascension being added to, or subtracted from the sidereal interval according as that right ascension is increasing or decreasing. The variations in the right ascension of the sun, the moon, or a planet, for a given interval of time, may be obtained from the Nautical Almanac, by methods which will be presently noticed (art. 316.). Conversely, if the time of the day be given, the sun's hour angle is immediately found on multiplying by 15 the ap- parent time elapsed since the preceding noon. When the time of night is given, the most simple way of finding the horary angle of a fixed star, a planet, or of the moon, is to take the difference between the right ascension of the mid- heaven, and of the celestial body (both of them being found for the given time); for the result, when multiplied by 15, gives the required angle in degrees. The following example will serve to illustrate the methods of finding the hour angle of a fixed star for any given time and station. Let it be required to find the hour angle of a Polaris for Sandhurst at 10 ho. 46 m. 21 sec. mean time, or 11 ho. 1 m. 13 sec. apparent time, No- vember 17. 1843. First Method. ho. , Right asc. a Polaris, Nov. 17. - - 1 4' 0.8 (Naut. Aim.) Right asc. sun (noon at Sandhurst) - 15 28 35.7 (Do.) Approximate time of culminating - 9 35 25.1 Variat. sun's right asc. in 9 ho. 35 m. - 1 39.4 Apparent time of culminating - - 9 33 45.7 Apparent time given - - 11 1 13 Star's hour angle in solar time - 1 27 27.3 Do. in sidereal time (by the table of time equivalents) - - 1 27 41.8 CHAP. XIV. HOUR ANGLES. 247 Second Method. ho. , ff Sidereal time at mean noon, Greenwich - 15 43 34*6 (Naut. Aim.) Variation for 3 min. diff. of long, (add) - .5 15 43 35.1 Sidereal time corresponding to lOh. 46' 21" (mean solar time at Sandhurst) - 10 48 7. Sidereal time at the instant, or right as- cension of the midheaven - - 2 31 42.1 Right ascension of the star - - 1 4 0.8 Star's hour angle in sidereal time - - 1 27 41.3 Note. If there be a sidereal clock at the station, the right ascension of the midheaven is given by it for the instant ; therefore, by taking the dif- ference between such right ascension and that of the star, the hour angle is immediately found. Third Method. n - / // Right ascension of the sun at apparent noon (Gr.) - - 15 28 35.2 (Naut. Aim.) Variation for difference of longitude (= & west) - .5 Right ascension of the sun at app. noon, Sandhurst - - - 15 28 35.7 Variation for 1 1 h. 1' 13" - - 1 54.1 Right asc. sun at the given time - 15 30 29.8 Apparent time at the instant (add) - 11 1 13 Right ascension of the midheaven - 2 31 42.8 Right asc. star - - - 1 4 0.8 Star's hour angle in sidereal time - 1 27 42 B 4 248 INTERPOLATION. CHAP. XV. CHAP. XV. INTERPOLATIONS PRECISION OF OBSERVATIONS. 316. THE longitudes and latitudes of the sun, moon, and planets, and the right ascensions and declinations of the sun and planets, with the semidiameter of the former, and the logarithms of the radii vectores, are given in the Nautical Almanac for the noon of every day at Greenwich, that is, for intervals of time equal to twenty-four hours ; the semidiameter and horizontal parallax of the moon for intervals of twelve hours, and the right ascensions and declinations of the moon for every hour : and certain corrections only are necessary, in order to obtain the values of the elements for any given time at the place of observation. For the ordinary purposes of practical astronomy, it will suffice to find, by a simple proportion, the variation of the element in the interval between the given instant, and the hour at Greenwich for which the value of the element is given in the almanac. The longitude of the station, from Greenwich, being known by estimation or otherwise, the time at Greenwich corresponding to the given instant can be found by subtracting the difference of longitude, in time, from the given time at the place, if the place be eastward of Greenwich, or adding the two together if westward ; for the given time being reckoned from the preceding noon, the re- mainder or the sum is the required time at Greenwich, reckoned also from the preceding noon. Now, in the Nautical Almanac, adjoining the column of the sun's right ascension, and also of his declination, there is given a column containing the hourly variation of the ele- ment ; and this variation being multiplied by the number of hours in a given interval, as that between the Greenwich noon and the time found as above, will be the required cor- rection of the element, which being applied to the value for Greenwich noon, the sum or difference will be the value for the given instant. In like manner, may the correction of the moon's right ascension and declination for a given interval be obtained from the Nautical Almanac ; the former, by means of the difference between the hourly right ascensions, and the latter by means of the column of variations for ten CHAP. XV. ORDERS OF DIFFERENCES. 249 minutes. Thus, for example, on the 7th of April, 1843, at 4 ho. 34' 30" P. M, mean time at a station whose longitude west of Greenwich is 1 ho. 10', the corresponding mean time at Greenwich is 5 ho. 44' 30" ; and, if it were required to have the correct declination for that instant, the variation for 10 min. at 5 P.M. being 63".45, the following proportion may be made : 10' : 63"-45 :: 44'-5 : - 282"-35, or - (4' 42"-32), which subtracted from 21 58' 1"*9, the declination at 5 P.M., leaves 21 53' 19"*55 for the corrected declination. Correc- tions thus found are said to be for first differences. The variations of a planet's geocentric right ascension and declination in any given interval of time, may be obtained from the columns containing the planet's right ascension and declination for the given month in the Nautical Almanac, by taking proportional parts of the daily differences. 317. But when the variations of the elements are consider- able, and when the values are required with great accuracy, the corrections for second, and occasionally for third, as well as for first differences become necessary : these may be ob- tained by means of the usual formula for interpolation, which is investigated in the following manner : Let y, the quantity to be interpolated, be considered as a function of a variable quantity m, and assume # A + Bm + Cra 1 + Dra 3 + &c. Now, let m have successively values represented by o, n, 2 n, 3 n, &c., then the corresponding values of y will be A ............. (1) A + J$n + Cri 2 + I)n* + &C ...... (2) A + B. 2/z-fc. (2rc) 2 + D (2rc) 3 + &c.. . (3) A + B . 3n + c (3rc) 2 + D (3rc) 3 + &c. . . (4) &c. Subtracting successively (1) from (2), (2) from (3), &c., and representing the remainders by P', Q', R', &c. we have n - &C ..... (5) 2 + &c. . . (6) & c . . (7) &c. &c. 250 FORMULA OF INTERPOLATION. CHAP. XV. Again, subtracting successively (5) from (6), (6) from (7), &c., and representing the remainders by P", Q", &c., we get p" = c + D.3ra+ &c (8) 2n v ' _ == c + r>.6w+ &c ..... (9) &c. &c. Next, subtracting (8) from (9), &c., and representing the re- mainders by P'", &c., we have But P'" = Q " P// : also Q" = - and p" = - ; 2n n n therefore P "' = (R/ ~ Q/ ^ Q/ ~ F/ \ and putting A '" for the numerator of this fraction, we have A'" Substituting this value of D in the equation (8), and trans- posing, r\ _ __ '^ . Q' P' but P" = , and putting A" for Q' p 7 , we get A" A"' 2 tf 2 2 ' Again, substituting these values of c and D in (5), and trans- posing, D I ~ n 2n + 2n or putting A ' for P x and simplifying, n 2n 3n ' Finally, substituting these values of B, c, and D in the as- sumed equation, we obtain n n n im/nP 3m \ *n\& n * 2 ) A ofwhich the three first terms will generally ffice. CHAP. XV. EXAMPLE. 251 Here y is the element required for the given time. A is the element for the noon, midnight or complete hour preceding the given instant. m is the given number of hours since noon or mid- night, or the number of minutes since the complete hour. n is either 24 h., 12 h., or 60 rii, according to the interval between the times for which the elements are given in the Nautical Almanac. A', A", &c., are the first, second, &c. differences of the elements in the almanac. EXAMPLE. Let it be required to find the moon's latitude for August 4th, 1842 at 16 h. 18' mean time at Greenwich, that is, at 4.3 h. after mean midnight. Moon's Lat. A' A" Mean Second Diff. O i a / a ^ // Aug. 4. noon + 45 48.1 Midnt. + 05 54.6 Aug. 5. noon 34 33.1 Midnt. 1 14 49.4 39 53.5 40 27.7 40 16.3 + 34.2 11.4 + 11-4 The latitudes are taken from the Nautical Almanac ; the positive sign indicates north latitude ; the negative sign, south latitude. Here A = 5' 54".6, A' = -40' 27".7, or -40.463. m. = ^f = 0.358, A' = - 14' 29".16, A" = + 11".4 - 1 = - 0.642, l n 2 n - l) A" = - n J 1-31 Therefore y 8' 35".87, the moon's correct latitude, south. 318. The horary motions of the sun, moon, and planets, when great accuracy is required, must also be taken from the almanac with an attention to second differences. When the elements are given in the almanac for every 12 hours; the first difference between them at the noon or midnight pre- ceding, and that which follows the given instant being di- vided by 12, will give the mean horary motion during the 12 hours : this must be corrected by applying to it with its pro- per sign the difference between the values of the third term in the formula for interpolation for the complete hour before, and for the complete hour after the given instant ; for this differ- ence will express the quantity by which the true differs from the mean hourly motion at the given time. Thus, wanting 252 INTEKPOLATION. CHAP. XV. the moon's hourly motion in latitude at the time given in the above example, we have A ' 4(V 27" 7 =- - _ - = - 3' 22".31 for the mean hourly motion : but the value of the third term in the series is, for 4 hours after midnight, = (11".4) = 1".27, for 5 hours = - fo (H".4) = - 1.39. The difference between these is /r .12, which added to the mean hourly motion, gives 3' 22".43 for the true hourly motion between 16 hours and 17 hours on the given day. When the moon's horary motion in declination is required for any given time, the mean horary motions must be taken out for at least three complete hours, including within them the given time (they may be obtained by multiplying each of the differences for 10 minutes, corresponding to those hours, by 6), and these may be considered as the horary motions for the middles of the intervals : the differences between them will be nearly constant, but a mean of the differences may be considered as a first difference, which being multiplied by 777 - will give the variation of the hourly motion ; and this must be added to, or subtracted from, the mean hourly motion in order to give the correct hourly motion at the given time. Here n 60', and m is the number of minutes between the given instant and the middle between the complete hour pre- ceding and following it. 319. The subjoined process for obtaining a series of inter- polated numbers by means of second differences will occa- sionally be found useful. Let the numbers in the column R below be the radii vectores of a comet corresponding to cer- tain given longitudes or anomalies, reckoned on the orbit, and differing from one another, for example, by one degree ; and let it be required to interpolate the numbers for every 10 minutes. Long, or Anom. R A' Sum of the A' 30 31 32 1-11072 1-13214 1-15427 2142 2213 4355 Then between 30 and 32 there wiU be twelve first dif- ferences, and their sum will be 4355. Let d be the first of these twelve first differences, and 8 the second difference, supposed to be constant : then the several first differences will be d, d + 8, d 4- 28, &c to d + 11 8; CHAP. XV. PERSONAL EQUATION. 253 and their sum will be I2d + (1 + 2 -h 3 +....+ 11) 8, or 12 d + (1 + 11) ^8: hence 12 d + 668 = 4355. In like manner the sum of the six first differences between 30 and 31 will be Gd + 15 S = 2142. From these equations we find ^ = 352.07,8 = 1.972: then for 30 we have K 1.11072 ; for 30 10', K' - R + d - 1.11424 ,< for 30 20', R" = R' + d + 8 = 1.11778 ; for 30 30', R'"=R" + rf + 28 = 1.12134 ; &c. 320. In the simple circumstance of observing a signal, as the flash of gunpowder or the occurrence of a celestial phe- nomenon, the estimates made by two persons of the instant at which the event took place will, in general, differ ; and the same person does not, on different occasions, observe phe- nomena, with respect to time, in like manner : in the bisection of a star by a wire in a transit telescope, such difference has amounted to several tenths of a second. The incongruity is conceived to arise from peculiarities in the organs of vision, or in the perceptions of different individuals, and from variations in the state of the nervous system, according, probably, as the person may be more or less fatigued at the time of making the observation. 321. The correction which should be applied, on account of this cause of error, to the observed time of the occurrence of a phenomenon is called the personal equation ; and, in the present state of Astronomy, it is incumbent on every observer, by comparing the results of his observations with those deduced from the observations of other persons, to determine the value of the correction which should be applied. At the Greenwich Observatory, when a difference exists between the personal equations of two observers, the parties determine the error of the sidereal clock at a certain instant, from the transits of stars observed by them on alternate days ; and the dif- ference between the errors is considered as a correction to be subtracted from the several errors determined by the observer whose personal equation is the greatest, in order to reduce them to the values which would have been determined from the observations made by the other. In general, when a ter- restrial signal or a celestial phenomenon is to be observed by different persons simultaneously, at every repetition of the observation the parties should interchange their stations, in order that the equations may be detected and determined. 322. It is customary for observers to estimate the goodness of a single observation according to a judgment formed at the 254 WEIGHTS OF OBSERVATIONS. CHAP. XV. time under existing circumstances ; and when they have made two or more observations which are presumed to differ in degrees of goodness, numbers expressing the relative values are multiplied into the numerical expressions of the observa- tions, in order that all the observations may be reduced to the same standard with respect to correctness. Thus, if one observation should appear to merit confidence twice as much as another, it would be multiplied by 2; and, in general, A 19 and A 2, representing the numerical expressions for two ob- servations whose relative merits are denoted by two numbers represented by w j and w 29 respectively, the two observations, when reduced to one standard, would be expressed by w t AJ and w 2 A 2 . The numbers w 19 w 2 , &c. are called weights ; and each, when applied to an observation, may be conceived to represent the number of observations of standard goodness to which the observation is equivalent. The weight due to several observations taken collectively is equal to the sum of the weights due to the observations separately : thus II 2 2 . . .7,7 - is taken to express the weighted mean W 1 + W 2 + &C. of the observations A 13 A 2 , &c. collectively. 323. If the analytical expression given by writers on Probabilities (De Morgan, Theory of Probabilities, in the Encyclop. Metropol., art. 100.) for the probability that the error in a single observation lies within certain limits ex- pressed by + e and e (e representing some small number) be made equal to J, which denotes equality of chance that an observed result may differ from the truth in excess and defect by equal quantities; and if from the equation so formed the value of e be obtained, it will be found to be equal to 0.476936. Let this be represented by e, and if it be sub- stituted in the expression for the limit of the probability of error, on taking the number representing the common average or arithmetical mean of several observations as the true mean (art. 115. in the work above quoted), that expression becomes, denoting the sum of several terms of the like kind by one such term having prefixed to it the symbol 2, 9 V i?2 iV^v*=i-, ...... '(A), n n or, which is proved to be an equivalent expression, ,2 /C 2.A 2 /2.Ax% , v s v / -Vi --- --- ) ] . . . . (B): n L n \ n I 3 here 5) . E 2 (representing E 2 X +E% -f&c.) is the sum of the squares of the presumed errors in the several observations, or of CHAP. XV. FORMULA FOB PRECISION. 255 the differences between the actual observations and the average or arithmetical mean of all ; - represents what is called the mean square of the numerical expressions for the several ob- servations A 1? A 2 , &c., or the sum of the squares of the separate observations divided by n, the number of observations, ^ A 2 and t \ is the square of the arithmetical mean of all the observations : in the formulae, s A/ 2 = 0.6 7449. 324. It is evident that the precision of the result obtained by taking the ordinary average of several observations as a true result, will be so much greater as either of the expressions (A) or (B) is smaller ; or, using the first expression, the degree of precision is inversely proportional to - \/ 2) . E 2 , or n * / \ directly proportional to f 2 . E 2 J , that is, to n lows also, from the expression (A) that in two sets consisting of an equal number of observations, the precision of the results 2. E 2 is inversely proportional to A/^.E 2 ; and if - -*-, the mean square of the errors, be the same in both, which implies that all the observations are equally good, the precision is directly proportional to \/n. Thus, if there be taken the ordinary average of two sets of observations unequal in number, but all equally good ; in order to reduce the averages to the same degree of precision, each must be multiplied by the square root of the number of observations from which it was obtained. 1*- 7Z 2 325. The term 2 . E 2 or ^ 2 is considered as represent- / E~" n ing the weight due to the average of a series of observations ; therefore the precision due to the result of a series of observ- ations varies with the square root of the weight. From this formula for the weight an observer may learn to determine the weight due to any single observation made by himself: for, having obtained the numerical values of several observ- ations (for example, the particular number of seconds indicated by a clock at the bisection of a star by a wire in a telescope), and taken the difference between each and an arithmetical mean of all, let him consider the several differences as so many errors ; then, on dividing the square of the number of ob- servations by the sum of the squares of the errors, the result 256 RELATIVE VALUES CHAP. XV. compared with like results obtained by other observers will afford an indication of the relative value of his observations. 326. From the known weights due to two independent observations there may be obtained the weight due to a phe- nomenon depending on those which were observed ; and the following example will serve to illustrate the process. Let a certain angle c depend on two other angles A and B, and let these last be actually observed while C is obtained merely by adding A and B together, or by subtracting one of them from the other as the case may be : it is required to find the weight due to C. In a single observation the precision is inversely propor- tional to the error : and since the weight is represented by the square of the precision (art. 325.),, w a -^ or E 2 a -. But, in any result obtained, like the angle c, from two ob- servations whose errors maybe represented by E A and E 2 , and weights by w 1} w 29 the precision varies with . g ^ ^, V(E j +E a ) or the weight with ^ ^~ ; substituting, therefore, in the E i 4" E last expression, - , for E , 2 and E A we get - for the M! w 2 w l -f w 2 weight due to C. In combining together several observations of circumpolar stars for determining the latitude of a station, let w l be the weight due to all the zenith distances observed when the star is above the pole, and w< 2 the weight due to all those which are observed when it is below : then the fraction just formed will be the weight due to the result of the whole series of observations. 327. Either of the formulae (A), (B) may be used to find the relative merits of two or more instruments : thus, let several zenith distances of a celestial body be observed with two different circles, and, omitting the degrees and minutes, let the seconds read on the head of a micrometer screw be as follow : No. 1. circle, 58".5, 56".33,59".25, 54".75; mean=57".2075. No.2.circle,40".25,38".5, 36".8 ,42".5 ; mean = 39".2625. The following errors are found by taking the differences between the mean and each observation, No. 1., + l".2925,-0".8775, + 2".0425,-2".4575. No. 2., + 0".9875, - 0".7625, 2".4625, + 3".2375. Taking the sum of the squares of the errors and dividing by CHAP. XV. OF INSTRUMENTS. 257 ^ E' 2 the number of observations, we have for , in No. 1., n 2.9629, and in No. 2., 4.5843, the square roots of which are 1.721 and 2.141: these divided by Vn ( = 2) give 0.8605 and 1.0705, which multiplied by sV2 ( = 0.67449) give finally the numbers 0.5804 and 0.7221. The degree of precision in the result obtained by taking the average of the first set of observations is greater than by taking that of the second in the ratio of 7221 to 5804 ; and the first instrument, including the skill and carefulness of the observers in both, is therefore better than the second in the ratio of those numbers. It is evident, however, that a much greater number of observations ought to be employed when considerable accuracy in the relative values is required. In like manner may the relative merits of two or more chrono- meters be determined, the daily rate of each for several days being considered as errors, or as observations, according as the first or second formula is employed. The results of observations made by direct view and by reflexion with the same instrument, are frequently found to differ from one another in goodness when tried by means similar to that which has been described ; and it is stated in the Introduction to the " Greenwich Observations " that, with the mural circle in use at the Observatory, the zenith distances of celestial bodies appear to be greater by reflexion-observa- tions than by the others, the difference in some cases amounting to four seconds ; near the horizon it is small or vanishes, and it is greatest when the zenith distance is about 30 or 40 degrees. In order to find the corrections for such discordances with small trouble, the following process is used : From any point in a straight line are set out, as abscissae, several zenith dis- tances or polar distances, and at right angles to the line, at the termination of each distance, is drawn an ordinate equal to the error or difference between the distances obtained from the direct and reflexion observations. A curve line being traced through the extremities of the ordinates so drawn ; the ordinate corresponding to any given zenith or polar distance being measured by the scale gives the value of the required error or difference, half of which is applied as a correction to the direct, and half to the reflexion observations. 328. The value of every element in physical science is determined from the results of numerous observations or ex- periments combined together, and, in the present state of astronomy in particular, it has become indispensable -to employ some method of making the combination so as to afford the 258 GROUPING CHAP. XV. most probable value of the required element. For this purpose there are formed, from the different observations, equations in each of which the true or most probable value of an element is made equal to that value which is deduced immediately from the observations, together with the cor- rections due to the several causes of error, those corrections involving, as unknown quantities, the most probable values of other elements. Thus, let T 1 + a T x + b x y, T 2 + 2 x + b 2 y, &c. be expressions formed from so many independent observations in which T 1? T 2 , &c. are the immediate results of the observ- ations, a l} 2 , &c., b l9 & 2 , &c., are quantities obtained from the nature of the elements, and x and y are unknown quantities, of which the most probable values are to be determined : the terms are either positive or negative, according to circum- stances. If T be the true value of the element represented by T,, T 2 , &c., and if all the results of observation were free from errors, we should have T! + ! x + bi y=T, or TJ- T + tfj x + b^ y T 2 + 2 ^ + ^ 2 y=:T, or T 2 T + a 2 x+b 2 y = 0, Sec. which might be put in the forms p l +a 1 x + b l y=0, P 2 +a< 2 x + b 2 y=0, &c. The nature of the terms which constitute such equations of condition will be subsequently shown (art. 450.). It is evident that if no errors existed it would be sufficient to have only as many equations as there are unknown quan- tities ; but, since errors do exist, the degree of correctness in the results will be greater as the number of equations is in- creased ; and it is now to be shown how any number of such equations may be most advantageously combined. 329. A method at present much in use among astronomers consists, after having, by the necessary transpositions, ren- dered the coefficients of one of the two unknown quantities, as x, positive in all the equations, in arranging the equations in two groups, one of them containing all those in which the coefficients of x are the highest numbers, and the other all those in which the coefficients of x are the lowest numbers. The equations in each group are then added together, and those which are formed of the two sums are divided by the coefficients of y in each respectively ; thus y has unity for a coefficient in both equations : then, if the signs of that quantity be alike, on subtracting that in which the coefficient of x is CHAP. XV. OF EQUATIONS. 259 the least from the other, y will be eliminated, and from the resulting equation the value of x may be found : if the signs of y be unlike, the two equations must be added together. The coefficient of x in the final equation being the greatest possible, the value of that quantity becomes the least that is consistent with the conditions, and therefore contains the smallest amount of error. If x had been determined from an ordinary mean of all the equations, its coefficients being positive in some equations and negative in others, they would, in part, have compensated each other, and thus have rendered the final coefficient less than it becomes when the process above described is employed. The value of x, found in the manner above mentioned, being substituted in the equations for y, and a mean of the resulting values of y taken ; the de- termination of this quantity will also, since it is deduced from the most correct value of x, be obtained with the least possible amount of error. This method is sometimes modified by transposing the quantities so as to render the coefficients of y positive in all the equations, and afterwards selecting from the whole those in which x has the highest positive and the highest negative coefficients : of these selected equations two groups are formed, one containing the positive coefficients of x and the other the negative coefficients of the same quantity. Then taking the sums of the two groups, and dividing each sum by the coefficients of y in it ; on subtracting one result from the other, y will be eliminated, and the coefficient of x in this final equation being comparatively great, the value of x will be more free from error than if it had been obtained from an ordinary mean of all. The value of y may then be found as before. If the equations be of the form T? + ax + by + cz = Q (con- taining three unknown quantities) they may be divided into two sets, in one of which all the coefficients of one of the unknown quantities, as x, are positive, and in the other all are negative, and of each of these sets there may be formed two groups, one of them containing those equations in which y, for example, has the greatest coefficients, and the other those in which it has the least ; thus there will be formed four groups : then, having added together the equations in each, and divided each sum by the coefficient of z in it, there will be four equations of the forms S 2 260 METHOD OF CHAP. XV. Subtracting the third of these from the first, and the fourth from the second, z will be eliminated, and there will arise two equations of the forms from which, y being eliminated, x may be found in the usual way ; and the coefficient of x being comparatively great, the value of that quantity will be determined with considerable accuracy. Let this value be substituted in the four equations above, and the resulting equations be formed into two pairs, in one of which the coefficients of y are the greatest, and in the other the least : then, subtracting the sum of the latter pair from that of the former, z will be eliminated ; and, in the resulting equation, y having a coefficient which is compara- tively great, it will be determined with considerable accuracy. With these values of x and y, that of z, found in the usual way, will also be advantageously determined. 330. A different, and, in some cases, a more accurate method of determining the most probable values of the unknown quantities, is that of least squares as it is called, which may be thus explained. In consequence of errors presumed to exist in the observa- tions, let E!, E 2 , &c. be put in place of zero in the second members of the equations of condition :=:0, &c. E 19 E 2 , &c. representing errors. Then imagining each equa- tion to be squared and all to be added together, the sum of all the first members would be the sum of the squares of the errors, and there would be obtained an equation of the form Since the degree of precision in the result obtained from a given number of observations is inversely proportional to v/ ^.E 2 (art. 324.), it follows that the precision will be the greatest when the sum of the squares of the errors is a minimum ; and in order to find the values of x, y, and z 9 con- sistently with this condition, the differentials of the first member taken relatively to x, y, and z must, by the theory of maxima and minima, be made separately equal to zero. Now, on so differentiating we should get z)a 1 +(P 2 + a< 2 x + b q y + c 2 z)a 2 + &c. =0 (p 2 + a< 2 x + b 2 y + C 2 z)c 2 -f &c.~ 0, CHAP. XV. LEAST SQUARES, 261 And from these equations the values of x 9 y, and z might be found by the usual algebraic process. It is manifest that, in the first of the three equations, the coefficients of x 9 y, and z must consist of the sums of the products of the several coefficients of x, y, and z in the original equations of condition when, in each separately, every term is multiplied by the coefficient of x in that equa- tion, and that the coefficient of x must be the sum of the squares of those coefficients : in the second, the coefficients of x } y, and z must consist of the sums of the products of the several coefficients of x, y, and z in the original equations when, in each separately, every term is multiplied by the coefficient of y ; and, in the third, the coefficients of x, y, and z must consist of the sums of the products of the several coefficients when, in each equation separately, every term is multiplied by the coefficient of z. Therefore, in applying the method of least squares, the given equations are to be so multiplied : and three equations of the form A + B x will be obtained by taking the sum of all those which were multiplied by the coefficients of x } the sum of all those which were multiplied by the coefficients of y, and the sum of all those which were multiplied by the coefficients of z. From the three equations the values of x, y, and z are to be found; and these will be the most probable values of the elements. Instead of the ordinary process for determining x, y, and z, from these last equations, the following method has been proposed (Gauss, Theoria combinationis &c. Gottingen, 1823. Galloway, Treatise on Probability, 1839, art. 159.). Let the second members of the preceding equations (the differentials of the sum of the squares of the presumed errors) be repre- sented by R 15 R 2 , and R 3 respectively ; and from the equations, by the usual rules of algebra, obtain thr :ee equations of the forms Rj +H 2 R 2 +K 2 R 3 M 3 Z = F 3 + G 3 R! -f H 3 R 2 + K 3 R 3 ; T^ T^ "IT 1 then the numbers represented by , , - will express JV1 j JM () JML 'r the most probable values of x, y, z respectively, and will coincide with those which would have been obtained from the three equations above mentioned: also , , - will G l H 2 K 3 be the weights due to x, y, and z ; and the constant number s 3 262 LEAST SQUARES. CHAP. XV. 0.476936 divided by the square roots of the several weights will give the relative values of the probable errors. In the method of least squares the values of x, y, and z are determined from equations in which the coefficients of those quantities are the sums of the squares of the proper coefficients of the same quantities in the original equations ; and as these squares are necessarily positive, their sum, in each equation, is greater than the coefficient in either of the other terms of the same equation, the latter coefficients being made up of the sums of quantities, some of which are positive and some negative : thus the unknown quantities, being determined by a division in which the divisors are the greatest possible consistently with the conditions of the subject, must contain the least possible amount of error. It should be observed that, in forming the equations of condition, every error which may be considered as constant, such as those which depend upon a false position of the instrument, must be previously corrected, or must have an allowance made for it ; so that, in each equation, equal deviations from truth in excess and defect may be equally probable, or that all the equations which are employed may have equal weights. Should the weights be unequal, the several equations must be reduced to the same standard in this respect (since the degree of precision is proportional to the square root of the weight), by being multiplied into the square root of its proper weight. The inaccuracy which frequently attends the estimate formed of the relative weights of the several equations is the chief cause of im- perfection in every method of determining the most probable values of elements from the equations of condition. CHAP. XVI. USE OF NAUTICAL ASTRONOMY. 263 CHAP. XVI. NAUTICAL ASTRONOMY. PROBLEMS FOR DETERMINING THE GEOGRAPHICAL POSITION OF A SHIP OR STATION, THE LOCAL TIME, AND THE DECLINATION OF THE MAGNETIC NEEDLE. 331. THE determination of the place of a ship at sea by the distance sailed and by the angle which the ship's course makes with the meridian is, from the nature of the means used to ascertain those elements, as well as from the devi- ations produced by currents, by the set of the waters, and other causes, too uncertain to be relied on when the ship has been long out of the sight of land. Even the magnetic needle and the machine for measuring time may fail in giving correctly the indications for which they were pro- vided ; but celestial observations skilfully made will always enable the mariner to find his place on the ocean with the requisite precision, and hence it becomes his duty to make such observations as often as circumstances will permit. The nature of the observations, and the manner of apply- ing to the purposes of navigation the formula deduced from the investigations of general astronomy, or the tabulated results of such formulae, constitute the subjects of that branch of the science which from the circumstances of its application is called nautical. The like observations and reductions are, however, equally requisite for the scientific traveller on land, in order that he may be enabled to fix the geographical positions of the remarkable stations at which he may arrive ; and hence the several problems which follow, though par- ticularly subservient to navigation, must be considered as belonging to a general course of practical astronomy. 332. On the hypothesis that the earth is at rest in the centre of the imaginary sphere of the fixed stars, the planes of the meridian and horizon of an observer, while he remains in one place, may be indicated by two fixed circles in the heavens ; and the diurnal rotation of the earth on its axis may be represented by a general rotation of the heavens from east to west. The sun, moon, and planets may, also, be considered as attached to the surfaces of spheres whose radii are their distances from the earth, and which, like the sphere of the 264 NAUTICAL ASTRONOMY. CHAP. XVI. fixed stars, have movements of rotation. Now, let it be imagined that the observer is on the surface of a small sphere or spheroid representing the earth in the centre of the sphere of the stars : then, if the celestial body which is the subject of observation be not in the plane of the meridian, a spherical triangle can be imagined to be formed by the colatitude of the observer's station, the polar distance of the body and its zenith distance, or the complement of its observed altitude above the horizon : the sides and angles of one such triangle, or of two or more combined, will constitute the data and the things required ; and these last may consequently be found by the rules of spherical trigonometry. Several corrections must however be applied to the observed altitude of a celestial body in order to reduce it to its true value, and these being alike, whatever be the nature of the problem of which the altitude is one of the data, it will be convenient to describe them before the particular problems are enunciated. 333. The octant, sextant, or circle may have an index error arising from the mirrors not being parallel to one another when the index of the vernier is at the zero of the graduations. At sea, the eye of the observer is above the plane of the sensible horizon by a quantity which depends upon the height of the ship; the refraction of light in the atmosphere causes the celestial body to appear too high, and finally, the effect of parallax is to make it appear too low. The index error of the instrument having been previously found (art. 136.) must be applied to the observed altitude as a correction, by addition or subtraction, as the case may be : the dip, or angular depression of the horizon, if the observa- tion be made at sea (art. 165.), must be taken from the tables and subtracted from the observed altitude ; or if an artificial horizon be employed, half the altitude, after correcting the index error, must be taken. The refraction may be had from the proper tables (art. 145.) ; and the horizontal parallax may be taken from the Nautical Almanac, but this term must be multiplied by the cosine of the observed altitude in order to reduce it to the value of the parallax for that altitude (art. 154.). It may be stated, however, that the parallaxes of the sun or of a planet are seldom used in reducing ordinary ob- servations, and a fixed star has no sensible parallax. The refraction is greater than the parallax, for the sun or a planet, but the parallax of the moon exceeds her refraction : hence the difference between the corrections for parallax and re- fraction must be subtracted from the apparent altitude of the sun or of a planet, and added to that of the moon ; and as it is proper to observe the altitude of the upper or lower limb of CHAP. XVI. CORRECTIONS OF ALTITUDES. 265 the sun or moon, the correction thus applied gives the true altitude of the observed limb. The angular semidiameter of the celestial body must then be added to, or subtracted from the altitude of the limb, according as the lower or upper has been observed, and this may be found in the Nautical Al- manac ; but if the celestial body be the moon, the augment- ation of her semidiameter on account of her altitude (art. 163.) must be added to that value of the semidiameter which is taken from the Almanac ; and thus the correct altitude of the centre of the luminary is obtained. The following examples will illustrate the process of cor- recting the observed altitudes of the sun and moon preparatory to the employment of such altitudes or the zenith distances in any geographical or nautical problem. Ex. 1. Sept. 30. 1842, at Sandhurst, there was observed, by reflexion from mercury, the double altitude of the sun's upper limb, which was found to be - 60 40' 40" Index error of the sextant (subtractive) - 8 40 2) 60 32 Observed altitude of the upper limb - 30 16 Refraction from tables - - - 1' 39".4j ,. ff Sun's parallax in altitude from tables - 7.4 J Correct altitude of the sun's upper limb - - - 30 14 28 Sun's semidiameter (Nautical Almanac) - 16 .0 True altitude of the sun's centre - - 29 58 28 Ex. 2. Nov. 29. 1843, at Sandhurst, at 6h. 39min. P.M. (Greenwich mean time) by reflexion from mercury, the double altitude of the moon's lower limb when on the meridian was found to be - 75 33' 30" Index error of the sextant (subtractive) - 42 2) 75 32 48 Altitude of the lower limb - - - - 37 46 24 Refraction from tables - - - - 1 14*8 37 45 9.2 Determination of moon's augmented semidiam. (Art. 163.) Moon's semidiam. (Naut. Alm.)=14 / 57". Therefore the altitude of the moon's centre is (nearly) 38. Log. sin. alt. moon's centre (nearly) - - 9.7893 Augmentation in the zenith - = 16"., log. = 1.2041 Augmentation - - 9".8= 0.9934 Add - - - 1457 Augmented semidiam. - -1568- - 15 6.8 Altitude of the moon's centre - - - - 38 16 266 NAUTICAL ASTRONOMY. CHAP. XVI- Determination of the moon's parallax in altitude. (Art. 154.) Moon's horizontal parallax at 6 ho. 36 min. (Naut. Aim.) 54' 51" or 3291" - log. = 3.517328 Cosine moon's altitude (38 0' 16") log. = 9.896527 Moon's parallax in altitude - - log. = 3.413855 = 2593. // 3, or 43' 13".3 43 13.3 Altitude, as above - 38 16 True altitude of the moon's centre - - 38 43 29.3 The manner of obtaining the declination of the sun or moon for a given time and place, from the value of that element in the Nautical Almanac, has been explained (art. 316.) ; and, as an example, let it be required to find the sun's declination for a station whose longitude from Greenwich is 50 degrees or 3 ho. 20' (= 3.33 ho.) in time, westward, at 4 ho. 15 X (= 4.25 ho.) P.M., apparent time at the station on the 4th day of May 1843. May 4. Declin., apparent noon, Greenw. (Naut. Aim.) = 15 5V 32".5 N. (Hourly variation = -f 43".37, Naut. Aim.) Variat. declin. for 4.25 hours diff. of longitude - = -f 3 4.32 Variat. declin. for. 3.33 ho. since noon - - = -f 2 24.57 Sun's declin. at 4 ho. 15' P.M., at the station - = 15 57 1.39 N. If the variations in the Almanac diminish from one hour to the next, that which is taken must be considered as negative , and, in applying the variation for the difference of longitude, it must* be observed that at a given physical instant, the hour at a place eastward of Greenwich is later in the day than the hour at Greenwich. Also, in applying the variation for a given number of hours after noon, that variation must be subtracted if the declination be decreasing ; while, for a given number of hours before noon, the variation must be sub- tracted if the decimation be increasing. PROS. I. 334. To find the latitude of a ship or station by means of an observed altitude of the sun when on the meridian. Let PZSH be part of the meridian in the heavens, c the centre of the earth, z the zenith of the station, p the pole of the equator, and E the intersection of the equator with the meridian. Then the correct altitude of the sun's centre being found as in Ex. 1. above, let it be represented by SH, its complement is zs the zenith distance. The sun's declination must be obtained from the Nautical Almanac, for the time of the observation, by an estimate of the distance of the ship, CHAP. XVI. LATITUDE, BY MEKIDIONAL ALTITUDES. 267 or station, in longitude from Greenwich ; let this be repre- sented by SE (the sun being supposed to have north declina- tion) : then it is evident that the sum of the arcs z s and s E will give ZE, the required latitude. If the sun's declination had been south, as E s', it must have been subtracted from the zenith distance zs' in order to give the latitude. If the earth were a sphere, this would be the geocentric latitude ; if a spheroid, it would express the angle between a normal at the station and the equator (art. 151.), and the correction, art. 152., may, if necessary, be applied in order to reduce it to the geocentric latitude. Ex. 1. At Sandhurst, July 28. 1843, by reflexion from mercury, the double altitude of the sun's upper limb was found to be - 116 5' 15" Index error of the sextant (subtr active) - 1 15 2) 116 4 Observed altitude of the sun's upper limb - - 58 2 The corrections on account of refraction, the sun's parallax in altitude and his semidiameter, found as above-men- tioned (art. 333.), amount to- - - - 1619 Correct altitude of the sun's centre - - - 57 45 41 90 Zenith distance of sun's centre - - - 32 14 19 To determine the sun's declination, the longitude of Sand- hurst from Greenwich being about three minutes (in time) westward. Sun's declination at Greenwich apparent noon, from the Naut. Almanac is - 19 6' 12 '.1 N., and the hourly variation is 35" : hence the variation of declination for 3 minutes westward is - - - - 1 . 75 Sun's declination at the time and place of observation (nearly) - -19 6 11 N. -19 6 11 Latitude of the station - - - 51 20 30 In the above example, and also in those which follow, the correction on account of refraction is supposed to have been taken from a table of refractions : the sun's parallax in alti- tude may also be obtained from a table, or by multiplying the horizontal parallax (Naut. Aim., p. 266.) by the cosine of the observed altitude. By a process exactly similar to that which is employed for the sun may the latitude of a station or ship be computed from a meridian altitude of the moon, a planet, or a fixed star. A fixed star has no sensible parallax, and that of Jupiter scarcely exceeds two seconds when greatest, but the horizontal parallax of Mars may amount to about nineteen seconds, and that of the moon to above sixty- 268 NAUTICAL ASTRONOMY. CHAP. XVI. one minutes ; therefore, when the altitude of either of these last celestial bodies is employed in a problem of practical astronomy, the parallax in altitude, or the product of the horizontal parallax and the cosine of the altitude, is to be sub- tracted from the altitude in order to reduce it to that which would have been obtained from an observation at the centre of the earth. Ex. 2. To find the latitude of a station by a meridional altitude of a planet. At Sandhurst, Nov. 27. 1843, at about 5 P.M., by reflexion from mercury, the double altitude of Mars was found to be - 44 IV 20"' Index error of the sextant (subtractive) - 42 2) 44 10 38 Loo-, hor. parallax of Mars (= 6".6 Naut. Altitude = 22 5 19 Aim.) - - 0.819 Log. cos. alt. of Mars (= 22) - - 9.967 Parallax in altitude - = 6" - 0.786 Eefraction - 2 22.6 2 16.6 diff. = - - - 2 16.6 Correct altitude of Mars - - 22 3 2.4 90 Zenith distance - - 67 56 57.6 Declination of Mars at 5 P.M. (Naut. Aim.) - 16 36 27(S) Latitude of the station - - - -512030.6 Ex. 3. To find the latitude of a station from an observed altitude of the moon when on the meridian. Nov. 29. 1843, at 6 ho. 36' mean time at the station, the correct al- titude of the moon's centre when on the meridian was (Ex. 2., art. 333.) 38 43' 29."3. Therefore the zenith distance = - 51 16' 30/'7 Determination of the moon's declination at 6 ho. 39', the Greenwich mean time of the observation. Moon's declin. at 6 ho. P.M. (Naut. Aim.) = 3' 43".2 (S.) Variation for 10 min. = 120".03 ; conse- quently the variation for 39 min. = - -j- 7 48 .1 + 4 4 .9 (N.) 4 4.9(N) Latitude of the station - - 51 20 35.6 335. Since a method of determining the latitude of a ship without taking the altitude of a celestial body may have some use when great accuracy is not required, it will be proper to mention here that an approximation may be made to a know- ledge of that element by observing with a watch the time in which the diameter of the sun ascends above, or descends below the horizon. Let this be done : and, in the annexed CHAP. XVI. LATITUDE, BY MEKIDIONAL ALTITUDES. 269 diagram, let A and B represent the places of the sun's centre when its upper edge and lower edge touch the horizon, as at a and b, in descending, for example ; and let c be the place of the sun's centre when in the horizon. The lines A, B, are semi-diameters of the sun, passing through the points of contact ; therefore perpendicular to the horizon, and passing through z, the zenith of the observer, if pro- duced. The right angled triangles A a C, B b c, are equal to each other, and on the parallel AB of declination, the arcs AC, Ad (the latter equal to A a) have to one another the same ratio as the angles ArC, Afd (p being the pole of the equator) ; that is the same as half the time in which the diameter descends has to half the time in which the diameter would pass, by the diurnal movement, over the meridian, or over any horary circle. Now, in the triangle A a c considered as plane, AC : Aa :: rad. : sin. CA; that is, the time of the semidiameter descending, is to the time of its transit over the meridian (Naut. Aim., p. I. of the month), as radius is to sin. aCA. But, in the spherical tri- angle c E t, formed by the equator E t, the horizon c E and the hour circle P C, in which triangle C t denotes the sun's decli- nation, and the angle E c t is the complement of a c A, we have (art. 60. (/)) Rad. cos. c E t = cos. c t sin. E c t. Thus there may be found the angle CE, which is measured by P z, the colatitude of the ship : the required latitude is therefore found. In this problem no attention has been paid to the effects of refraction. 336. In the arctic or antarctic regions, when the sun, the moon, or a star has considerable declination towards the ele- vated pole, it is visible at its culmination below as well as above the pole ; and in this case the latitude of a station or ship can be obtained by the meridional alti- tudes in both situations. Thus the benefit of having two observations is gained, and the knowledge of the declination can be dispensed with. Let the primitive circle QR be the horizon, z the zenith, and P the pole of the equator ; also let A and B be the places of any celestial body at the times of culmina- tion above and below the pole, the observed altitudes having 270 NAUTICAL ASTRONOMY. CHAP. XVI. been corrected on account of parallax and refraction : then, the polar distances PA and PB being equal, we have AZ + ZP = ZB ZP; whence 2zp = ZB ZA: that is, the colatitude ZP is equal to half the difference be- tween the two zenith distances, or between the two altitudes. On land, in the tropical regions, the chief difficulty attend- ing the determination of the latitude by meridional observ- ations with an artificial horizon, arises from the sun, moon, and planets being, at the time, very near the zenith; on which account the ordinary reflecting sextants or circles, from the great obliquity of the mirrors to each other, cannot be used to take angles equal to twice the altitude of the celestial body above the horizon: also the sun, moon, and stars then change their altitudes very slowly, so that it is difficult to ascertain the moment when they are in the meridian. In this case, recourse must be had to observations taken at times considerably before or after the time of cul- mination. Observations taken at such times are, in fact, most generally employed in all climates, as many circum- stances, particularly a cloudy sky, may prevent the celestial body from being observed on the meridian : but before the formulae relating to such observations are investigated, it will be proper to give the problems for determining at any instant the hour of the day or night at the station or ship. PROB. II. 337. To find the hour of the day by an observed altitude of the sun ; the latitude of the station, and the sun's declina- tion being known. N. B. It is to be understood that the time of the day should be previously estimated within an hour of the truth, and also that the longitude of the station is known or can be ascertained within a few degrees, in order that the sun's declination may be found for the time, from the Nautical Almanac, with sufficient precision. Let the diagram represent a projection of the sphere on the plane of the horizon of the sta- tion ; then z, the centre, will be the zenith, and a diameter as PZH will represent the meridian. The altitude of the upper or lower limb of the sun being observed, and the corrections made by which the altitude of the sun's centre is obtained, the com- plement of that altitude may be ex- pressed by the arc z s in the sphere. CHAP. XVI. TIME FOUND BY ALTITUDES. 271 The colatitude of the station is represented by the arc ZP of the meridian, and the arc PS represents the sun's north, or south polar distance, which is either the difference or sum of 90 degrees and his declination, according as the latitude of the station and the declination are on the same, or on oppo- site sides of the equator. Then, in the spherical triangle ZPS the three sides are given ; and the hour angle ZPS may be found by either of the formulae (i.), (n.), (ni.), art - 66. Suppose the first ; then . , sin. (iP PS) sin. (P PZ) , , . sin 2 i z p s = - ^ ; -- -. - - .... (A), in sin. P s sin. P z which P represents the perimeter of the triangle. The value of ZPS is thus obtained in degrees, and being divided by 15, the result expresses the number of solar hours between the time of the observation and apparent noon : hence the apparent time of day is found, and the equation of time being applied, according to its sign, there is obtained the required mean time of the observation ; which being com- pared with the time indicated by the watch when the observ- ation was made, the difference, if there be any, will be the error of the observation or of the watch. But the above formula is not that which is usually found in treatises of Navigation ; and the formula which is given in the " Tables requisite to be used with the Nautical Almanac," is derived from it in the following manner : Since P = PS + PZ + ZS, and considering PS as equal to PD, the sun's polar distance at noon, Jp PS = ^(ZS ZD) and |p PZ = (ZS + ZD); therefore 1 z P s = sb ' K zs ~ zp ) sin. KZS + ZD) 2 sin. P s sin. p z \ / sin. Again (PL Trigon., arts. 36, 41.), sin. 2 JZPS = Jvers. sin. ZPS, and cos. z D cos. z s = 2 sin. |-(z s z D) sin. ^(z s + z D) ; therefore cos. z D cos. z s vers. sin. z p s = ~. = sin. P s sm. P z (cos. z D cos. z s) cosec. PS cosec. p z . . . . (c) The value of this last member being obtained from the data and then sought in the table called Log Rising (Requi- site Tables, tab. xvi.), the value of z p s in solar time is found by inspection. The formula is not strictly correct, since in 272 NAUTICAL ASTKONOMY. CHAP. XVI. considering P s as equal to P D, the sun's declination at the time of the observation is supposed to be equal to his declina- tion at noon, which is not the case ; the error is, however, very small. The formula (B) is also given in treatises of navigation. cos. z s cos. P s cos. P z The expression cos. z p s - -. ; cor- sin. P s sin. p z responding to (a), (), or (c) in art. 60., may be put in the form COS. Z S COS. P S COS. P Z COS. Z P S = -a : : i - ; sin. P s sin. p z sin. p s sm. P z .,, 1 -, .COS. PS COS. PZ therefore, if the values of -s -. - and of -. -. sm. PS sm. PZ sm. PS sin. PZ were computed and arranged in two tables having for their arguments given values of P S and p z or their complements ; the values of those fractions might be obtained by inspection : then, the former being multiplied by the natural cosine of zs, and the latter taken from the product, the result would be the natural cosine of the required hour angle. The " Spherical Traverse Tables " in Raper's Navigation are of the kind here alluded to. 338. It is evident that either of the formulae (i), (n), and (m), in art. 66., may be used for determining the value of the hour angle ZPS in degrees when the altitude of a fixed star or a planet has been observed ; and that, subsequently, the hour of the night can be determined with nearly the same accuracy as the hour of the day is found from an observed altitude of the sun. If the celestial body be a fixed star, the angle ZPS when divided by 15, expresses, in sidereal time, the hours between the time of the observation and that at which the star cul- minates, or comes to the meridian of the station : this interval being, therefore, converted into solar time, either by sub- tracting the acceleration or by means of the table of " time equivalents," and added to, or subtracted from the time of culminating, according as the star is on the western, or on the eastern side of the meridian, the sum or difference will express the hour of the night in apparent time ; and by applying the equation of time, with its proper sign, there will be obtained the mean time of the observation. When the altitude of a planet is observed, the angle ZPS being divided by 15, must be considered as expressed in sidereal time ; then the variation of the planet's right ascen- sion for that time being found from the table of the geocentric CHAP. XVI. TIME FOUND BY AN ALTITUDE. 273 right ascensions in the Nautical Almanac, it must be added to, or subtracted from the same time according as the right ascensions are increasing or decreasing (art. 315.). The sum or difference being converted into solar time by subtracting the acceleration, must be added to or subtracted from the apparent time at which the planet culminates, according as the latter is westward or eastward of the meridian ; and the result will be the apparent time of the observation. Ex. 1. July 15. 1843, at 9 ho. 23m. 39 sec. by a watch, the double alti- tude of the sun's upper limb was observed by reflexion from mercury to be 94 57' 20"; the index error of the sextant was 7min. subtractive, the colatitude of the station was 38 39' 27" N. and its approximate longitude from Greenwich, 3' in time, westward ; it is required to find the correct time of the observation and the error of the watch. On applying as before the corrections for the index error, refraction, the sun's parallax in altitude and its semidiameter, it is found that the true altitude of the sun's centre is Consequently its true zenith distance is - - The presumed mean time at Greenwich being 9 ho. 26' 39" Equation of time (subtractive) - - 5 32 The apparent time at Greenwich = 9 21 7 Time before apparent noon (Gr.) - 2 38 53 Then : sun's declin. at 'apparent noon Gr. 21 36 / 52".4 N. Variat. of declin. for 2 ho. 38 m. 53 sec. (add) 1 3.1 Sun's declin. at time of observation 47 42 51 36".7 23 .3 21 90 37 55 .5 N". Sun's north polar distance - - 68 22 4 .5 To find the hour angle from the above formula (A), o ^ (i) art. 66. zs = 42 51 23.3 PS = 68 22 4.5 PZ = 38 39 27 log. sin. co-ar. 0.0317183 log. sin co.-ar. 0.2043537 Perim. 149 52 55 i p = 74 56 27.5 FS = 6 34 23 PZ = 3 17 0.5 log. sin. i ZPS = 19 58' 17' log. sin. log. sin. - 9.0586916 - 9.7721593 19.0669229 - 9.5334614 ZPS = 39 56 34 2 39 46 = ZPS in time. 12 9 20 14 = apparent time of the observation. 5 32 = equation of time (add). 9 25 46 = mean time of the observation. 9 23 39 = ditto by the watch. 2 7 = error of the watch (too slow). T 274 NAUTICAL ASTRONOMY. CHAP. XVI. The same example worked by formula (c). psorpD = 68 23' 7". 6 PZ = 38 39 27 ZD =29 43 40.6Nat.cos.=86839(therad.ofthetables=100000) zs =42 51 23.3 Nat. cos.=73304 (do.) diff. = 13535 log. = 4.13146 log. cosec. PS, = 10.03165 log. cosec. PZ, = 10.20434 vers. sin. ZPS (in time) or log-rising = 4.36745 Therefore from the table designated log-rising, ZPS, in time, = 2 ho. 39 m. 42 sec. ; and hence the apparent time of the observation is 9 ho. 20' 1 8 ." By the formula (A) the apparent time was found to be 9 ho. 20' 14." Ex. 2. August 1. 1843, at 10 ho. 37 min. 7 sec. by the watch, there was observed by reflexion from mercury, the double altitude of the star a Andromedse near the prime vertical, = 67 3' 40", the index error of the sextant being 6' 30" subtractive. After making the corrections for the index error and for refraction, the true altitude was found to be 33 27' 7" ; and the star's zenith distance, 56 32' 53". The declination of the star, from the Naut. Aim., = 28 13' 45" N. ; therefore the north polar distance = 61 46' 15" The co- latitude of the station is 38 39' 27". Then by the formula (in) in art. 66. z s = 56 32' 53" us = 61 46 15 PZ= 38 39 27 2) 156 58 35 p = 78 29 17.5 log. sin. co-ar. 0.0088256 P zs = 21 56 24.5 log. sin. co-ar. 0.4275520 p ps=16 43 2.5 log. sin. 9.4588620 P p Z = 39 49 50.5 lo. sin. 9.8065322 . 2) 19.7017718 log. tan., 9.8508859, 35 21' 5" (= i ZPS) 2 15) 70 42 10 (= ZPS) 4 42 48.7 do. in sid. time- (By the table of time equivalents) = 4 42 2.4 in solar time. h- / // Eight asc. star =00 20.7 (Naut. Aim.) Right asc. sun at noon = 8 43 51.4 (do.) 15 16 29.3 Approximate time of the star culminating. 2 27.5 Variat. sun's right asc. in 15 ho. 16m. 29 sec. 15 14 1.8 Time of culminating (apparent). 4 42 2.4 The star's hour angle as above. 10 31 59.4 Apparent time of the observation. 6 1.7 Equation of time (add). 10 38 1.1 Mean time of the observation. 10 37 7 Do. by watch. 54.1 Error of the watch (too slow.) CHAP. XVI. TIME FOUND BY AN ALTITUDE. 275 The star's hour angle in sidereal time may be found by formula (c) as that of the sun was found in the preceding example. Ex. 3. April 27. 1844, at 8 ho. 56' 35" P.M. by the watch, there was ob- served by reflexion from mercury the double altitude of the centre of Venus, the colatitude and the approximate longitude of the station being as in Exs. 1,2.; consequently the approximate Greenwich mean time of the observation was 9 hours nearly. o / // Double altitude of the planet's centre by observation - 41 40 30 Index error (additive) - - _ 30 2) 41 41 Altitude of the planet's centre - 20 50 30 Refraction - - - 2 3L2 Diff. ref.l Q Hor. parallax Venus and par. j~ (N. A.) 10"1, log. = 1.0043 Cosine alt. Venus, log. = 9.9706 Correct alt. 20 48 8.2 Par. in alt. Venus, 9".4 - 0.9749 9.4 90 2 21.8 Zen. dist. 69 11 51.8 (=zs). Geocentric declin., Venus at Gr. mean noon (N. A.) - 26 4 10.2 Increase of declination in 24 hours = 5' 38" Increase of declination in 9 hours = (^ of 5' 38" j 2 5 Declination at the observation - - - 26 6 15.2 90 North polar distance, Venus - - - 63 53 44.8 (=PS). Colatitude of the station = 38 39' 27" (= PZ). By either of the formulae (i), (11) or (in) art. 66., with zs, PS, and PZ, we have the star's horary angle ZPS, equal to 88 49' 12" ; or, in sidereal time - - - - - - 5h.55' 17" Increase of the planet's rt. asc. in 24 h. = 4 / 42."13 (N. A.) Increase in 5h. 55' 17" ( 5 b> ^ l7 " (4'42".13)) - 1 10.2 Planet's horary angle at the observation, in sid. time - 5 56 27.2 Do. in solar time (by the table of time equiv.) - - 5 55 28.7 Geocentric right asc. Venus at apparent h. , /t noon at the station (N. A.) - 5 23 34.35 Right asc. sun at the same time - - 2 19 47.19 Approximate time of Venus culminating - 3 3 47.16 Increase of sun's r. asc. in 3 ho. 3' 47"=28"9 Increase of Venus's right asc. in ditto =34. 7 Excess of planet's right asc. above that of the sun - - - 5.8 (add) 5.8 Apparent time of culminating - - 3 3 52.96 Horary angle above - - - 5 55 28.7 Apparent hour of the night - - 8 59 21.66 Equation of time (subtract) - 3 32.8 Mean time of night '- - 85548.86 Do. by the watch - 8 56 35. Watch too fast - 46.14 276 NAUTICAL ASTRONOMY. CHAP. XVI. 339. It is sometimes advantageous to ascertain the value of a small error or variation in the time of an observation, arising from a small error or variation in the observed altitude of a celestial body ; and conversely, to ascertain the variation in the altitude, consequent on a small variation in the time. A formula for this purpose may be investigated in the follow- ing manner. PBOB. III. Having (fig. to Prob. 2.) the colatitude PZ of a station and the polar distance P s of a celestial body with the observed altitude of the latter (the complement of zs) and the azi- muthal angle P z s, either by computation or observation ; to find a relation between a small variation in the altitude and the corresponding variation of the hour angle. On substituting z, s, and P, for A, B, and c respectively in the equation preceding (a) in art. 60., and transposing, we have in the triangle ZPS, cos. zs =r cos. ZPS sin. PZ sin. PS -f cos. PZ cos. PS ; and differentiating, PZ and PS being constant, sin. zse?zs = sin. zpsdzps sin. PZsin. PS: sin. z P &sin. p z sin. P s sin. zs or again, since (art. 61.) sin. ZPS : sin. zs :: sin. PZS : sin. PS ; , . sin. ZPS sin. PZS that is, ; -- == - . , sin. z s sin. p s we have , , sin. PZS sin, PZsin. PS rfzs =rfZPS- , rfzps sin. pzssm.pz; SID. P o whence dzps = . ** ___ , or SPS' = - sm. pzssm. PZ sm.pzssm.pz That is, if the distance of the celestial body from the zenith were increased or diminished by a small quantity as ss', which should not exceed 20 or 30 minutes of a degree, the hour angle would be increased or diminished by this value of SPS'; or the time from noon would be increased or diminished by ^-Jj-^ . This fraction is evidently a minimum when sin. pzs is equal to radius, or when PZS, CHAP. XVL- AZIMUTHS. 277 the azimuth, is a right angle ; and consequently, when the time of day or night is to be found from an observed altitude of a celestial body, the most favourable position of the body is on the prime vertical. If the altitude be observed when the celestial body is in, or near that situation, so that PZS may without sensible error be considered as a right angle, the value of SPS' becomes ss' sinTpz' Hence in determining the time of day or night from a series of altitudes observed near the prime vertical, after the horary angle has been computed from one of the observed altitudes, the angles corresponding to the other observed altitudes may be readily obtained, and a mean of these will express the angle very near the truth. 340. The needle of a compass preserving, within the extent of a few miles about a given point on the earth's surface, a nearly parallel direction, it is frequently employed for deter- mining the relative positions of objects in a topographical survey, and the seaman trusts to it for his guidance on the ocean. The determination of its position with respect to the geographical meridian of a station or ship, is therefore an. object of the utmost importance, and should be made by means of astronomical observations as often as circumstances will permit. The process for determining the variation, or, as it is called, the declination of the needle by an observed altitude of the sun, a fixed star, or a planet is similar to that which has been explained in the problem for finding the error of a watch by a like observation ; the azimuthal, instead of the horary angle being computed. PROB. IV. By an observed altitude of the sun, a fixed star, or a planet, to determine its azimuth, and the variation, or decli- nation, of the needle ; the latitude of the station being given together with an approximate knowledge of its longitude and of the time. Let the primitive circle ANT represent the horizon; z its centre, the zenith ; N p z the direction of the meridian, and p the pole of the equator. Then, s representing the place of the sun, we have in the spherical triangle PZS, the colatitude PZ of the station, the distance PS of the sun from the pole, or T 3 278 NAUTICAL ASTRONOMY. CHAP. XVI. the complement of its declination, and the distance zs of the sun from the zenith ; this last being obtained from the observed altitude after having made the ne- cessary corrections as in the former problem : with these data the azi- muthal angle pzs may be found by either of the formula? (i), (n), or (ill), in art. 66. Suppose the last ; then p being the perimeter, fractions Or, as in Prob. 2., the values of the and cos. ZP cos. zs being arranged in sm. ZP sin. zs sm. ZP sin. zs tables having for arguments ZP and zs or their complements, those values may be taken by inspection : then the tabular value of the first fraction being multiplied by the natural cosine of PS (attention being paid to the sign of the cosine, which is negative when PS is greater than a quadrant), and the tabular value of the second fraction being subtracted from the product ; the result will be the natural cosine of the re- quired azimuthal angle PZS. Now at the moment when the altitude of the sun or star was taken, the alidad AB of the compass, or the line of the sights, being directed to the centre s of the celestial body, and nzs being the direction of the needle, the angle TZZS might be read on the rim of the box, or on the card if the latter be fixed to the needle : this is the azimuth, or bearing, of the sun from the magnetic meridian. Therefore if the angle SZP be made equal to the computed azimuth, PZ will repre- sent the direction of the terrestrial meridian, and WZP, the difference between the angles, will express the required vari- ation or declination of the needle. Ex. Sept. 30. 1842, at 9 ho. 41 m. 45 sec. A.M. by the watch, the co- latitude of the station being 38 39' 27", the double altitude of the sun's upper limb, as observed by reflexion from mercury, was found to be 60 40' 40" ; the index error of the sextant being 8' 40" subtractive : also the azimuth of the sun's centre observed with the compass was S. 12 15' E. After making the corrections for the index error, refraction, the sun's parallax in altitude and his semi-diameter, the true altitude of the sun's centre was found to be 29 58' 28" ; consequently the true zenith distance = 60 V 32". CHAP. XVI. DECLINATION OF THE NEEDLE. 279 The sun's declination, app. noon Greenwich - Variation for 2i hours before noon Sun's declination at the time of observation - Sun's north-polar distance PS 92 42 1.2 zs = 60 1 32 PZ = 38 39 27 2 44 17.4S. 2 16.2 2 42 90 1.2S, - = 92 42 1.2 2) 191 23 0.2 P = 95 41 30 log. sin. co. ar. 0.0021464 | p PS= 2 59 29 log. sin. co. ar. 1.2824470 p zs = 35 39 58 log. sin. 9.7657138 |p PZ= 57 2 3 log. sin. 9.9237595 2) 20.9740667 log. tan. = 10.4870333 71 57' 13" (PZS) Sun's true azimuth, N. E. or, S. E. Sun's azimuth by the compass - 143 54 26 (PZS) -36 5 34 (s'zs) - 12 15 (*zs) Variation, or declination, of the needle (N. W.) - 23 50 34=5 zs 7 or WZP. 341. The true azimuth of a fixed terrestrial object, and from thence the trace of a meridian line on the ground, may be easily obtained by this problem : thus, if M be such an object, the azimuthal angle TZZM being observed with the compass and the value of WZP, the variation, found as above, being subtracted from it (the object being situated as in the figure), the remainder PZM will be the azimuth required. But if the observer be on land, the azimuth may be more correctly determined by means of a good theodolite, or an altitude and azimuth instrument; for having directed the telescope to M with the index of the horizontal circle at zero, let the telescope be turned from that position to the sun and the eastern or west- ern limb of the latter be placed in contact with the vertical wire at the moment that the altitude of the sun's upper or lower limb is observed with a sextant, or with the vertical circle of the same instrument. Then, the horizontal angle M z s between the point M and the sun's centre being obtained, and the altitude, or zenith distance, of the sun's centre found ; also, having computed the azimuthal angle p z s as above, on subtracting from it the angle M z s, the remainder (the positions being as in the diagram) will be the azimuth of the terrestrial object. T 4 280 NAUTICAL ASTRONOMY. CHAP. XVI. If the altitude of the upper or lower limb of the sun be ob- served, that altitude, after being corrected for refraction, &c., must be diminished or increased by the value of the sun's semidiameter in the Nautical Almanac. And if the bearing of either limb of the sun from the terrestrial object be ob- served by means of the horizontal circle, there must evidently be added to or subtracted from this bearing, the angle sub- tended at the zenith by the horizontal semidiameter of the sun, or the corresponding arc of the horizon, in order to have the bearing of the sun's centre. This angle or arc may be found by the rule in art. 70. : thus a denoting the sun's altitude, r the angular measure of the sun's semidiameter (from the Naut. Aim.), and z the required arc on the horizon, or the angle at the zenith, we have z = . cos. a The azimuth of a terrestrial object may also be determined by observing the zenith distance of the sun's centre when near the prime vertical, the zenith distance or altitude of the object, and the angular distance between the centre of the sun and the object : thus there will be obtained the three sides of a spherical triangle, and with them the angle at the zenith may be computed. This angle being added to, or subtracted from the sun's computed azimuth, will give the azimuth of the terrestrial object. It should be remarked that the two zenith distances are not to be corrected on account of refraction nor the sun's zenith distance for parallax, since the observed angular distance is also affected by those causes of error, and the angle at the zenith is the same as it would be if they did not exist. Formulae for determining the difference of latitude and difference of longitude between two stations as z and M, when there are given the distance of one from the other, the latitude of either and the bearing, or azimuthal angle PZM, will be investigated in the chapter on Geodesy (art. 410.). 342. The declination of the needle may also be found by means of an observed amplitude of the sun, that is the arc of the horizon between the place of the sun's centre at rising or setting, and the eastern or western point of the compass card ; the amplitude so observed being compared with the computed amplitude of the sun, from the eastern or western point of the horizon : this last amplitude may be obtained by the following problem. CHAP. XVI. THE SUN'S AMPLITUDE. 281 PROB. V. Having the latitude of the station or ship, the day of the month, and an approximate knowledge of the apparent time of sun-rising or setting ; to find the sun's amplitude. Let WNE represent the horizon of the station, z its centre the zenith ; w and E the east and west points, and w M E the equator, of which let P be the pole ; also let D s be part of a parallel of de- clination passing through s the true place of the sun at rising. Then, imagining the great circle p s T to be drawn ; in the right angled spherical triangle SET we have ST the sun's declination (which may be found from the Nautical Almanac sufficiently near the truth by means of the approximate knowledge of the time of rising or setting) the angle SET, which being measured by the arc MN of the meridian is equal to the colatitude of the station, and the right angle at T; therefore SE, the true amplitude, may be found (art. 60. ( ~* at-a't : "T- From the equation for T' we obtain a value of T + T' the time by the watch when the sun was on the meridian ; and from that for A' A we get a value of A', the meridional alti- tude: consequently, by means of the north-polar distance, the latitude of the station. Ex. At Sandhurst, July 28. 1843, the following observations were made : h' / // o / // 16 20 P.M. by the watch; double alt. sun's upper limb - 115 55 45 17 48p.M. ditto - 115 53 20 24P.M. ditto - 115 47 20: the index error of the sextant being 1 minute subtractive. After making the several corrections for the index error, the refraction, the sun's parallax in altitude and his semidiameter, the three correct altitudes are found to be 57 41' 3" (= A), 57 39' 41".2, 57 36' 50".2: hence t = 88" and tf = 244" ; also a = 81".8 and a! = 252".8. Then t* = 7744, and a't z = 1957683.2 ; also f z = 59536, and at'* 4870044.8. Therefore at* -aft* 2912361.6. Again, a't = 22246.4 and af = 19959.2 ; therefore aft at' = 2287.2. Hence ^-I^ = 1273" ; the half of which is 636".5 (= T'), the time from noon to the instant of the first observation. Then log.T /2 = (636.5) 2 = 5.6069142 log. a (= 81.8) = 1.9127533 log. 2T' t z (=104280) = 4.9817573 co. ar. 2.5014248 = 317".27, or 5' 17. x/ 3 (=A'-A) 57 41 3 (=A) The sun's meridian altitude - 57 46 20.3 (=A X ) The sun's declination (Naut. Aim.) - 19 6 14 Colatitude of the station - 38 40 6.3 354. When from any circumstance, as from a temporary stoppage of the watch, the hour of the day at any station is CHAP. XVI. LATITUDE, BY TWO ALTITUDES. 301 not known, when yet the watch being set going is capable of giving the interval, in time, between the instants of making two observations of the sun's altitude; the latitude of the station may be found in the following manner. PROB. VIII. To find the latitude of a ship or station having the alti- tudes of the sun observed at two instants in one day, with the times of making the observations and an approximate knowledge of the observer's longitude. Let s and s' represent the places of the sun at the times of observation, and, P being the pole of the equator, let the angle s p s' be the hour angle, or the interval in apparent solar time reduced to de- grees: the intersections z or z' of two small circles described about s and s' as poles with distances equal to the complements of the observed altitudes, will be the place of the zenith, and the arc P z or p z' will be the colatitude of the station. The problem is unavoidably ambiguous except when the latitude of the station, or the position of the zenith, is known approximatively. The two observed altitudes must be corrected as usual, and the true zenith distances found ; and the small circle s s' is supposed to be a parallel of declination at a distance from p equal to the sun's polar distance, which must be found from the Nautical Almanac by means of the estimated time at the middle of the interval between the observations ; the varia- tion of the declination between that middle time and the two times of observation being disregarded. Imagine a great circle to pass through s and s', and draw pp to bisect the angle SPS'; that arc will bisect ss' at p and cut it at right angles : let fall also z q perpendicularly on P s ; then by spherical trigonometry, In the right angled triangle p sp ; Bad. cos. PS rrcotan. pps cotan. PS/> (art. 62. (d'J) also (art. 60. (?)), Had. sin. p s =. sin. P s sin. p P s ; and 2 p s =: s s'. In the triangle zss', P being put for the perimeter, we have (art. 66. (in), r. sin. (|P z s) sin. (^p S s') tan. - 4- z s s' = ^. . ' .. : sm. P sin. ( P z S') 302 NAUTICAL ASTRONOMY. CHAP. XVI. Also in the right angled triangle zsq, r. cos. z$q r=cotan. zs tan. sq (art. 62. (/'), and in the triangle p z s, cos. zs cos. P^ cos. s<7 cos. PZ. (3 Cor. art. 60.) But the rule given in some treatises of navigation for work- ing this problem is obtained from the formula (a), (b) or (c\ (art. 60.) for the angles ZPS and z p s', in the following manner cos. z s cos. P s cos. P z cos. z P s -. -. sin. p s sm. P z COS. Z S' COS. P S 7 COS. P Z and cos. z p s' = ; -. -. . sin. P s' sin. p z Subtracting one of these from the other, and observing that p s = P s', we have cos. zs' cos. zs COS. ZPS' COS. Z P S = -: j ; sin. p s sin. p z But (PL Trigonom., art. 41.) the first member is equal to 2 sin. \ (z P S + ZPS') sin. ^ (z p s + z p s'), and dividing both members by 2 sin. \ (z p s + z p s'), that is by 2 sin. s P s', we have cos. zs' cos. zs 1 In the second member of this equation substituting for p z the given approximate value of the colatitude, there will be from thence obtained the half difference or the half sum of the angles ZPS, z p s' according as the observed altitudes were taken on opposite sides, or on the same side of the meridian. In either case, the half sum and half difference of the angles ZPS, ZPS' being known, we have the angle ZPS or z p s'. Now, imagining p z to be produced if necessary, to cut the parallel of declination in m so that Pm PZ, or PS PZ, is equal to zm, we have, as in art. 337., cos. zm cos. zs vers. sm. z P s = ; -. , sin. PS sin. PZ or vers. sin. ZPS sin. PS sin. PZ = cos. zm cos. zs . . . (B) ; or again, vers. sin. ZPS sin. PS sin. PZ + cos. zs = cos. zm . . . (c). But zm is the meridional zenith distance of the sun; there- fore, this term being found, by means of the approximate value of the colatitude, and being subtracted from i y m or PS, the sun's polar distance ; the remainder will be the colatitude of the station. Note, the log. of ; , or of , ?, r , sin. P z sin. p s cos. lat. cos. sun s decl. CHAP. XVI. CORKECTION FOK CHANGE OF PLACE. 303 that is, log. sec. lat. + log. sec. declin., is the term which in the " Requisite Tables " is called log. ratio : also log. ^ -. z sin. g" SP s or log. 2 cosecant of half the hour angle between the places of the sun at the times of observation is the logarithm in the column of half the elapsed time : and from the formula (A), (B), and (c) is obtained the following rule, which is given for the solution of the problem in the tables just mentioned : Let log. sec. lat. + log. sec. declin. + log. (cos. zs' cos. zs) + log. J elapsed time be represented by L, and take the time corresponding to L in the column headed " Middle Time " (Tab. XVI.) ; then, since L is the logarithm of the second member in equation (A), this time will express the half sum, or half difference of the angles ZPS, ZPS'; and hence, as above, the angle ZPS or ZPS' may be found. Next, find the logarithm corresponding to ZPS or ZPS' in the column headed log. rising, that is log. vers. sin. ZPS, and subtract from it the log. ratio ; the result is the logarithm of the first member in equation (B) and is called the logarithm of N. Lastly, from equation (c) we have N 4- nat. sine of sun's altitude at s or s' nat. sine of sun's meridional altitude ; and thus, as above, the latitude or colatitude may be found. 355. In order to show how the latitude of a ship at sea, when computed by means of two altitudes, is to be corrected on account of the movement of the ship during the interval between the observations ; let z be the zenith of the ship and s the place of the sun at the first observation, z' the zenith and s' the place of the sun at the second ob- servation. Then the angle z' z s will represent the bearing of the ship's path, in the interval between the observations, from a vertical circle passing through the sun, and zz' the distance sailed in the same in- terval : the bearing must be obtained by means of the compass, and the distance must be expressed in minutes of a degree, by the log. Imagine z' 't to be let fall perpendicularly on zs; then, unless the distance zz' be very great, an approximation suffi- ciently near the truth may be obtained (in the case indicated by the figure) by subtracting z T! cos. z' z t, the value of z t, from z S, and considering the remainder as equal to z' s. This value of z's must be used with z's' in working out the 304 NAUTICAL ASTRONOMY. CHAP. XVI. problem ; which may then be done as if the ship were sta- tionary. The correction zt must be added to zs in order to give z's if the ship were proceeding from z towards P instead of proceeding towards s', as in the figure. 356. By differentiating the formulae cos. zs = cos. ZPS sin. PZ sin. PS + cos. PZ cos. PS, cos. z's' = &c. (art. 60.), considering zs and z's', ZPS and Z'PS', PZ and PZ' as variables, and comparing together the values of d P z, d P z' (the errors in the latitudes), it is ascertained that, if the lati- tude be determined from the observation made when the sun is nearest to the meridian, that error is less than when the latitude is determined from the observation which is more remote ; and that the difference between the latitudes is greatest when this observation is made near the prime vertical. (Delambre, Astronomic, Ch. XXXVI. Nos. 79, 80.) The same analysis shows that if both the altitudes were observed near the prime vertical, the latitude of the ship being nearly equal to the sun's declination, the error in the computed lati- tude might become very great. In general, it may be stated that, when it is known whether the latitude of the ship or place is greater or less than the sun's declination, the direct method of solving the problem will succeed ; otherwise, the case is ambiguous. For should the sun pass the meridian between the zenith z (fig. to art. 354.) and the elevated pole, as it may at a station within the tropics, the angle z s s' must be added to P s s' in order to have the angle PSZ. Should it pass the meridian between the zenith z and the depressed pole, as in the figure, the dif- ference between the angles zss' and PSS' would give the angle PSZ. In general, also, it may be observed that when both observations are made in the forenoon, or both in the afternoon, the result will be most precise when one of the altitudes is observed near the meridian and the other near the prime vertical : and that when one observation is made before, and the other after noon, both should be made when the sun is near, but not less than one hour from the meridian. The latitude determined from the observations will be, more- over, very inaccurate if the sun pass the meridian within four or five degrees from the zenith ; and the distances run by the ship during the interval between the two observations should not exceed thirty or forty miles, on account of the risk of an erroneous estimate being made of its change of place in longitude. 357. The latitude of a station may also be found by two alti- tudes of the sun or of a star, with the sum or difference of the CHAP. XVI. LATITUDE, BY TWO ALTITUDES. 305 azimuths observed by the compass or otherwise at the times when the altitudes were taken. For, in the above figure, z 7 representing the zenith of the station, let s and s 7 be the true places of the celestial body at the two instants of observation; then the angle sz 7 s' will represent the azimuthal interval between those places, and, after the observed altitudes have been corrected as usual, Z 7 s and z's' may represent the two zenith distances. The arcs p s and p s', which may be considered as equal to one another, denote the polar distance of the celestial body at the middle of the interval between the times of observation : the great circle PZ'M represents the meridian, and PZ' the colatitude of the station. In the spherical triangle S z' s' there are given z 7 s, z' s' and the angle sz 7 s 7 ; to find ss' and the angle z's's. Imagining p<7 to be let fall perpendicularly on ss 7 , the latter arc is bisected in q; and in the right-angled triangle PS'<7 there are given PS 7 and s^; to find the angle PS 7 <7 or PS'S. Lastly, in the triangle PZ 7 s' there are given PS 7 , z's' and the angle P s 7 z ; to find p z 7 , the colatitude required. The latitude may also be found by means of the altitudes of two stars observed at the same instant. For the polar distances PS, PS 7 of the stars being given in the Nautical Almanac, and also the angle SPS 7 , which is equal to the difference between their right ascensions; there are in the triangle p s s 7 sufficient data for finding s s 7 and the angle PS 7 s. Then, in the triangle Z 7 s 7 s there are given the three sides ; and consequently the angle z 7 s 7 s may be computed : lastly, in the triangle PS 7 z 7 , there are given PS 7 , Z 7 s 7 and the angle PS 7 z 7 , to find P z 7 , the colatitude. Ex. At Sandhurst, August 23. 1841, the following observations were made : At 9 ho. 44' 33" A.M. by the watch, the double altitude of the sun's upper limb - - - - - = 83 30 40 and at Oho. 39' 45" P.M. - - - - - = 99 20 the index error of the sextant being 6' 30" subtractive. (Fig. to art. 354.) The interval between the two times of observation is 2 ho. 55' 12"; hence the angle SPS' 43 48', and | SPS', or svp = 21 54'. The two observed altitudes when corrected as in the former examples give for the true altitudes of the sun's centre 41 25' 15".5, and 49 20' 10".4; whence zs = 48 34' 44".5 and zs' = 40 39' 49".6. The middle between the times of observation is 11 ho. 12' 9" A.M. (subject to the error of the watch, supposed to be unknown) ; or, the lon- gitude of the place from Greenwich being 3', in time, westward, the cor- responding mean time at Greenwich is 11 ho. 15' 9", or 44' 51" before the time of mean noon at Greenwich. Then, X 306 NAUTICAL ASTRONOMY. CHAP. XVI. Sun's declinat. at mean noon, Greenwich - 11 26 18.5 N. (Naut. Aim.) Variation for 44' 51", additive - 38.4 Sun's declmat. at the middle time - 11 26 56.9 Hence the north polar distance at the middle time = 78 33' 3".l (=rs or P'S). In the right angled triangle SPJP, -log. cos. PS (78 33' 3") - 9.2977571 -log. cotan. svp (21 54') 10.3957769 = log. cotan. rsp - - 8.9019802 or PSS' = 85 26' 16". log. sin. PS =2.9912709 log. sin. sp/> = 9.5716946 log. sin. sp = 9.5629655 =21 26' 32'' 2 42 53 4 = ss'. In the triangle zss' (P representing the perimeter), zs' = 40 39 49.6 zs = 48 34 44.5 ss' = 42 53 4 p= 132 7 38.1 IP' = 66 3 49 log. co. ar. = 0.0390555 TT |p-zs' = 25 23 59 log. co. ar. = 0.3676128 Hencezss / = 57 33' 10 AP zs = 1729 4 log. =9.4777677 |p ss' = 23 10 45 log. =9.5950535 PSZ =27 53 6 2) 19.4794895 log. tan. zss' (28 46' 35") 9.7397447 In the triangle PSZ, log. cos. PSZ (27 53' 6") 9.9463973 -log. cotan. zs (48 34 44.5) 9.9456033 =log. tan. sq = 45 3' 9" 10.0007940 PS = 78 33 3 po = 33 29 54 log. cos. zs (4 8 34' 44". 5) 9.8205877 + log. cos. P g (33 29 54) 9.9211149 + log. cos. s q (45 3 9)co. ar. 0.1509133 = log. cos. PZ = 38 39' 17" 9.8926159 The required co-latitude. The same example by the indirect method in the " Requisite Tables," as above (art. 354.). Interval between the observations in time = 2 55 12 as above Half interval (= " half elapsed time ") - = 1 27 36 log. half elapsed time (from the tables) = 0.42830 log. secant, lat. by account (suppose 51 30') = 10.20585 \ = 0.21458 +log. secant, sun's declination (11 26' 56") = 10.00873 J log. ratio) Nat. sin. sun's greatest al. (49 20 10) =75854 Do. least (41 25 15)=66158 (as whole numbers) 9696, log.=3.98659 =log. half difference ZPS, ZPS' 4.62947=0 h. 49' 12 //P (designated L, and found in the col. headed " middle time.") Half sum of ZPS, ZPS' (= SPS', in time) - 1 27 36 = Smaller angle, ZPS' in time", or the angle corresponding to the place of the sun when nearest to the meridian Oho. 38 24 Log. Oho. 38' 24" in the column headed "log-rising " = 3.14625 Log. ratio (above) - - = 0.21548 CHAP. XVI. ALTITUDES FOUND. 307 = Log. of the number represented by N - = 2.93077 Whence N - = 853 +Nat. sin. sun's altitude at s' (=49 20' 10") = 75854 =Nat. sin. sun's merid. alt. (=50 5 30) = 76707 Sun's declinat. - =11 26 56 =Colatitude required - =38 38 34 358. In determining the longitude of a station or ship by means of the distance of the moon from the sun or a star, it is necessary to have the altitudes of the two luminaries ; and since the circumstances may be such as to prevent the al- titudes from being observed, the following problem, by which, when the latitude of the station and the time there are known, one or both may be computed, is introduced. PKOB. IX. To find the altitude of the sun, the latitude of the station and the time of day being given. From the given time there must be obtained the sun's horary angle and also his declination, or polar distance ; and then, in the triangle ZPS, there are given PZ the colatitude, ZPS the hour angle, and PS the north polar distance ; to find z S, the true distance of the sun from the zenith. Let fall z perpendicularly on PS; then, by Spher. Trigo., in the triangle pz*(art. 62. (/'))/ Bad. cos. P cotan. PZ tan. p, and in the triangle p z s (3 Cor. art. 60.), cos. PZCOS. #8= cos. Pt cos. zs. Thus z s, the complement of the sun's true altitude, is found. But the rule frequently given in treatises of navigation is obtained from the formula cos. zs^cos. ZPS sin. ZP sin. PS + cos. PS cos. PZ (art. 60.), by transforming it, as before shown (art. 354. (B)), into cos. zs^rcos. (PS PZ) vers. sin. ZPS sin. PS sin. PZ. The term cos. (PS PZ) is the sine of the sun's meridional altitude, and log. vers. sin. ZPS is designated log. rising, which may be found in the " Requisite Tables" (tab. XVI.) for the given hour angle in time : therefore, the natural number corresponding to log. rising + log. cosin. latitude f log. cosin., sun's declination x 2 308 NAUTICAL ASTRONOMY. CHAP. XVI. being represented by N, we have the sine of the sun's meridian altitude N = sine of the required altitude. The processes are exactly similar when the celestial body is the moon or a star. The zenith distance or altitude found by either of the methods is not affected by parallax or refraction ; and since, in the problem for finding the longitude, the apparent zenith distance or altitude is required, it becomes necessary to apply to the computed zenith distance, for example, the difference between the sun's refraction and parallax. Now the sun's refraction is greater than his parallax, so that the sun appears always elevated above his true place ; therefore that difference must be subtracted from the value of zs above found in order to have the apparent zenith distance. The apparent place of a fixed star or planet is also higher than the true place ; but the apparent place of the moon is lower, because the moon's parallax is greater than her refraction; therefore the dif- ference between her refraction and her parallax in altitude must be added to the computed value of zs in order to have the apparent zenith distance of the moon. Ex. 1. At Sandhurst, Sept. 4. 1843, at 9 ho. 35' 55" A.M., mean time, the longitude of the station from Greenwich being about 3', in time, and its latitude equal to 51 20' 33"; it is required to find the sun's altitude. Mean time + Equation of time = Apparent time, "} Sandhurst J Time before noon at Sandhurst. ho. , 9 35 55 54.8 9 36 49.8 ; hence the apparent time at Greenwich is 9 ho. 39' 49".8, or 2 ho. 20' 10".2 12 before apparent noon. 2 23 10.2 = 35 47' the sun's hour angle The sun's declination apparent noon, Gr. 7 2'o 44.2 (N. Xaut Aim ) -f Variation for 2 ho. 20' 10".2 - - 2 9.3 ^= Sun's declination at the given instant - 7 90 22 53.5 Sun's north polar distance (= p s) - 82 In the spherical triangle ZPS, log. cos. ZPS (35 47' 33") 9.9090956 log. cotan. PZ (38 39 27 ) 0.0969459 37 6.5 log. tan. rt = 32 58' 41" PS = 82 37 6.5 9.8121497 ts = 49 38 25.5 log. cos. PZ + log. cos. ts + log. cos. Ft, co. ar.- =log. cos. z s = 52 55' 39", the true of the sun's centre. - 9.8925921 - 9.8112965 - 0.0763007 9.7801893 zenith distance CHAP. XVI. ALTITUDES COMPUTED. 309 The same example, by the rule in the " Eequisite Tables." Time before noon as above 2 ho. 23' 10'', log. rising 4.27612 +Log. sin. PS (82 37' 6") - - - 9.99638 +Log. sin. PZ (38 39 27 ) - - - 9.79565 = Log. of 11699 - 4.06815 9") \ or nat. sin. sun's meridian alt. J Nat, cos. (PS PZ =43 57' 39") 60282 = Nat. cos. 52 55' 40", the sun's true zenith distance. Ex.2. At Sandhurst, April 13. 1840, at 10 ho. 5' 25".6 P.M. mean time, or 10 ho. 5' 4 // .3 apparent time, the altitude of /3, Geminorum (Pollux), was required. ho. , Right ascension of the star - 7 35 32.6 (Naut. Aim.) Right ascension of the sun at noon 1 27 25.8 Approximate time of culminating 68 6.8 Variation of sun's rt. asc. in 6 ho. 8' 56.5 Apparent time of culminating - 6 7 10.3 Apparent time at the station - 10 5 4.3 Solar time elapsed since culmi- nating - - - - 3 57 54 Sidereal time corresponding = 3 58 33, by the table of time equi- 15 valents. (Naut. Aim.) o The star's horary angle - = 59 38 16.5 The star's north polar distance = 61 35 30 (Naut. Aim.) Then in the triangle ZPS, log. cos. ZPS (.59 38' 16".5) 9.7036910 -log cotan. PZ (38 39 27) 0.0969599 = log. tan. ft = 22 053 9.6067311 PS = 61 35 30 ts = 39 34 37 log. cos. ZP - 9.8925911 + log. cos. ts 9.8869268 + log. cos. P* - 0.0328792 co. ar. =log. cos. zs - 9.8123971 =49 31' 0", the star's true zenith distance. Ex. 3. At Sandhurst, April 13. 1840, at 10 ho. 5' 25".6 mean time, or lOho. 5' 4". 3 apparent time, the altitude of the moon's centre was re- quired. Computation of the moon's right ascension and declination. The estimated difference of longitude between the station and Greenwich being 3', as above, the mean tune at Greenwich was 10 ho. 8' 25".6. ho. / The moon's right asc. at 10 ho. mean time Gr. 11 23 57.2 (Naut. Aim.) + Variation for 8' 25' .6 since 10 ho. - 15.7 Moon's right asc. at the given instant - = 11 24 12.9 Moon's declin. at 10 ho. mean time Gr. 2 26 3.8 (N. Naut. Aim.) Variation for 8' 25".6 since 10 ho. - 2 5.9 Moon's declinat. at the given instant - = 2 23 57.9 90 Moon's north polar distance (=PS) = 87 36 2.1 x 3 310 NAUTICAL ASTRONOMY. CHAP. XVI. ho. , R. ascens. of the sun at apparent noon - 1 27 25.8 (Naut. Aim.) Variation in 10 ho. 5' 4".3 1 32.8 R. ascens. of the sun at the given instant - 1 28 58.6 Apparent time given - - 10 5 4.3 R. ascens. of the meridian - - 11 34 2.9 R. ascens. of the moon - - - 11 24 12.9 The moon's hour angle, in time \- " 950 15 Do. in degrees, (= ZPS) - , ,, -1- - 22730 In the triangle ZPS (s in the figure, representing the true place of the moon), log. cos. ZPS (2 27' 30") 9.9996001 log. cotan. PZ (38 39 27) 0.0969459 =log. tan. Tt =38 37 54 9.9026542 rs =87 36 2.1 ts =48 58 8.1 log. cos. ZP - - 9.8925911 + log. cos. ts . ..i ^ - 9.8172116 + log. cos. et, co. ar. - 0.1072496 =log. cos, zs - 9.8170523 = 48 59' 15", the true zenith dis- tance of the moon's centre. 359. The longitude of a station may be determined by various methods which will be presently explained, but that of a ship which is beyond the sight of land can be ascertained astronomically only by what are called lunar observations, or by obtaining the mean time of the day or night from the observed altitude of a celestial body, and comparing it with the time shown by a chronometer which is supposed to ex- press, at every instant, the mean solar time at Greenwich. The principle on which the processes are founded is the same in all cases, and consists in a determination of the time at Greenwich and at the station or ship at the same physical instant ; the difference between those times being equal to the difference, in longitude, between Greenwich and the ship. This is obvious; for since apparent time at any place is expressed by the angle (in time) between the meridian of the place and a horary circle passing through the sun at the instant, the difference between the apparent times, and con- sequently between the mean times, at two places at the same instant, must be equal to the angle, in time, between the meridians of the two places. Whatever phenomenon, there- fore, constitutes a signal which may be observed at two places at the same instant becomes a means of determining the difference between the longitudes of those places ; and many such phenomena present themselves in the heavens, as the eclipses of the sun and moon, the immersions and emersions of Jupiter's satellites, &c. The moon's proper motion from west to east, is more rapid than the like motion in the sun or a planet, and the fixed stars have no such motion ; hence, the angular distance between the moon and any other celestial body varying continually, the occurrence CHAP. XVI. LONGITUDE, BY LUNAR DISTANCES. 311 of a particular distance becomes also an instantaneous phe- nomenon ; and as such may be employed to determine the times, at the instant, at any two places where it may have been observed : or if a series of such distances have been cal- culated for given times, and for any one place, as Greenwich, and a distance have been observed at another place ; an iden- tity of distance will serve to determine the time of the observa- tion according to the reckoning of time at the former place. PROB. X. 360. To find the longitude of a station or ship by means of the observed angular distance between the moon and the sun or a star. The latitude is supposed to be well known ; %nd, in order that the polar distance of the sun may be found for the purpose of ascertaining the correct time at the place or ship, the longitude, as well as the hour of the day, must be known approximatively. In general, in making a lunar observation, one person observes the angular distance between the moon and sun, or the moon and a star, and at the same time two other observers take the altitudes of the celestial bodies. If the moon and sun be observed, the dis- tance between either the eastern or west- ern limb of the latter, and the enlight- ened limb of the former is taken ; also the altitude of the upper or lower limb of the sun, and that of the enlightened limb of the moon. Now, after ap- plying the corrections for the index errors of the instruments, and the dip of the horizon if at sea, or taking the halves of the altitudes if an artificial horizon have been used ; let the angular semidiameter of the sun, and the augmented semi-diameter of the moon be applied, by addition or subtraction, so as to reduce the observed altitudes and distance to the altitudes and distance of the centres. Then the two zenith distances and the direct distance of the centres will form in the heavens a triangle, as z s M, of which all the three sides will be affected by parallax and refraction. With the three sides thus found compute the angle SZM by the rules of spherical trigonometry: then, imagining Ss to be the star's refraction, or the difference, in a vertical direc- tion, between the sun's parallax and refraction, and Mm to be the difference, also in a vertical direction, between the moon's parallax and refraction (the former being added to zs because the sun's refraction is greater than his parallax, and x 4 312 NAUTICAL ASTRONOMY. CHAP. XVI. the latter being subtracted from ZM for a contrary reason), the points s and m become the corrected places of the sun, or star, and the moon, or the places which the luminaries would appear to occupy in the heavens if seen from the centre of the earth and were not affected by atmospherical refraction. Therefore, in the triangle zsm } having two sides and the included angle, the remaining side sm, which is the true dis- tance of the luminaries from each other, may be computed. This is the direct, and the most accurate method of find- ing the true distance between the centres of the sun and moon, or between a star and the centre of the moon; but the computation is laborious, and it requires great atten- tion to the logarithms in determining the angle at z. It would be eatier, and sufficiently correct, to proceed in the following manner. By means of the three sides ZM, zs and MS, compute (art. 66.) the two angles ZMS and ZSM; and, ss being equal to the difference first mentioned above, let fall s t perpendicularly on MS produced if necessary: then in the triangle S s t considered as plane we should have S/ Ss cos. ZSM orirss cos. sst. If the angle ZSM were obtuse, st would fall on the other side of s. Next, M m being equal to the difference between the moon's refraction and her parallax in altitude, imagine about s or t as a pole a portion mp of a small circle to be described, so that sm may be considered as equal to tp ; then the value of Mp may be found by a formula similar to that which was used in determining the latitude of a place by an altitude of the pole star (art. 352.) : thus, Mm and Mp being expressed in seconds of a degree, and using only two terms (see art. 410.), Mp z=Mm cos. ZMS (Mm) 2 sin. 2 ZMS cotan. MS sin. 1". The value of Mp must, if the angle at M be acute, be sub- tracted from M^, that is from MS + stfin order to have the value of p t or its presumed equivalent m s, which is the correct distance between the luminaries. If the angle at M be obtuse, the value of Mp must be added to Mt. The process of finding the angles at S and M would be facilitated by the use of spherical traverse tables correspond- ing to those which are given in Raper's " Navigation : " the formation of such tables has been indicated in art. 337. Having thus obtained the correct distance sm, there must be found in the Nautical Almanac the time at Greenwich when the distance between the sun, or star, and the moon is equal to that which has been found ; then that time being compared with the mean time, at the station or ship, deduced from the sun's horary angle SPZ at the instant of the ob- servation ; the difference between them will be the required CHAP. XVI. CORRECTION OF THE DISTANCE. 313 difference of longitude in time. And the place or ship will be westward or eastward of Greenwich according as the mean time is earlier or later in the day than that at Greenwich. Since the computed distance sm will seldom be exactly found in the Nautical Almanac, a proportion must be made by means of the difference between the distances for three hours ; and in this operation the proportional or logistic logarithms (" Requisite Tables," tab. XV.), as they are called, are gene- rally used. Ex. At Sandhurst, April 13. 1840, at 10 ho. 4' 48" by the watch, the following observations were made : Double altitude of , Geminorum, by reflexion - - = 81 1 20 Double altitude of the moon's upper limb - -= 81 14 50 Distance between the star and the moon's nearest limb - = 60 42 10 The error of the watch was 37". 6 (too slow) ; therefore the mean time of the observation was 10 ho. 5' 25 // .6. o / Double alt. moon's up. limb 81 14 50 Index error, subtractive - 5 29 2) 81 9 21 App. alt. moon's upper limb 40 34 40.5 Moon's semidiameter - 15 14. 3 Altitude of moon's centre, ap- proximative - - 40 19 26.2 Augmentation - - 10.3 App. alt. moon's centre - 40 19 15.9 90 log. sin. moon's alt. =40 19' 9.81076 + log. 16" ... 1.20412 = log. augmentation of moon's semidiameter - - 1.01488 Augmentation = 10". 3 Double alt. star - - 81 I 20 Index error (subtr.) - - 14 2) 81 16 Apparent altitude - - 40 30 8 90 Apparent zenith dist. star - 49 29 52 Apparent zenith distance of the moon's centre (=ZM) 49 40 44.1 Observed distance of the star from the moon's nearest limb = 60 42 10 Index error, subtractive - - - - - 23 19.4 60 18 51.6 Moon's augmented semidiameter - - - 15 24.6 Apparent distance of the star from the moon's centre =60 34 16.2 Computation of the angle M z s in the spherical triangle M z s (P repre- senting the perimeter). o / // MS = 60 34 16.2 zs = 49 29 52 ZM = 49 40 44.1 159 44 52.3 P = 79 52 26.2 log. sin. CO. ar. 0.0068182 P MS 19 18 10 log. sin. CO. ar. 0.4807495 P ZS = 30 22 34.2 log. sin. 9.7038707 P ZM = 30 11 42.1 log. sin. 9.7015200 2) 19.8929584 lo. tan. MZS = 41 28' 43" 9.9464792 314 NAUTICAL ASTRONOMY. CHAP. XVI. Therefore MZS = 82 57 26. Corrections for parallax and refraction in the altitudes or zenith dis- tances. log. cos. moon's app. alt. (40 19' 16) 9.882200 -}-log. moon's horizontal parallax 55' 34" or 3354" (Naut. Aim.) - 3.525563 =log. moon's parallax in altitude - 3.407763 = 2557".2 Refraction = 68 .5 Difference between the moon's parallax and re- fraction (= M m) - - - ' 2488.7 or 41' 28".7 Apparent zenith distance of the moon (ZM) ---./. - =49 40 44 .1 True zenith distance of the moon (= zm) - - = 48 59 15 .4 The star's refraction - - = 1' 8" Apparent zenith distance of the star = 49 29 52 True zenith distance of the star (zs) = 49 31 In the spherical triangle mzs. log. cos. mzs (82 57' 26") 9.0885272 log. cotan. zs (49 31) 9.9312431 = log. tan. zt = 8 10' 27" 9.1572841 zm=48 59 15.4 = 40 48 48.4 log. cos. mt 4- log. cos. zs ' C S " ^ C ' =log. cos. ms - 9.6958370 = 60 14' 17", the true distance of the star from the moon's centre. By the Nautical Almanac, the true distances of the star from the moon's centre were at ix P. M. Gr. time, 59 38 16 59 38 16 at xii P. M. 61 13 7 True dist. by the observation = 60 14 17 Difference = 1 34 51 Difference - 36 1 Proportional log. 1 34' 51" = 2782 Do. 36 1 = 6988 Difference - = 4206 = Proportional log. llio. 8' 20" Add 9 Mean time at Greenwich corresponding to the distance obtained from the observation - 10 8 20 Mean time at the station - 10 5 25.6 Difference of longitude, in time, = 2 54.4 (Sandhurst west of Greenwich.) 361. The proportional or logistic logarithms here employed are merely the common logarithms of the fractions arisino* from the division of the number of seconds in 3 hours, or 3 degrees, by the four terms of a proportion none of which exceeds that number of seconds ; and they are convenient on many occasions similar to that which occurs in the above proposition. In order to explain their nature, let R represent any number, and let A : B : : c : D be any proportion ; then CHAP. XVI. LONGITUDE, BY THE MOON'S ALTITUDE. 315 T> T> T> T> we shall have - :-::-: -. Now if R = 10800 (the number A B C D of seconds in three hours) and none of the terms A, B, c, D R R exceed that number, the common logarithms of -, -, &c. will .A. X3 be those which are called the proportional logarithms of A, B, &c., and if three such logarithms be taken from a table, the proportional logarithm of the fourth term may be found exactly as when common logarithms are employed in working a proportion. If one of the terms, as A or B, be equal to 10800, its proportional logarithm will be zero ; and the operation will then consist in merely adding together, or in subtracting one from another, the two proportional logarithms which have been taken from the tables ; the sum or dif- ference will be the proportional logarithm of the required term. Similar logarithms are formed by making R equal to 3600 (the number of seconds in one hour, or one degree), and they are used in a similar manner. 362. Besides a method which was proposed by Borda, Delambre has given in his ee Astronomic " three formulae for finding the correct lunar distance by logarithms, two formulas requiring only the addition of natural cosines, two requiring natural versed sines, and two requiring both natural and logarithmic sines : he has also given a considerable number of formulas by which the difference between the true and apparent distances may be computed directly.* The method of determining longitudes by lunar distances appears to have been first used by La Caille in his voyage to the Cape of Good Hope ; and that astronomer gave certain formula? by which the calculations might be abridged : the method did not, however, become general till the year 1767, when Dr. Maskeline first published the British Nautical Almanac, in which the distances of the moon from the sun and from certain stars are inserted, by computation, for every three hours according to the time at Greenwich. Previously to the publication of that work the longitude had been generally determined by means of the moon's horary angle. This method consisted in observing the altitude of the moon's enlightened limb and reducing it as usual to the altitude of the centre; then, knowing the latitude of the station and finding the moon's declination in the ephemeris for the time of observation (an approximate estimate of the longitude of * The practical seaman, in determining the longitude of his ship by. lunar observations, will find the labour considerably diminished by the use of Mrs. Taylor's " Lunar Tables," third edition. 316 NAUTICAL ASTRONOMY. CHAP. XVI. the station being made), the moon's horary angle z p m was computed: the sun's horary angle ZPS for the same time was also computed, and hence was obtained the difference between the right ascensions of the sun and moon. The right ascension of the moon, in mean time at the station or ship, was thus known : finding then in an ephemeris the time at the first meridian, as Greenwich, when the moon's right ascension was the same ; the difference between the times became the required difference of longitude in time. The quantities employed were, however, too variable to allow sufficient precision in the result ; and if, on comparing the moon's declination obtained from the estimated time at the station with that which was obtained from the time determined by the computation, the two were found to dis- agree, it was necessary to repeat the computation with the last mentioned time. This repetition rendered the operation extremely laborious, and the method has been long since abandoned. 363. A chronometer being set so as to indicate, at a given instant, the mean solar time at Greenwich, may, by its uniform motion, be supposed to continue in every part of the world to indicate at any instant the hour at that place: conse- quently the mean time at any station or ship being obtained by an altitude of the sun or a star, or by two equal altitudes of either, the mere comparison of that time with the time indicated by the chronometer will give the difference of longitude between the station and Greenwich. 364. In 1763, Mr. Harrison completed a time-keeper which was found -to fulfil the required conditions in deter- mining the longitude of a ship, and the improvements since made in horology have led to the construction of machines possessing still greater uniformity of motion. Yet in every machine there may, from insensible defects in the materials, exist causes of derangement which the artist cannot anticipate ; therefore the best warrant must not be allowed to supersede the employment of every means which astronomy furnishes for ascertaining the error, in order that due allowance may be made in finding from it the time at Greenwich. On land, the time indicated by the chronometer at the instant of an observed eclipse of one of Jupiter's satellites being compared with the time set down in the Nautical Almanac, gives at once the error. If a transit telescope be set up, or even a good theodolite, and the successive transits of the same star over the vertical wire of the telescope be observed; or even if the nightly disappearance of a star behind any terrestrial object be observed, the spectator CHAP. XVI. LONGITUDE, BY A CHRONOMETER. 317 looking from any fixed point, as through a small perforation in a wall, should the interval between any two transits or disappearances be exactly 24 hours in sidereal time, or 23 ho. 56' 4 // .09 in solar time, the time-keeper goes correctly: if otherwise, the difference is the error; and thus the daily error, or the daily rate, may be easily found. In default of such means, the time of mean noon at a place, given by some watch, may be compared with mean noon found by equal altitudes of the sun or a star, and thus the error of such watch may be found and corrected : then, if the longitude of the station be known, On applying to the corrected time of noon, by the watch, the difference of lon- gitude between the station and Greenwich, the result is the time of mean noon at the latter place ; and this result ought to agree with the chronometer. Observing the altitude of the sun or a star for the purpose of determining the time at a station, and subsequently comparing that time with the time given by the chronometer, is what is familiarly called taking sights for the chronometer. The error, and daily rate of a chronometer may be found by means of lunar observation in the following manner. When the contact of the sun and moon has been observed, let the time shown by the chronometer be marked ; and then, with the corrected lunar distance, find the Greenwich time from the Nautical Almanac : the difference between that time and the time shown by the chronometer gives the error of the machine. The error found by any of these means, from observations repeated after one day or several days, will serve to determine the required rate. The error of the chronometer may also be found by what is called the method of cross-bearings. This consists in taking, on board a ship, suppose at s, the bearings from the true or magnetic meridian, of two objects, as A and B, on the shore (the mutual distance of these objects, with their bearing MAB or NBA, frome one another, and both the lati- tude and longitude of one of them being supposed to be known): then in the triangle ABS, from the parallelism of the meridians, there are data sufficient for the computation, by plane trigonometry, of the A ; distance s A or SB ; and letting fall SM or SN perpendicularly on the meridian of A or B, the distance SM or SN may be found in the right- angled triangle ASM or BSN. One of these distances, as SM, being expressed in geogra- phical miles, or equatorial minutes, and divided by the cosine of the latitude of s, the result will be the difference 318 NAUTICAL ASTEONOMY. CHAP. XVI. between the longitudes of the ship and of the station A, whose geographical position is supposed to be known. Thus the longitude of the ship from Greenwich is found : then the time at the ship being obtained by an observation, the time at Greenwich becomes known ; and this being compared with the time indicated by the chronometer, the difference is the error of the latter. Meridional distances as they are called, that is the differences of longitude between places, are usually obtained, when chronometers are employed for the purpose, by finding the absolute longitudes of the stations, from comparisons of the computed times with the corre- sponding chronometer times, and taking the differences. 365. Among the methods which may be put in practice for determining the longitude of a station are those which depend on observed meridional transits of the moon, and on observed transits of the moon combined with those of certain fixed stars, as proposed by the late Mr. Baily in the " Memoirs of the Astronomical Society " (Yol. II.). The last method in par- ticular, by the help of the tables which have been published in the Nautical Almanacs since the year 1834, and with due care in the observations, unites the advantages of affording considerable accuracy in the results, with great facility in the computation. If it were required to find the longitude of a station by an observed transit of the moon over the meridian of the station, the following process must be used. Take from the Nautical Almanac, in the table of " moon-culminating stars," the right ascension of the moon's centre ; and having observed the transit of that centre over the meridian of the station, which, for example, may be westward of Greenwich, let E be a point in the meridian of the station when the moon's centre at C culminates at Greenwich. Now while the western meridian is revolving to the moon, the centre of the latter will, by the proper movement of the luminary, have advanced to some point D, where it will culminate on that meridian ; conse- quently the difference between the sidereal times of the transits, or culminations, which is the difference between the right ascension of the moon's centre in the Nautical Almanac and the observed sidereal time of the transit at the station, or is the change in the moon's right ascension while the meridian of the station is revolving from E to D, may be represented by CD. But the variation of the moon's right ascension cor- responding to one hour of longitude on the earth, that is in the time that a meridian of the earth is revolving through an CHAP. XVI. LONGITUDE, BY TRANSITS OF THE MOON. 319 angle of 15 degrees, is given in the Nautical Almanac; there- fore, v representing the variation of the moon's right ascension (in time) corresponding to one hour, v : 1 hour :: CD : ED (in time), and ED CDinEC, the difference of longitude required. It has been supposed, for simplicity, that the transit of the moon's centre was observed at the station ; but, in fact, the transit of the moon's enlightened limb is that which ought to be observed, and the time of such transit should be reduced to the transit of the centre by adding or subtracting the time in which the moon's semidiameter passes over the meridian wire of the telescope. This redaction is not necessary when the station is near the meridian of Greenwich, because the variation in the moon's proper movement would then be in- sensible between the times of the transit at Greenwich and at the station : but if the latter be so far distant in longitude that the difference between the times of passing the wire be- comes sensible, the reduction must be made. 366. In order to determine the longitude of a station by the culminations of the moon and certain stars having nearly the same right ascension and declination, the following process is employed. Let t represent the sidereal time at Greenwich, when, by the revolution of the earth on its axis, the meridian of that place passes through a star, supposed to be at A ; and imagine that when the same meridian, in revolving, arrives at the en- lightened edge of the moon at c, the time is also shown by the sidereal clock. To this last time apply, by addition or sub- traction, according as the first or second limb of the moon is on the meridian, the time in which the moon's semidiameter passes the meridian, and let the sum or difference be repre- sented by T ; then, the moon being supposed, for example, to culminate first, t T will express the difference (represented by AC) between the right ascensions of the moon's centre and of the star when the former culminates at Greenwich.) (These right ascensions may be taken from the Nautical Almanac, from the table of " moon-culminating stars.") Let the meridian of the station pass through some point E when the moon culminates at Greenwich ; then while that meridian is revolving to the moon the latter will have moved to some point D. Now let the transits of the star at A, and of the moon's limb be observed, and adding to the latter the time in which the semidiameter passes the meridian, let the first term be represented by t' and the last by T' ; then t? T' (repre- sented by AD) will be the difference between the right ascen- 320 NAUTICAL ASTRONOMY. CHAP. XVI. sions of the moon's centre and of the star when the former culminates at the station. The difference c D, between AD and AC, is therefore known; and consequently, by the proportion above mentioned, we may have ED. This value of ED expresses, in time, the angle through which the meridian of the station revolves while the moon moves from C to D by her proper motion. It is evidently equal to E c + c D ; and E c is, in time, the required difference in longitude between Greenwich and the station. In the above description it has been supposed that Greenwich is east- ward of the observer's station, but the process would have been the same in the contrary case. Ex. 1. At Sandhurst, October 8. 1840, there was observed by the si- dereal clock, the transit of the moon's first limb on the meridian. ho. , 23 1 9.2 Error of the clock (too slow) - - - - 7.75 Correct time of the transit - - 23 1 16.95 Right ascension of the moon's first limb (from the table of moon-culminating stars in the Naut. Aim.) - - 23 1 23.31 Difference (= CD) - - 6.36 Then, the variation of the moon's right ascension for one hour being 122".59, 122".59 : 1 hour : : 6".36 : 3 '6".8 (=ED) 6 .36 (=CD) Difference of longitude, in time, - 3 .44 (=EC) Ex. 2. At Sandhurst, Oct. 8. 1840, there were observed the transits ho - / i, of the moon's first limb -23 1 9.2 of the star K, Piscium - 23 18 27.6 Difference *' T', or AD) = 17 18.4 ho. , R. asc. moon's first 1. ( N. Al.) 23 1 23.31 R. asc. K, Piscium (do.) 23 18 47.98 Difference (<-T, or AC) = 17 24.67 AD = 1718.4 CD = 6.27 Then by proportion, as above, we get E D = 3' 4" ; and subtracting the present value of CD, there remain, for the difference of longitude, in time, 2' 57".73 (= EC.) 367. The difference between the longitudes of two stations may be determined by means of signals, which may be visible at both at the same instant, and such a signal is the flash of fired gunpowder, the extinction of a lamp, or the explosion of a rocket. Let, for example, a rocket be fired from some place between the stations, and let the instant of explosion be ob- served by persons at those stations : then, the times of the explosion being given by watches previously adjusted so as to indicate the mean times proper to the stations, the difference between them will be, in time, the required difference of longitude. If the stations be so remote that the rocket fired CHAP. XVI. LONGITUDE, BY SIGNALS. 321 between them cannot be seen from both, a chain consisting of three or more signal stations may be formed between the two places whose difference of longitude is required ; and in this case, the observer at each of the two extremities of the chain registers the time indicated by his chronometer or sidereal clock at the instant of the explosion at the signal station nearest to him, while the person at each of the intermediate stations registers the time indicated by his chronometer at the instant of the explosion which takes place on each side of him. In the year 1825 the difference in longitude between the meridians of Greenwich and Paris was determined by means of rockets which, at intermediate stations, were fired at regulated times ; and the method then pursued may be conveniently employed on all similar occasions. Fairlight Down in Sussex, and La Canche on the opposite coast of Normandy, were stations separated from each other by about 50 miles of sea, and Paris was connected with the latter by two intermediate stations, Mont Javoult and Lignieres : on the English side, Fairlight, by an intermediate station at Wrotham, was connected with Greenwich; the several stations, in the order here mentioned, from Paris to Greenwich, lying successively westward of one another. The azimuths, or bear- ings, of the several stations from one another were previously determined by computation, and, at the appointed times, by means of azimuth circles, telescopes were placed in the proper directions, in order that the explosions of the rockets, at the instants of their greatest altitudes, might be observed in the fields of view. The observers being prepared, on a certain night a rocket was fired at Mont Javoult and observed at Paris to explode at 18 ho. 39' 52".5, sidereal time (let this time be represented by A), and the same explosion was ob- served at Lignieres at 10 ho. 49' 41", mean solar time for that station, by a chronometer ; let this time be represented by B. About 5 minutes afterwards a rocket fired at La Canche was observed at Lignieres at 10 ho. 54' 53".2 (call this time c), and at Fairlight station at 10 ho. 46' 37". 5, mean solar times at the two places ; represent the latter time by D. Finally, about 5 minutes later, a rocket was fired at Wrotham, which was observed at Fairlight at 10 ho. 51' 59". 4 mean solar time (call it E), and at Greenwich at 18 ho. 41' 7". 11 sidereal time ; let this last be represented by F. The intervals between the discharges of the successive rockets were pur- posely small, in order that the errors in the rates of the chro- nometers might be inconsiderable. Now c B = 5'12".2 and E-DZI 5'21".9; therefore their 322 NAUTICAL ASTRONOMY. CHAP. XVI. sum, 10'34".l in solar time, or 10'35".83 in sidereal time, was the interval between the observation of the first rocket at Paris and of the third rocket at Greenwich. This interval being added to the time A gives 18 ho. 50'28".33, the sidereal time which would be reckoned at Paris at the instant that the sidereal time at Greenwich was expressed by r : the difference between these times, 9' 21 ".22, is therefore the distance between the two meridians, in time. In the above example it has been supposed that the mean time chrono- meters at the stations between Paris and Greenwich were subject to no gain or loss on sidereal time ; but this will seldom be the case, and the rates of the chronometers with respect to sidereal time were determined in the following manner. Rockets being fired on two successive days at Mont Javoult, for example, were observed ho. t /x ho. , at Paris, the first day, 18 32 21.88 ; the second, 18 19 41.83 sid. times ; at Lignieres - 10 46 13.6 ; 10 28 33.93 mean times by chronom. the differences were 7 46 8.28 and 7 50 7.9 The excess of the latter difference above the former (=3' 5 9". 62.) was the gain of the chronometer on sidereal time during the interval ( = 23 ho. 47' 19''.95.) between the sidereal times of the two observations ; this interval being found by adding 24 hours to the time on the second day, and sub- tracting from the sum the time on the first day. But 23ho. 47 / 19' / .95 : 3'59".62 :: 24ho. : 4 / l // .74, and this last term was the gain of the chronometer in one sidereal day : now the solar day exceeds the sidereal day by 3'55".91 ; therefore the difference (5". 83) was the gain of the chronometer on one solar day. Hence 24ho. : 5".83 ::-c B( = 5'12".2) : // .02, and the last term subtracted from 5' 12". 2 gives the correct value of c B, In like manner might be found the correc- tion of E D, which may also be considered as equal to 0".02 : thus the interval between the observations of the rockets at Paris and Greenwich becomes 10'34".06 or, in sidereal time, 10' 3 5 ".7 9, and the difference of the longitudes becomes 9 / 21 // .18. The signals are made by previous agreement on several different nights, in order that a mean of the results may be taken, and because the observations often fail. During the seasons in which the operations here alluded to were carried CHAP. XVI. LONGITUDE, BY SIGNALS. 323 on the rockets sometimes exploded without ascending, at times they exploded twice; and one rocket passed through the field of view without exploding. The details of all the operations for connecting, by means of signals, the meridians of Paris and Greenwich may be seen in Mr. (Sir John) Herschel's paper in the "Philosophical Transactions " for 1826, part 2, v 2 324 ECLIPSES. CHAP. XVII. CHAP. XVII. ECLIPSES. OUTLINES OP THE METHODS OF COMPUTING THE OCCURRENCE OF THE PRINCIPAL PHENOMENA RELATING TO AN ECLIPSE OF THE MOON, AN ECLIPSE OF THE SUN FOR A PARTICULAR PLACE, THE OCCULTATION OF A STAR OR PLANET BY THE MOON, AND THE TRANSITS OF MERCURY AND VENUS OVER THE SUN'S DISK. THE LONGITUDES CF STATIONS FOUND BY ECLIPSES AND OC- CULTATIONS. 368. AN Appendix to the Nautical Almanac for 1836, by Mr. Woolhouse, contains investigations of all the formulae necessary for computing the phenomena of eclipses of the moon and sun, the occultations of stars by the moon, and the transits of Mercury and Venus over the sun's disk: the formulae relating to eclipses of the sun consisting of such as serve to determine the phenomena with respect to the earth generally, the terrestrial limits of the phases, the phenomena for particular places, and of such as serve to reduce the phenomena for one station to those which correspond to them for another. It is intended in the present chapter to follow, nearly, the methods employed in that essay, but outlines only will be given of the processes for determining the commence- ment and end of an eclipse of the moon; and, for particular stations, those of eclipses of the sun, occultations of stars and the transits of planets : to these will be added the rules for computing terrestrial longitudes from observed phenomena of eclipses and occultations. There exists, beyond the earth with respect to the sun, a space within which the rays proceeding from the sun's disk and touching the surface of the earth do not enter ; and this space, which is of a conical form, has its vertex beyond the region of the moon. There exists, likewise beyond the earth, and on the exterior of that umbra or shadow, a space within which the sun's rays only partially enter : this, which is called the penumbra, is bounded by rays which coming from every part of the circumference of the sun's disk cross one another at a point between the sun and earth, and diverging from thence touch the surface of the latter; and in determining the phenomena of eclipses of the sun and moon, it is neces- CHAP. XVII. THE EARTH'S SHADOW. 325 sary first to find the semidiameters of the shadow and penumbra in a plane passing through the moon's centre per- pendicularly to the common axis of the cones, or the line joining the centres of the sun, earth and moon, it being- supposed here that those centres are in one right line. There- fore, let OR be a diameter of the sun, whose centre is s ; ET a semidiameter of the earth, and Mm a semidiameter of the moon, supposed to be in direct opposition, so that s, E and M are in a straight line : also let a line touching the surface of the sun at o and of the earth at T meet the line s M produced in c ; then the triangle TCE will represent half a longitudinal section through the cone of shadow beyond the earth, c being the vertex of the cone, and MN will be a semidiameter of a transverse section in the region of the moon. If a line touching the surfaces of the sun at R and of the earth at T meet MN produced in Q ; then MQ will be a semidiameter of a tranverse section of the penumbra in the same region. Imagine the other lines in the figure to be drawn, In the Nautical Almanac there will be found, at page XII. of each month, the day and hour (in mean time) of full moon, or of the opposition of the sun and moon in longitude. Therefore, for any convenient hour of that day, suppose that which is nearest to the mean time of opposition, take from thence the equatorial horizontal parallaxes P and p of the moon and sun, and the semidiameters of the luminaries, and find the horizontal parallax of the moon for the station of the observer, its latitude / being given. This latter parallax is sin 2 l\ expressed (art. 157.) by p (1 ' ) putting p, in seconds, oUO ' for sin. P : let this expression be represented by P'. The angle TCS is equal to OTS TSE, or to the difference between the horizontal parallax and the angle subtended at T, or at E nearly, by the semidiameter of the sun ; therefore if s denote the angular semidiameter of the sun (in seconds) the angle TCS =5 p. Now the angle NEM is that which is subtended at E by the semidiameter of the earth's shadow, and it is equal to TNE TCS: but TNE may be considered as equal to TME, the moon's horizontal parallax; therefore NEM p' s + p nearly. Y 3 326 ECLIPSE CHAP. XVII But the angles NEM, N#M are inversely proportional to EM, *M nearly, and tM. : EM :: 60 : 61 nearly; therefore fi i N #M = ^r (P' s + p), and this may be considered as the angular measure, at the station, of a semidiameter of the earth's shadow in the region of the moon. Again, OTR or Q T N is the angle subtended by the sun's diameter ; therefore it may be represented by 2 s ; and Q T N may be considered as equal to QfN nearly; therefore the angle MQ, or the /> -t angular semidiameter of the penumbra, is equal to -^ (p' s + p) -f 2 s 9 nearly. 369. In order to determine the phenomena of an eclipse of the moon, there must be found from the Nautical Alma- nac, for the time above mentioned, the right ascension of the moon, in degrees, and the right ascension, in degrees, of the centre of the shadow ; the latter being the sum or difference of the sun's right ascension and 180 degrees: also the decli- nation of the moon, and of the centre of the shadow ; the latter being the sun's declination with a contrary name, or on the opposite side of the equator. Now the portion of the orbit apparently described by the moon during an eclipse may be con- sidered without sensible error as a straight line ; therefore let XY be such portion, and let M be the moon's place in it at the time above mentioned : also let s be the place of the centre of the shadow at the same time, and PS a declination circle passing through it. Let A, in seconds, represent the difference between the right ascension of the moon and of the shadow's centre, that is the angle SPM or the corresponding arc of the equator; and let D be the moon's declination : then A cos. D is equal (in seconds) to the arc M^ of a parallel of declination passing through M. Find, for the same assumed time, the hourly motions of the sun and moon, both in right ascension and declination (the former as well as the latter in seconds of a degree) : let the difference between the hourly motions in right ascension, or the moon's relative hourly motion in right ascension, be represented by a, and the moon's relative hourly motion in declination by b. The term a cos. D will express the moon's relative hourly motion on M^. CHAP. XVII. OF THE MOON. 327 Imagine the line m n to be drawn parallel to P s ; then in the right angled plane triangle Mmw, Mm and mn may represent, "respectively, the relative horary motions of the moon upon the parallel of declination and in declination ; therefore these are known, and we have Mm : m n : : rad. ; tan. m M n : again, cos. mmn : rad. :: Mm : Mra; thus Mtt is found, and its value is the moon's relative horary motion in her orbit. The arc 8*7 represents the difference between the moon's declination and that of the centre of the shadow, that is the sum of the declinations of the sun and moon ; and in the right angled triangle MSy, considered as plane, we have s<7 : M<7 :: rad. : tan. M 8*7; also sin. M 8 q : rad. : : M q ^: MS: thus MS and the angle MS^ are found. If Su be let fall perpendicularly on XY, the angle will be equal to PSV; therefore the angles MS s-\O therefore -^ may be represented by _ or 14 29' or 52140". This value being multiplied by sin. I" becomes the length of an equivalent circular arc in terms of the radius. If t, in decimals of an hour, express the interval between the instants of true conjunction in right ascension and of 330 ECLIPSE CHAP. XVII. apparent conjunction at the place; then the expression 3- 1, or its equivalent r d A? . ,.. sin. P' cos. / dr . , ,.-1 \ -j sin. 1" cos. T -= sin. 1 \ t, \. a t cos. D at J becomes, at the latter instant, the apparent difference of right ascension, or the apparent relative parallax, expressed in arc. But this expression is equivalent to the value of sin. a (art. 161. (i)) when the horary angle ZPS' in degrees is equal to r+ 15 t ; and, in an approximation, for r-f- 15 t may be put r; therefore the above expression may be considered , sin. P' cos. I . _, . , as equal to sin. T. Equating these expressions and putting P' sin. 1" for sin. P', we have, in decimals of an hour, _ sin. T 6 ' ' ~ ' dAf . , cos. D dr . _ -y- sin. \" r. =-,,- j -j sin. \' f cos. T: dt P' sin. 1 cos. I dt here -=- (in seconds of a degree), the difference between the horary motions of the sun and moon in right ascension, may be taken from the Nautical Almanac for the Greenwich mean time of true conjunction in right ascension : -= sin. \" (a constant) is equal to 52140 sin. 1", or to 0.25278 ; and its logarithm 9.40274. The value of t thus found being added to the Greenwich mean time of true conjunction in right ascension will give, approximatively, the Greenwich mean time of apparent con- junction in right ascension: let the latter be represented by T. . For the time T thus found there must now be obtained from the Nautical Almanac the following elements : The equatorial horizontal parallaxes of the sun and moon : the difference between these parallaxes is represented by P'. The horary motions of the sun and moon in right ascension : the difference between them is represented by Y . The horary motions of the sun and moon in declination : the difference between them is -. . at The true right ascensions of the sun and moon : the difference between them is A'. CHAP. XVII. OF THE SUN. 331 The true decimation d of the sun, and D of the moon (these are to be considered as positive if north, and negative if south) : the difference between them is D'. Also the true semidiameters of the sun and moon. Find next (art. 315.) the horary angle of the sun and moon, or the apparent time at the place, for the approximate instant T of apparent conjunction in right ascension ; this angle must be considered as positive if the moon is on the west of the meridian, and negative if on the east : let it be represented by T'. With the known latitude / of the place, and the values of p', D and r', find, from the formula (n) art. 161., the value a of the relative parallax in right ascension at the time T : again, with the same data and the value of a just obtained find, from the formula (v) or (vi) art. 162., the value 8 of the relative parallax in declination at the time T. Then A' a will express the apparent difference between the right ascensions, and D' 8, the apparent difference between the declinations of the sun and moon at the time T ; and it may be observed that the former difference is not zero because T is not exactly the time of apparent conjunction in right ascension. There must now be obtained (art. 358.) the moon's appa- rent altitude or zenith distance, and with this element there must be computed (art. 163.) the augmentation of the moon's apparent semidiameter, which augmentation being added to the semidiameter of the moon, taken from the Nautical Almanac, the sum will be the moon's apparent semidiameter. The sum of the semidiameters of the sun and moon will be required if the eclipse is partial, and their difference if total or annular. 372. Now (fig. to art. 369.) let sand M be the apparent places of the sun and moon at the time T, or the approximate time of apparent conjunction : let x Y drawn through M be a portion of the moon's apparent orbit, which, for a time equal to the duration of the eclipse, may be considered as a straight line ; and, for a partial eclipse, let sx and SY be each equal to the sum of the apparent semidiameters of the sun and moon. Then x and Y, in the orbit, will be the places of the centre of the moon at the commencement and end of the eclipse respectively. Let p be the pole of the equator, and PS a horary circle passing through the sun at the time T. Draw an arc M;? of a great circle perpendicular to PS, and let Mq be part of a parallel of declination passing through the moon at M. Draw also the straight line s v perpendicular to x Y ; then 332 ECLIPSE CHAP. XVII. v will be the place of the moon's centre at the middle of the eclipse, or the instant of greatest phase. The angle SPM is equal to A.' a, and the apparent decli- nation of the moon at M is D 8 : also the arc Mp of the great circle and Mq of the small circle may be considered as equal to one another; therefore (art. 70.) MP=(A / a) cos. (D 8). This value of M.p will be expressed in seconds if A' a be in seconds. Now (art. 71.) we have pq (in seconds) =i( A' a)- sin. 1" sin. 2 (D 8); therefore sp ( = Sq + pq) = -D'S + (A'ay sin. I" sin. 2 (0-8). 373. There must next be found the apparent relative horary motions in right ascension and declination; viz. dA.' d.dA.' dv' d.drt there must also be found the value of (, --- J cos. V dt dt i (D 8), the apparent relative horary motion on the parallel Mq of declination. Now j and are found from the Nautical Almanac, at dt as stated above ; and it has been shown that ^ - n: a t p r cos. / 0.2528 - cos. r: this value will be in seconds of a cos. D degree if p' be expressed in seconds, and, in the compu- tation, the values of D and r found for the time T must be used. Again, d D, the value of the true relative parallax in declination, being the value of 8 in the formula (vi) art. 162., may with sufficient correctness for the present purpose be expressed by P' (sin. Z cos. D cos. / cos. T sin. D), disregarding ^ a in the term sin. (r + % a) and considering cos. % a as equal to unity. Then ' becomes (T only being a function of the time, and D 8 being put for D) p' cos. / sin. (08) sin. T -j- sin. 1", which will be in seconds if P' be expressed in seconds: and thus there may be obtained the value of the apparent relative horary motion in declination. Imagine next, the line m n to be drawn parallel to P s : then these apparent relative horary motions upon the parallel of declination, and in declination, may be represented by Mm and CHAP. XVII. OF THE SUN. 333 mn respectively; therefore the values of these lines are known, and in the plane triangle Mmn we have Mm : wz/z::rad. : tan.wzMTz; thus the angle mMn or PSv may be found. Again, cos. mM.n : rad. ::Mm : M?z; thus Mtt is found, and it will represent the moon's apparent relative horary motion in her orbit. In the triangle MSJO, which may be considered as recti- linear, and right angled at p, we have $p : M/> : : rad. : tan. M s p, and cos. M Sp : rad. :: Sp : s M : thus the angle M sp and the distance s M may be found. The steps to be taken for obtaining the times of the com- mencement and end of the eclipse, and the time of the greatest phase, are similar to those which have been given in the investigation concerning the phenomena of an eclipse of the moon (art, 369.). 374. The moon, by her proper motion, occasionally passes between the earth and some planet or fixed star ; in which case the planet or star is suddenly concealed behind her disk, or as suddenly reappears after having been for a time in- visible : the disappearance is called an immersion) and the re- appearance an emersion ; and both phenomena are designated by the general term occultation. The conditions under which an occultation of a planet or star by the moon may be visible at any station are that the difference between the declinations of the moon and star should be less than 1 30' ; that the time of the conjunction should be more than two days before or after the day of new moon ; that the sun should be below, or very near the horizon, and that the star should be above it, In the Nautical Almanac there is given a table of the elements for occultations, from which may be obtained the Greenwich mean time of the conjunction of the moon and star in right ascension, as it would appear if a spectator were at the earth's centre ; the true right ascensions and declinations of the moon and star at the same time ; also the geographical parallels of latitude between which the occultation will take place. And the process of determining, for any given place, the time of immersion and emersion when a star is occulted by the moon may be as follows : 375. Find the right ascension of the midheaven of the station at the instant of true conjunction in right ascension, and subtract from it the star's right ascension; there will 334 OCCULTATION CHAP. XVII. remain, in time, the horary angle of the moon and star at the instant of true conjunction in right ascension: let it be represented, in degrees, by r. From the table of elements for occultations in the Nautical Almanac; that is, for the Greenwich mean time of true conjunction in right ascension, take the difference between the declinations of the moon and star, and the true declination of the star : let the latter be represented by d. Take the moon's equatorial parallax ; and, if the star be a planet, take its horizontal parallax: in the latter case, the difference between these will be the moon's relative horizontal parallax: then compute the absolute or the relative geo- centric horizontal parallax of the moon, according as a fixed star or a planet is used ; and let its value be represented by P'. Take out also the moon's horary motion in right ascension when the star is fixed ; if a planet, find the difference between the moon's horary motion in right ascension and the planet's horary motion in geocentric right ascension : let this be re- , , , dAf presented by -5 . Compute the parallax in right ascension from formula (i) (art. 161.), viz. - ^V sin. r, which is the value of a 01 of dA?-, cos. d also compute the value of * by the formula p'cos. I dr . , 0>T , N T cos. T -a* sin. I" (art. 371.). cos. d dt ^ In these formulae d, the declination of the star, is used instead of D, the declination of the moon, in order to obtain the parallaxes with respect to that part of the moon's limb which is in contact with the star at the times of immersion and emersion : hence, at those times, the distance of the star from the centre of the moon will be equal to the moon's true semi- diameter ; and thus the necessity of computing the augment- ation of that semidiameter is avoided. In the latter formula, T- represents the hourly motion of the earth on its axis with o f*f\ respect to a fixed star; viz - 2 3ho 56" r 15 2/28 "> or 54148 /x , which multiplied by sin. 1" gives -j- sin. 1" = 0.2625, whose logarithm is 9.41916. The above values being substituted in the formula for t in the investigation relating to eclipses of the sun (art. 371.), the value of t will be obtained, and we shall have r + t or T ; CHAP. XVII. OF A STAR OR PLANET. 335 that is, in Greenwich time, the approximate mean time of apparent conjunction of the moon and star in right ascension at the station. 376. For this time T there must then be found the common horary angle of the moon and star, as in the investigation above mentioned (arts. 371, 375.): let it be represented by r : and for the same time, compute as above the geocentric horizontal parallax p' ; also the true right ascensions and declinations, and the horary motions in right ascension and declination. The difference between the true right ascensions of the moon and star at the time T being called A?, subtract from it the parallax in right ascension (the value of a or of dAf above) ; the remainder A? a being multiplied by cos. d will give the value of M/> or M.q in the figure to art. 369., M being the apparent place of the moon's centre at the tune T. The difference between the declinations of the moon and star at the time T being called D', subtract from it the parallax in declination (formula (vi) art. 162.), viz. 8 = p' (sin. I cos. d cos. I sin. d cos. T') : the remainder D' 8 will be the value of sp or 8g t s being the apparent place of the star or planet. The values of -= and =- , or the true hourly motions in at at right ascension and declination, being found from the Nautical Almanac for the time T ; by differentiating the above values of a and 8, we have that of d.dA! , . d.dV , , . . . f dr j , as above, and =- = P cos.Z sin. a sm.r-y- sm. 1". at at at Thus there may be obtained the values of ( -^ ^ ) \ at at J cos. d for the apparent hourly motion of the moon on the parallel M^ of declination ; and -y- ~ for the apparent hourly motion of the moon in declination. These may be represented by Mw and mn\ and from thence the moon's horary motion Mn in her orbit may be computed. Now, s being the supposed place of the star or planet, and x and Y the centres of the moon when her limb is in contact with the star, sx and s Y will, each, be equal to the true semi- diameter of the moon, or to the sum of the semidiameters of the moon and planet: then XM and MY may be calculated as in the investigations concerning the lunar and solar eclipse. Lastly, by means of the horary motion in the orbit the times of describing XM and MY may be found ; and the moon being 336 TRANSIT CHAP. XVII. at M, at the time T, there may evidently be obtained the times of the immersion and emersion. 377. When the two inferior planets happen to be in con- junction with the earth and sun near the nodes of their orbits they appear to pass across the sun's disk: this phe- nomenon is frequently presented by Mercury, but the last time at which Venus was so seen was in the year 1769 ; and this planet will not be again in a like position with respect to the sun till the month of December in 1874. On all such occasions the observation of the phenomenon may be made subservient to the determination of the difference between the parallaxes of the planet and the sun ; and subsequently, the ratio between the distances of the planet and sun from the earth being known, to the determination of the absolute parallaxes and distances. Of the two planets, Venus is that which is the most favourably situated for affording precise values of those elements ; and the computation of the time at which, for a particular station, the phenomenon will occur may be briefly indicated in the following manner. The process consists in finding the times of ingress or egress, or those at which the planet enters and quits the sun's disk, as if the spectator were at the centre of the earth ; and then, in finding with the present knowledge of the relative parallaxes, the correc- tions which are to be made to that time on account of his position on the earth's surface. Comparing together the geocentric right ascension of the planet and the right ascension of the sun, as they are given in the Nautical Almanac for the noons of the two days between which noons a conjunction in right ascension must take place, there may be found approximatively (by means of the hourly motions which in this case may be considered as uniform) the time of the conjunction of the sun and planet in right ascension. This will be expressed in Greenwich mean time, and will be within one or two minutes of the true time of conjunction : let it be represented by T. For this time find from the Nautical Almanac, with an attention to second differences, the following elements: 1. The right ascension of the sun and the geocentric right ascension of the planet : take the latter from the former, and let the difference be represented by A?. 2. The declination of the sun and the geocentric declination D of the planet : take the latter from the former, and let the difference be represented by D' ; these last elements are to be considered as positive if north, and negative if south. 3. Find the semidiameters of the sun and planet. CHAP. XVII. OF AN INFERIOR PLANET. 337 Let s (fig. to art. 369.) be the place of the sun, and M the place of the planet at the time T: let XY be part of the planet's orbit, and PS a horary circle passing through the sun : also let MJP be an arc of a great circle perpendicular to p S and M q an arc of a parallel of declination passing through the planet. Then Mq or Mp A' cos. D ; and, neglecting pq, which is very small, Sp = D' (in seconds). From the Nautical Almanac, with an attention to second differences, take the relative geocentric hourly motion of the planet in right ascension ; which, since at the time of transit the movement of the planet in its orbit is retrograde, will be equal to the sum of the hourly motion of the sun and the geocentric hourly motion of the planet : let this be expressed in minutes of a degree and be designated by a. Then a cos. D will be the relative horary motion of the planet on the parallel M q of declination. Imagine m n to be drawn parallel to PS; then a cos. D may be represented by Mm. Take in like manner, with second differences, the relative hourly motion of the planet in declination, which is either the dif- ference or sum of the hourly motion of the sun and the geocentric hourly motion of the planet according as the variations of declination take place in the same, or in contrary directions : let this hourly motion be represented by m n in the figure, and designated d. 378. Proceeding next as in the investigations concerning eclipses of the moon and sun (arts. 369, 373.) we have / mn \ tan. mMn ( =. - J = - V Mm/ a cos. D and representing the angle mMn or QSw by I, / Mm*. \ a cos. D j. r jj I - I _ \~cos. mi&nJ ~" cos. I ' Now, to find Sv and Mv : T / A' cos. D\ In the triangle MpQ, cos. I : rad. :: Mp : MQ I = ), ( sin. i\ and cos. I : sin. I :: M : Qp :=A'COS. D - j : V cos. I' . , sin. I but sp Qp Q s = D A' cos. D , and in the triangle SQv, SQ cos. irrsv; therefore Su D r cos. I A' cos. D sin. I ; sin.' 2 i also, Q v ( SQ sin. i) = D' sm. I A' cos. D , cos. i 338 TRANSIT CHAP. XVII. sin. 2 I I . and Mv ( = Qv +MQ) = D' sin. I - A.' cos. ^ or = D' sin. i + A' cos. D cos. I. Let x and Y be the places of the planet when, as seen from the centre of the earth, the limbs of the sun and planet are in contact at the ingress and egress ; so that, at the ingress, sx and SY are, each, equal to the sum of the semidiameters of the sun and planet : then / &v\ sx : su::rad. : cos.xsw, ( = , V SX/ this angle is therefore found; and from it there may be obtained the value of xv or vY ( = xs sin. xsv). Now if, as in the figure, the time T, at which the planet is at M, precedes the instant of conjunction in right ascension, XMI^XV M.V and MY=XV + MV; and XV + MV may be expressed by xs sin. xs v + (D' sin. I + A' cos. D cos. i). This quantity being divided by --- - , the relative horary motion in the orbit, will give, in decimals of an hour, or (after such division) being multiplied by 3600, it will give in seconds, .the time t, which must be subtracted from T in order to give the instant of ingress, and added to T to give the time of egress ; we have, therefore, for a spectator at the centre of the earth, _ 3600 cos. i r . _, . . ,-, T + n: T + - jx S sm. X S v + (D'sm. I + A' cos. D cos. I) { . a cos. D c 379. Now, for a spectator on the earth's surface, let the relative parallaxes in right ascension and declination be re- presented by a and S : then M^> considered as a parallel of declination when affected by parallax will become, as in art. 372., (A' a) cos. D; and, neglecting pq, s/? will become D 7 8 : hence, substituting A' a for A', and D'- S for D' in the above expressions for sv and MV, we have s v = (D' 8) cos. i (A' a) cos. D sin. i, and MV = (D' S) sin. i -f- (A X ) cos. D cos. i. Now the values of a and S (the parallaxes in right ascension and declination) may be obtained from the formulae (n) art. 161. and (v) art. 162., on putting for D the geocentric declination of the planet, for r the sun's hour angle at the time T, and for p r the difference (supposed to be known) between the geocentric horizontal parallaxes of the sun and planet. This last value of s?; being divided by sx (the known sum or difference of their semi-diameters according as the CHAP. XVII. OF AN INFERIOR PLANET. 339 time of an exterior or of an interior contact is required), there will be obtained a value of cos. xsu, from which the angle and consequently its sine may be found. This sine, together with the last value of MU, may then be substituted in the second member of the equation for T + t ; and the result will be the time of either contact, at ingress or egress, for the given station on the surface of the earth. 380. If it be required to determine from the instants of ingress and egress observed at places on the earth's surface, the value of P', the difference between the parallaxes of the Sun and Venus, the process may be as follows. Let t and f be the intervals of time between the instants of ingress and egress and the time T of conjunction in right ascension ; then, neglecting the effects of parallax on the horary motions, t. a cos. D will express the relative motion of Venus and the Sun on the parallel of declination, from the hour circle pass- ing through P and x (fig. to art. 369.) to that which passes through P and M, while t.d will be the relative motion in declination from the parallel passing through x to that which passes through M. These being added to the above values of Mp and sp give (A' a) cos. D + t.a cos. D and D' 8 + t.d; of which the former may be represented by xv and the latter by sv: in like manner, t'.a cos. r> (A' ) cos. D and t'.d (D' S) may be represented by V'Y and sv'. Then, neglecting the errors in the tabular values of the right ascen- sions and declinations and in the semidiameters of the Sun and Venus, {(A' a) cos. D + t.a cos. D} 2 + {D' 8 + t.d}' 2 = sx\ and a corresponding equation may be found for SY 2 . But the parallaxes and horary motions being small quantities, in developing the first members of these equations, the second powers and the products of a, 8, a and d may be neglected, and we shall have {A' 2 2A' (a ta)} COS. 2 D = XV 2 , D /2 2D'(8 td) = SV 2 ; whence xs 2 = and a corresponding equation containing V may be obtained for YS 2 . In these equations substituting the values of t and V, or the differences between the observed times of ingress and egress and the computed time T of conjunction in right ascension, with the values of x s or Y s, the sum or difference of the semidiameters, the numerical equivalents of a and 8 z 2 340 LONGITUDE, CHAP. XVII. may be found ; and from these, by the formulae in arts. 161, 162. the value of P' may be obtained. Now, by Kepler's law (art. 254.), the ratio between the distances of the Earth and Yenus from the Sun is known ; let this be as 1 to r : then the ratio between the distances of the Sun and Venus from the Earth will be as 1 to 1 r. But the horizontal parallaxes being angles subtended at the centres of the Sun and Yenus by the semidiameter of the Earth, this ratio is the reciprocal of the ratio of the parallaxes : hence, if p represent the horizontal parallax of the Sun and p' that of 1 z/ Yenus, ^ is equal to -; which is therefore known. 9 1 r p Now, the value of P', supposed to have been found above, is the equivalent of p f p ; and from these two equations the separate values of p' and p may be found. From the last transit of Yenus, the parallax of the Sun, at his mean distance from the Earth, was found to be 8". 702. 381. Eclipses of the moon are of 110 value as means of determining the longitude of a station, it being impossible to observe the commencement or end of the obscuration with sufficient accuracy on account of the ill-defined edge of the earth's shadow on the moon; were it otherwise, the occurrence of either of these phenomena expressed in mean time at the station, and compared with the Greenwich time found by computation, would show at once the longitude of the station. With respect to an eclipse of the sun, the time of the commencement or end may be observed with con- siderable precision : hence, though the operation of correcting the right ascension and declination of the luminaries, or the longitude and latitude of the moon, on account of parallax, is laborious, the longitude of a station on the earth may be determined by it with accuracy, and the following process may be employed for the purpose. 382. The instant at which the phenomenon occurs may be expressed in sidereal or in mean solar time ; for example, let it be the latter, and let such mean time be represented by T. The difference between the longitudes of Greenwich and the station being approximatively known by estimation or other- wise, this difference (in time) must be added to, or subtracted from the mean time of the observation, according as the station is westward or eastward from Greenwich, and the result will be the approximate mean time which would be reckoned at Greenwich at the instant of the observation being made at the station. Let this be represented by r'. CHAP. XVII. BY AN ECLIPSE OF THE SUN. 341 For the time T' find from the Nautical Almanac the moon's right ascension and declination; and for the time T find the right ascension (art. 312.) of the meridian of the station : the diiference between this right ascension and that of the moon being multiplied by 15 gives the moon's hour angle at the station for the instant of the observation. For the time T' find from the Nautical Almanac the right ascension and declination of the sun ; also the true horary motions of the sun and moon, both in right ascension and declination. The geocentric latitude of the station is sup- posed to be known, or it may be found (art. 152.) from the geographical latitude. With the moon's declination, her hour angle, and the geocentric latitude of the station, find the moon's true zenith distance; and with this, find (art. 163.) the augmentation of the moon's semidiameter : the augmentation being added to the semidiameter which is in the Nautical Almanac will give her apparent semidiameter. Find in the Almanac the semidiameter of the sun; and let the sum or difference of these semidiameters be represented by s. For the time T' find from the Nautical Almanac the moon's equatorial horizontal parallax, and subtracting from it the horizontal parallax of the sun (N. A. p. 266.), the re- mainder is the relative equatorial horizontal parallax. Reduce this (art. 157.) to the relative horizontal parallax for the lati- tude of the station, and let it be represented, in seconds, by P'. With P', the geocentric latitude, the moon's hour angle, and her declination; find (art. 161. (n)) a, the relative parallax in right ascension, also find (art. 162. (v)) S, the relative parallax in declination. Applying 8 by subtraction or addition to the moon's true declination, observing that 8 always increases the distance of the moon from the elevated pole, there will be obtained the moon's apparent declination ; and adding to the moon's true hour angle found above the value of a, in time, the sum will be the moon's apparent hour angle. With p', the moon's apparent hour angle, her de- clination, and the geocentric latitude of the station ; find (arts. 371, 373.) the values of d -j and d-j 9 the variations of the CL 6 Ct 6 relative hourly motions in right ascension and declination. The differences between the true horary motions of the sun and moon in right ascension and declination, respectively, give the true relative horary motions ; and the variations of the relative horary motions being subtracted from the true horary motions, the results will express the apparent relative horary motions in right ascension and declination : the former being mul- z 3 342 LONGITUDE, CHAP. XVII. tiplied by the cosine of the moon's decimation gives the apparent relative horary motion on the moon's parallel of declination. Let the last horary motion be represented by p, and let the apparent horary motion in declination be represented by q.- The difference between the true right ascension of the sun and moon, found as above for the time T', being taken, if to this be applied the value of a, by addition or subtraction ; the result will be the apparent distance in right ascension between the centres of the sun and moon at the same time T', that is at the time of the observation being made at the station. In like manner the value of 8 being applied to the difference between the true declinations of the sun and moon at the time T' will give the apparent distance in declination between the centres : let this last be represented, in seconds, by n. The apparent distance in right ascension being multiplied by the cosine of the moon's declination will give the apparent distance between the centres on the parallel of declination passing through the apparent place of the moon : let this be represented, in seconds of a degree, by m. Let s be the apparent centre of the sun, and M that of the moon, at the instant of the observation, that is the instant at which the disks of the sun and moon are in contact (the commencement of the eclipse for ex- M_^--^r /' ample) ; then, if the estimated difference in longitude between Greenwich and the station were correct, it is evident that in the right-angled triangle MNS, the sides MN and NS might be represented, respectively, by m and n; and M s being represented by s, we should have m 2 + ri 2 = s 2 . But since there will, in general, be an error in that estimated longitude, let t (in decimals of an hour) denote that error ; then m + p t being substituted for m, and n + q t for n in the equation, there may be from thence found the value of t. This value, according to its sign, being added to, or subtracted from the estimated difference of longitude, will give (in time) the required difference. CHAP. XVII. BY AN ECLIPSE OF THE SUN. 343 Ex. May 15. 1826, at Sandhurst, the commencement of the solar eclipse was ob- served, by mean time, at 1 ho. 47' 1" (T). Estimated longitude from Greenwich (in time), 3'. Greenwich mean time of the observation at Sandhurst, 1 ho. 50' 1" (T'). Sandhurst mean time Acceleration ho. / 1 47 1 17.6 Sidereal interval since mean noon - - 1 47 18.6 Sidereal time at noon - 3 32 57.98 Right asc. mid-heaven - 5 20 16.58 Geograph. lat. Sandhurst - 51 20 33 Reduction (art. 152.) - 11 1 Geocentric lat. Colatitude - 51 9 32 90 - 38 50 28 Moon's equat. hor. parallax (N.A.)- Variat. in 1 ho. 50' 1" 54 25.6 1.21 54 24.39 8.48 54 15.91 6.5 Sun's hor. par. (N. A.) Relative hor. par. Reduction (art. 157.) Hor. par. for the station - 54 9.41 = 3249". 41 (P') With the elements above found, we have by spherical tri- gonometry, as in art. 358., the moon's true altitude at the station for the time T', = 51 18' 23. ho. R. asc. moon at 2 p. M. (N.A.) - 3 28 42.26 Variation in 9' 59" before 2 P. M. 20.06 R. asc. moon at the inst. T' 3 28 22. 2 R. asc. mid-heaven - 5 20 16.58 Difference - 1 51 54.38 15 Moon's hour angle west , n . of meridian - - 27 58 35.7 Declinat. moon at 2 P.M. 19 21 36.1(n) Variation in 9' 59" before 2 P, M. - - 1 39.6 Decl. moon at T' - 19 19 56.5 90 Moon's north polar dist. 70 40 3.5 Sin. moon's alt. - 14" (art. 163.) - 10".93 - - 9.89237 - 1.14613 - 1.03850 Augmentation of moon's semi- diameter Moon's semidiam. (JSTaut. Aim.) - - 14 49.6 Augmentation - - 10.93 Moon's augmented semidi. 15 0.53 Sun's semidiameter - 15 49.9 Sum of the semidiameters 30 50.43 or ]850".43 On putting in numbers, with the above elements, the for- mula for tan. a (art. 161.), we obtain, in angle, = 17' 2". 5 ; and, in time, =l / 8 // .16. Again, on putting in numbers the formula for tan. S (art. 162.), neglecting cos. \a, we have Also, on putting in numbers the formulae arts. 371. and 373., we find d.-^ 1 480".07 in angle, or 32".004 in time, and z 4 344 LONGITUDE, CHAP. XVII. d -5- 78 // .35. In these operations for finding the pa- dt rallaxes, logarithms with six decimal places will suffice. True right asc. sun at Gr. mean moon (Naut. Aim.) Variat. in 1 ho. 50' 1 '' - True rt. asc. sun at the time T' True rt. asc. moon at the same time True distance in right as- cension, moon west the sun Relative parallax in right ascension in time (a) - Apparent distance in rt. ascension, moon west of sun at the time T' (in time) - in angle, 31' 30". 45, or 1890".45 - log. = Cos. moon's apparent dec. (18 49' 41") - Apparent dist. in right as. on the moon's parallel of declination, at T', in arc = 178 9". 3 True dec. sun at Gr. mean noon (N. Aim.) Variat. in 1 ho. 50' 1" True dejcl. sun, at T' Moon's true hor. motion in declination at 1 ho. 50' 1 (Naut. Aim.) - Sun's true hor. mot. in decl. (Naut. Aim.) - Relative true hor. motion in declination Variation of horary motion in decl., or d dt Apparent relat. hor mot. in declin., 48 5". 85, or Now, putting t (in decimals of an hour) for the error in the estimated difference in longitude between Sandhurst and Greenwich, the equation m 2 + n 2 = s' 2 becomes (1789.3 + 1115.29*) 2 + (478.27 + 485.85 f) 2 = (1850.43) 2 , from which we obtain t (in hours) = 0.0014, or (in se- conds of time) 5".04 ; and therefore the difference between the longitudes is, in time, 2' 55" nearly. This determination is, however, subject to an uncertainty which always exists h. t. 3 / // 29 1.93 18.14 True dec!, moon at T' (above) - - 19 True distance in decl. moon north of sun - Relative par. in dec. (8) / // 19 56.5 22 16.93 30 15.2 e 3 e 3 29 20.07 28 22.2 Apparent dist. in decl. moon south of sun, or 478". 27 True declin. moon - 19 Relative parallax in declination 7 58.27 19 56.5 30 15.2 is- of t 57.87 1 8.16 Moon's apparent decl. 18 Moon's true horary motion in rt. asc. at 1 ho. 50' I" in time (Naut. Aim.) Sun's true horary mo- tion in right ascen- sion in time (N. A.) 49 41.3 / // 2 0.45 9.89 )f 11 2 6.03 >r 3.2765652 9.9761167 Relative true horary motion in right as- cension in time Variation of the ho- rary motion in right ascension, or d dt in time Apparent relative ho- rary motion in right ascension (in time) In angle 11 78". 34 Apparent relative horary mo- tion in right asc., in angle, 1178".34- Cos. moon's apparent declinat. (18 49' 41") 1 50.56 32.O04 i n 3.2526819 in 1856'35".9N. 1 3 .67 18 57 in >' 1" lecl. n in tion t. in 39 .57 / // 9 58.8 34.6 1 18.556 3.0712707 9.9761167 9 24.2 1 18.35 8 5.85 Moon's apparent relative hor. mot. on her parallel of decl., in angle, 1115" .29 3.0473874 CHAP. XVII. BY AN OCCULTATION. 345 in estimating the precise instant when an eclipse com- mences. 383. When the moon passes between the earth and a planet or fixed star, if either the immersion or the emersion take place at the unenlightened edge of the moon's disk, the in- stant of its occurrence may be easily distinguished from certain parts of the earth's surface ; and hence the longitude of the station may be determined with considerable precision. The process for determining the longitude of a station from the observed commencement or termination of the occupation of a fixed star or planet by the moon is very similar to that which has been just described, but is more simple since the parallax and the proper motion of a planet are very small, and those of a fixed star are insensible or may be disregarded. The instant (T) at which the immersion or emersion occurs may be expressed in mean time at the station, and there must be added to it, or subtracted from it, the estimated distance of the station in longitude (in time) from Greenwich : let the result be represented by T'. For the time T' find, from the Nautical Almanac, the moon's right ascension and declination; and, for the time T, compute the right ascension of the mid-heaven : the difference between these right ascensions being multiplied by 15 gives the moon's hour angle at the station for the instant of the observation. For the time T' there must be found from the Nautical Almanac the true horary motions of the moon in right as- cension and declination; and also the right ascension and declination of the star. With the geocentric colatitude of the station, the moon's declination and her hour angle, find the moon's true zenith distance, and subsequently the augmentation of her semi- diameter : let the augmented semidiameter of the moon be represented by s. For the time T' find, from the Nautical Almanac, the moon's equatorial horizontal parallax, and reduce it by art. 157. to the horizontal parallax (p') for the latitude of the station. With p', the geocentric latitude, the moon's hour angle and her declination, find (arts. 161. (n) and 162. (v)) the values of a and S, which are the moon's parallaxes in right ascension and declination ; and hence obtain the moon's apparent right ascension and declination. Find also the values of d -y-, and d -^, which are the variations of the moon's at at horary movements in right ascension and declination. 346 LONGITUDE, CHAP. XVII. From the true horary motions of the moon in right as- cension and declination subtract these variations; the re- mainders will be the apparent horary motions of the moon in right ascension and declination, and the former, multiplied by the cosine of the moon's apparent declination, gives her apparent motion on her parallel of declination : let this last be represented by p, and her apparent motion in declination by/- The differences between the true right ascensions and de- clinations of the moon and star found as above for the time T' being taken ; if to these be applied, by addition or sub- traction, the values of a and $ respectively, and the first of the two results expressed in seconds of a degree be multiplied by the cosine of the moon's apparent declination, there will be obtained the apparent distances between the star and the centre of the moon, on the parallel of declination and on a horary circle passing through the star: let these be repre- sented by m and n. Then, if the estimated difference between Greenwich and the station, in longitude, were correct, we should have, as in the process for a solar eclipse, m 2 + n 1 = s 2 . But t (in de- cimals of an hour) denoting the error, there must be sub- stituted in the equation, m + pt for m and n + qt for n ; and from the equation thence arising the value of t may be found. Finally, the difference between T' + t and T will express (in time) the required longitude. Ex. March 15. 1840. The emersion of a Leonis from the moon was observed, by meantime, at - - - - 8 ho. 17' 4" (T) Estimated longitude from Greenwich, in time, *. ' . . - - 3 Greenwich mean time of the observation ho. , Sandhurst mean time - 8174 Acceleration 1 21.25 - 8 Moon's r. asc. at 8 ho. Variat. for 20' 4" - Moon's r. asc. at T' R. asc. mid-heaven Moon's hour angle east of meridian Moon's decl. atSho. (N.A.) Variat. for 2O 7 4" - Moon's declin. at T' North polar distance 20 4 (T/) ho. , 9 58 43.85 41.13 Sidereal interval - 8 18 25.65 Sid. time at noon - - 23 32 27.3 9 59 24.98 7 50 52.95 R. asc. mid-heaven - 7 50 52.95 o / // Geocentric colat. Sandhurst 38 50 28 Moon's equatorial hor. par. noon(N. A.) - - 57 19.7 Variat. in 8 ho. 2 or a gam, by division, * Ct cos. z = cos. c a b + J- ( 2 + Z 2 ) cos. c. Next, let z be represented by C + h ; then (PI. Trigo., art. 32.) cos. z = cos. c cos. h sin. c sin. hi but because h is small, the arc may be substituted for its sine and its cosine may be considered as equal to unity ; therefore cos. z =: cos. c h sin. c : equating these values of cos. z, we get h sin. c = ab \ (a? + & 2 ) cos. c, or h ~ -2 -- ^(a 2 + fl 2 ) cotan. c. sin. c In this equation, a, b, and h are supposed to be expressed in arcs ; therefore, in order to have the value of h in seconds CHAP. XVIII. FORMULAE FOR THE SIDES. 363 when a and b are expressed in seconds, each of the two terms in the second member of the equation must be multiplied by sin. \". Hence the value of z may be found. If one of the stations as B were in the horizontal line C B", a? we should have b~ , and in this case h cotan. c. 398. In computing the sides of triangles formed by the principal stations on the earth's surface, when all the angles have been observed and one side measured or previously determined, three methods have been adopted. The first consists in treating the triangles as if they were on the surface of the sphere and employing the rules of spherical trigonometry. In the second method the three points of each triangle are imagined to be joined by lines so as to form a plane triangle ; then, the given side being reduced to its chord and the spherical angles to those which would be con- tained by such chords, the other chord lines are computed by plane trigonometry, and subsequently converted into the corresponding arcs of the terrestrial sphere or spheroid. The third method consists in subtracting from each spherical angle one third of the spherical excess, and thus reducing the sum of the three angles of each triangle to two right angles ; then, with the given terrestrial arc as one side of a plane triangle, computing the remaining sides by plane trigono- metry, and considering the sides thus computed as the lengths of the terrestrial arcs between the stations. The first and third methods present the greatest facilities in practice, and all may be considered as possessing equal accuracy. 399. The computations of the sides of the triangles by the first of the above methods might be performed by making the sines of the sides proportional to the sines of the opposite angles : but in so doing a difficulty is felt on account of the imperfection of the logarithmic tables; for the sides of the triangles being small, the increments of the logarithmic sines are so great as to render it necessary that second differences should be used in forming the correct logarithms. This labour may be avoided by using a particular formula, which is thus investigated : Let a be any terrestrial arc expressed in terms of radius ( = 1) ; then (PL Trigo., art. 46.) sin. a a \ a 3 (rejecting higher powers of a) sin. a or - I a 2 , a Now cos. a = (1 sin. 2 a)* ; hence cos.* a = (1 sin.' 2 a)* : 364 GEODESY. CHAP. XVIII. or, by the binomial theorem, cos.* a 1 a 1 (rejecting as before). Therefore ^ = cos.* a very nearly : and if a were expressed in seconds, sin. a ; = cos.3 a ; or sin. a = a sin. 1" cos.* a ; a sm. V or, in logarithms, log. sin. a log. a + log. sin. 1" 4- log .cos. ... (A). From this formula, while a (in seconds) is less than 4 de- grees we may obtain the value of log. sin. a with great pre- cision as far as seven places of decimals. Therefore, in any terrestrial triangle whose sides (in seconds) are represented by , b, and c, and whose angles, opposite to those sides, are A, B, and c, if a were a given side, and all the angles were given by observation or otherwise, we should have (in order to find one of the other sides, as b, from the theorem sin. A : sin. a :: sin. B : sin. b (art. 61.)), by substituting the above value of log. sin. #, log. a + log. sin. \" + ^ log. cos. a + log. sin. B log. sin. A = log. sin. b ; in which, since a is very small, its cosines vary by very small differences, and log. cos. a may be taken by inspection from the common tables. From the value of log. sin. b so found, that of b (in seconds) may be obtained by an equation corre- sponding to (A) above : thus log. b log. sin. b log. sin. 1" log. cos. b ; in which, since b is very small, log. cos. b may be found in the common tables, that number being taken in the column of cosines, which is opposite to the nearest value of log. sin. b in the column of sines. If a were expressed in feet it might be converted into the corresponding arc in terms of radius (== 1) on dividing it by R, the earth's mean radius in feet : and into seconds . by the formula a (in feet) -^-j r=r a (m seconds). K sm. 1" 400. A formula for the reduction of any small arc of the terrestrial sphere to its chord may be investigated in the fol- lowing manner : Let a be a small arc expressed in terms of radius ( = 1); then (PI. Trigon., art. 46.) we have o sin. \a \a (rejecting powers of a above the third) ; CHAP. XVIII. ANGLES BETWEEN CHOKDS. 365 hence 2 sin. I a, or chord . a, = a ^V ft3 > an d fl chord . a ^ a 3 : an equation which holds good whether a and chord . a be ex- pressed in terms of radius (=1) or in feet. If a were given in seconds we should have -^ (a sin. I") 3 equal to the difference between the arc and its chord in terms of radius ( r= 1) ; or in logarithms, 3 (log. a + log. sin. 1") log. 24 = log. (a chord . a) in terms of rad. ( = 1 ) ; (a \ -) for the It / value of the same difference ; or in logarithms, 3 (log. a log. R) log. 24 = log. (a chord . a) in terms of rad. (=1). 401. The reduction of an angle of a spherical triangle to the corresponding angle between the chords of the sides which contain it, may be thus effected : Let the curves AB, AC be two terrestrial arcs constituting sides of the triangle ABC, and let their chords be the right lines AB, AC : let o be the centre of the sphere, and draw OG, OH parallel to AB, AC : draw also OD and OE to bisect the arcs in D and E, cutting the chords AB and AC in d and e. Then, since the angles A OB, AOC are bisected by od and Oe, the angles Ado and Aeo are right angles ; therefore the alternate angles DOG, EOH will also be right angles, and DG, EH will each be a quadrant; also the arc GH or the angle GOH will be equal to the plane angle BAC between the chord lines. The sides AG, AH and the included sphe- rical angle at A being known, the arc GH which measures the reduced or plane angle BAC may be computed by the rules of spherical trigonometry (as in art. 64.) ; and in like manner the angles between the chords at B and c might be computed. The excess of each spherical angle above the cor- responding angle of the plane triangle formed by the chords of the terrestrial arcs is thus separately found ; and it is evident that the sum of the three reduced angles will be equal to two right angles if the spherical angles have been correctly observed. 402. The third method of computing the sides of terrestrial triangles is the application of a formula which was inves- 366 GEODESY. CHAP. XVIII. tigated by Legendre, who, in seeking what must be the angles of a plane rectilineal triangle having its sides equal to those of a triangle on the terrestrial sphere, arrived at the con- clusion that, neglecting powers of the sides higher than the third, each of the angles of the former triangle should be equal to the corresponding angle of the spherical triangle diminished by one third of the spherical excess found as above shown (art. 396.). This proposition may be demon- strated in the following manner. (See Woodhouse, " Trigo- nometry.") Let A, B, and c, as in the above figure, be the angles of a spherical triangle, and a, b, c expressed in terms of radius ( = 1) be the sides opposite to those angles ; also let p represent half the sum of those sides. Then, (art. 66. (n)), , ,- N sin. p sin. (p a) cos. 2 i A, or I (1-f cos. A) = -- j-4^ -- '- : sin. b sm. c developing (PI. Trigon., art. 46.) the second member as far as the third powers of p, a, b, c, we get , (1 + co , A) . (p or = or again, =l^^{l- $( p * + ( p _ )_ ^ - c *)}. But p \ (a + b + c) and p a \ (b + c a) : substituting these values in the second co-efficient of ^ ^ - - , b c the numerator of that co-efficient will be found to be equal to twice the product of ~ (a + c b) and | (a + b c), that is to 2 (p b) (p c) : therefore 1/1 \ P(p a ) P(P a )(p b)(p c} J(1+COS. A)=^__Z_0 '^-J^ J ... A) Now, in a plane triangle whose sides are expressed by a, b, c, and of which the angles opposite to those sides are repre- sented by A', B', c r , we have (PI. Trigon., art, 57.) cos. 2 J A', or I- (1 + cos. A') = P(Pa) 5 also, by the rules of mensuration, the square of its area is CHAP. xvin. LEGENDRE'S THEOREM. 367 equal to J 5 2 c 2 sin. 2 A', and to p (p a) (/> 5) (p c) ; therefore J # 2 c 2 sin. 2 A? =: p (p CL) (p Z>) ( jt? c), or y 1 ^ 5 c sin. 2 A' 35c ' Substituting the equivalents in equation (A) we have |(1+ cos. A) = (1 + cos. A 7 ) T2 # c sin. 2 A', or cos. A = cos. A 7 -J- Z> c sin. 2 A?. Again, assuming A A' + A, we have (PI. Trigon., art. 32.) cos. A = cos. A' cos. A sin. A? sin. A, or, since h is very small, cos. A = cos. A.'h sin. A? : hence h sin. A' = - Z>c sin. 2 A 7 , or A = be sin. A'. But, as above, \ b c sin. A? represents the area of a plane triangle of which the curve lines a, b, c, considered as straight are the sides : consequently h ^ (area), and A = A' + ^ (area) or A' A ^ (area). In like manner B' = B (area), and c' C ^ (area). Now the area of such triangle considered as on the surface of the sphere, and the radius of the latter being unity, has been shown to be equal to the excess of the arcs which measure the spherical angles, above half the circumference of a circle (the arcs being expressed in terms of radius = 1): hence, in employing the third method above mentioned, each angle of the spherical triangle on the surface of the earth must be diminished by one third of the spherical excess, in order to obtain the corresponding angle of the plane triangle, in which the lengths of the straight sides are equal to the terrestrial arcs whether expressed in seconds or in feet. 403. When a base line has been measured, or when any side of a triangle has been computed, it becomes necessary to reduce it to an arc of the meridian passing through one ex- tremity of the base or side ; and therefore the angle which the base or side makes with such meridian must be observed. For this purpose, as well as with the view of obtaining the latitude of a station by the meridian altitude, or zenith dis- tance, of a celestial body, or of making any other of the ob- servations which depend on the meridian of the station, the position of that meridian should be determined with as much accuracy as possible. Since the Nautical Almanacs now give at once the apparent polar distances and right ascensions of the principal stars, it is easy (art. 312.) to compute the moment at which any one of these will culminate ; and a first 368 GEODESY. CHAP. XVIII. approximation to the position of a meridian line on the earth's surface may be made in the following manner. A well-adjusted theodolite having a horizontal and a ver- tical wire in the focus of the object glass may have its telescope directed to the star a little before the time so com- puted ; then, causing the vertical wire to bisect the star, keep the latter so bisected by a slow motion of the instrument in azimuth till the moment of culmination. At that instant the telescope is in the plane of the meridian ; and the direction of the latter on the ground may be immediately indicated by two pickets planted vertically in the direction of the telescope when the latter is brought to a horizontal position. But, employing a transit telescope or the great theodolite with which the terrestrial angles for geodetical surveys are taken, a more correct method of obtaining the position of the meridian is that of causing the central vertical wire of the telescope to bisect the star Polaris at the time of its greatest eastern or western elongation from the pole ; that is, about six hours before or after the computed time of cul- minating ; and then, having calculated the azimuthal devia- tion of the star from the meridian, that deviation will be the angle between the plane of the meridian and the vertical plane in which the telescope moves. Consequently, by the azimuthal circle of the theodolite, the telescope can be moved into the plane of the meridian ; whose position may then, if necessary, be correctly fixed by permanent marks. - 404. To obtain the star's azimuth at the time of its greatest elongation from the pole, let N p z be the direction of the meridian in the heavens ; z s that of the vertical circle in which the telescope moves at the time of such elongation : let P be the pole, S the star, and z the zenith of the station. Then, in the spherical triangle ZPS, right angled at s, we have (art. 60. (e*)), Bad. sin. p s = sin. P z sin. p z s, and P zs is the required azimuth. The above method possesses the advantage of being free from any inac- curacy which, in the former method, might arise on account of the error of the clock ; since the star at S appears for a moment stationary in the telescope, and consequently it can be bisected with precision at that moment by the meridional wire. The method of bringing a transit telescope, or that of a great theodolite, accurately to the meridian, at any time, or of determining its deviation from thence, has been explained in arts. 94, 95, 96. 405. When a meridian line had not been previously de- CHAP. XVIII. REDUCTION TO THE MERIDIAN. 369 termined, the method employed in the English surveys for obtaining the angle which the measured base, or any side of a triangle, made with the meridian, was to compute from the elements in the Nautical Almanac the moment when the star a Polaris was on the meridian, or when it was at the greatest eastern or western elongation ; then, at the place where the angle was to be observed, having first directed the telescope of the theodolite to the star, at the moment, the telescope was subsequently turned till the intersection of the wires fell on the object which marked the other extremity of the base or side. With a telescope capable of showing the star during the day, should it come in the plane of the meridian while the sun is above the horizon, this angle may be thus taken ; or it may be obtained at night, when the star culminates after sun-set, if a luminous disk be used to in- dicate the place of the station whose bearing from the meridian is required. When the observation was made at the time of either elongation, the azimuthal deviation of the star, computed as above, was either added to, or subtracted from the observed azimuth according as the star was on the same, or on the opposite side of the meridian with respect to the station ; and the sum, or difference, was of course the required azimuth of the latter : but if the bearings of the station were observed at both elongations of the star, half their sum ex- pressed immediately the required bearing. Another method of obtaining the azimuth of a terrestrial object is given in art. 341. By some of the continental geodists a well-defined mark or, by night, a fire-signal was set up very near the meridian of the station whose azimuth was to be obtained ; then, by means of circumpolar stars or otherwise, they ob- tained the correct azimuthal deviation of that mark from the meridian ; and the sum or difference of the deviation, and the observed angle between the mark and the station, was con- sequently, equal to the required azimuth. 406. Different processes have been employed for reducing the sides of the triangles to the direction of a meridian passing through a station at one extremity of the series. Among the most simple is that which was adopted by M. Struve in measuring an arc of the meridian from Jacobstadt on the Dwina to Hochland in the Gulf of Finland. It consisted in computing all the sides of the triangles as if they were arcs of great circles of the sphere, either by the rules of spherical trigonometry or by the theorem of Legendre above men- tioned (arts. 398, 402.), one of them as AB being the measured base. Then, C, D, E, r, &c. being stations, imagining A and F, A and M, &c. to be joined by great circles ; in the B B 370 GEODESY. CHAP. XVIII. triangles ADF, AFM, &c. each of the arcs AF, were found by means of the two sides previously determined and the included angles ADF, AFM. Finally, the azi- muthal angle PAD having been obtained by observations, the angle PAM was computed : and imagining a great circle Mp to be let fall perpendicularly on the meridian AP, the meridional arc Ap was calculated in the right-angled spherical triangle AMp. A second process is that of computing by spherical trigonometry, or by the method of Legendre (art. 402.), the sides of the triangles ; and then, in like manner, the lengths of the meridional arcs AB', B 7 c 7 , &c. between the points where the sides of the triangles, pro- duced if necessary, would cut the meri- dian. For the latter purpose an azi- muthal angle, as P A B, must be observed : then, in the spherical triangle ABB', there would be given AB and the two angles at A and B ; to find A B', B B' and the angle B 7 . Again, in the tri- angle B 7 DC 7 , there would be given DB 7 (the difference between the computed sides B D and B #) the observed angle at D and the computed angle at B 7 ; to find C 7 B 7 , DC 7 and the angle c 7 . In like manner, in the triangle DC 7 E 7 , may be found DE 7 , C 7 E 7 and the angle E 7 . In the triangle E 7 F r D, formed by pro- ducing DF till it cuts the meridian in F 7 ; withDE 7 and the angles at D and E 7 may be computed E 7 F', DF' and the angle F 7 . Lastly, letting fall F k perpendicularly on the meridian, in the right angled triangle FF', with FF 7 (r=DF 7 DF) and the angles at F and F', the arc &F' may be determined ; and thus with ~s!k ( E 7 F 7 F'A) and AB X , B'C', C 7 E 7 , before computed, the length of the meridional arc from A to k is obtained. 407. When the sides of the triangles are the chords of the spherical arcs, and are computed by the rules of plane trigo- nometry, the following process is used for reducing those sides to the meridian. After calculating the chords AB, AD, BD, &c., the distance AB 7 and the angle AB 7 B are determined by means of the angle PAB. Subsequently, with the angle DB 7 c 7 ( = AB'B), the side B 7 c 7 is computed in the triangle B 7 DC 7 , and this is to be considered as a continuation of the former line AB 7 , which is supposed to have nearly the position of a chord within the earth's surface ; but the plane of the triangle ABD being considered as horizontal, that of the next triangle BCD will, on account of the curvature of the CHAP. XVIII. EEDUCTION TO HORIZONTAL PLANES, 371 earth, be inclined in a small angle to the former plane : therefore, in order to reduce the computed line B'c'to the plane of BCD, and allow AB' and the reduced line to retain the character of being two small portions of the geodetical meridian, that line B'C/ may be supposed to turn on the point B' as if it moved on the surface of a cone of which B' is the vertex and DB' or B'B the axis (that is, so as not to change the angle which it makes with D B') till it falls into the plane BCD. Thus let AM be part of the periphery of the terrestrial meridian passing through A in the former figure, and let AB' be the position of the chord AB'; then the first computed value of B'C' may, in the annexed figure, be repre- sented by B'C', which is in the plane of the meridian and of the triangle A B D. And when, by the conical movement above mentioned, the line B'C' comes into the plane of the triangle BCD in the former figure, it will have nearly the position of a chord line, and may be represented by B' N which terminates at N in a line imagined to be drawn from c' to the centre of the earth.* Since the inclination of the plane ABD to that * It must be remembered that (agreeably to what is stated in art. 387.) a geodetical meridian is a curve line the plane of which is every where perpendicular to the tangent plane or the horizon, at every point on the earth's surface through which it passes ; and unless the earth be con- sidered as a solid of revolution, the geodetical meridian is a curve of double curvature : the error which arises from considering it as a plane curve is, however, not sensible. Now, if AB', B'C' (in the above figure, and in the fig. to art. 406.) be considered as two small portions of the geodetical meridian, the vertical planes passing through these lines should be, respectively, perpendicular to the horizons at the middle points of the triangles ABD, BCD. Let it be granted that the vertical plane passing through A B' is perpendicular to the plane of the triangle ABD; and let it be required to prove that while B'C' (fig. to art. 406.) in the plane BCD makes the angle D B'C' equal to the angle A B'B, in the plane ABD, the vertical plane passing through A B' and B' c' may be considered as perpendicular to the plane BCD. Imagine, in the annexed figure, a sphere to exist having its centre at B' and any radius as B'D ; and let DTW, T>n be aics of great circles on such sphere, the former in the plane ABD and the latter as much below the plane BCD as Dm is above it : let also B' m be the prolongation of AB' in the plane ABD produced. Then, by the manner in which n'm was supposed to revolve to the position B'W (keeping the angle DB'/W or A B'B equal to DB'W) the spherical triangle Dm n is evidently isosceles, and a great circle passing through D, bisecting mn, will cut inn at right angles. Let B'N be in the plane of this circle; it will also be in the plane BCD, A and the latter will be cut perpendicularly by the plane passing through B B 2 372 GEODESY. CHAP. XVIII. of BCD is very small, the triangle B'NC' may be considered as right angled at N, and the angle C'B'N as that between a tangent at B' and a chord line drawn from the same point ; consequently (Euc. 32. 3.) as equal to half the angle B'EN, or half the estimated difference of latitude between the points B' and N. Therefore B' N (the reduced value of B' c) can be found ; and in a similar manner the reduced values of C' E ', &c. may be computed. The first station A is on the surface of the earth ; but the points B', c', &c., after the above reductions, are evidently below the surface : therefore the meridional arcs appertaining to the chords AB', B'N', &c. should be in- creased by quantities which are due to the distances of the points B', N, &c. from the said surface. When the sides of the triangles have been computed by Legendre's method (art. 402.), and the azimuthal angle between a station line and the plane of the meridian passing through one of its extremities has been observed ; if perpen- diculars be let fall from the stations to the meridian (the stations not being very remote from thence on the eastern or western side), the lengths of the perpendiculars and of the meridional arc intercepted between any station as A, and the foot of each perpendicular, may also be computed by the rules of plane trigonometry. For since the computed lengths of the station lines are equal to the real values of those lines on the surface of the earth, though the lines be considered as straight, the lengths of the arcs A, A, Ba, &c. (fig. to art. 406.), computed from them (one third of the spherical excess for each triangle being subtracted from each angle in the triangle), will be the true values of those arcs. Consequently the whole length of the meridional arc Ap will be correct. The following is an outline of the steps to be taken for the determination of the length of a meridional arc, as Ap, by perpendicular arcs let fall upon it from the principal stations and by arcs coinciding with the meridian, or let fall perpendicularly on the others from the several stations. Let AB (fig. to art. 406.) be the measured base; PAB the azimuthal angle observed at A ; and let B a, cb, &c. be the perpendiculars let fall from the stations B, c, &c. on the meridian AP : then, in the triangle AB, we have AB (supposed to be expressed in feet), the angle AB and the right angle at A, B', m, n : or the plane u'rnw, which is the plane of the geodetical meridian passing through B'C', may be considered as at right angles to the plane BCD. In like manner the plane passing through the next portion C'E' of the geodetical curve, in the plane of the triangle CDE, may be considered as at right angles to the plane of that triangle'; and so on. CHAP. XVIII. MERIDIONAL ARC DETERMINED. 373 a ; to find A a and Ba. In like manner in the triangle DAC, we have AD, the right angle at c and the angle DAC (equal to the difference between the angles DAB and PAB); to find DC and AC. Again, imagining BC? to be drawn parallel to AP, in the triangle C dB we have B C, the right angle at d and the angle CB(/(=ABC AB Bc7, the last being a right angle) ; to find cd and Bd; thus we obtain Ab (=Aa + Bd) and cb (cd + a B). In the triangle D C e, we have D C, the right angle at e and the angle D c e (the complement of D c ) ; to find D e andce: hence we obtain AC and DC a second time. The values may be compared with those which were determined before ; and if any difference should exist, a mean may be taken. In the like manner the computation may be carried on to the end of the survey ; and the whole extent of the meridional arc from A to p as well as the lengths of the several perpendiculars may be found. But at intervals in the course of the survey other azimuthal angles as P M F must be obtained by observation : then, since the angle r MJO will have been found from the preceding com- putations, and the angle PMJP by the solution of the right angled spherical triangle PM/> ; the sum of these two may be compared with the observed azimuth, and the accuracy of the preceding observations may thus be proved. The angle PHM being computed in the spherical triangle PHM, that azimuthal angle may be employed to obtain the meridional arcs and the perpendiculars beyond the point H. The process above described is particularly advantageous when it is intended to make a trigonometrical survey of a country as well as to determine the length of an extensive meridional arc ; for the spherical latitudes and longitudes of the stations A, B, c, &c. might be found from the above com- putations, and thus the situations of the principal objects in the country might be fixed. For this purpose it is con- venient to imagine several meridian lines to be traced at intervals from each other of 30 or 40 miles ; and to refer to each, by perpendiculars, the several stations in the neighbour- hood. The lengths of these perpendiculars will not, then, be so great as to render of any importance the errors arising from a neglect of the spherical excess in employing the rules of plane trigonometry for the purpose of making the re- ductions to the several meridian lines. 408. If a chain of triangles be carried out nearly in the direction of an arc perpendicular to any meridian, the situ- ations of the stations may, in like manner, be referred to that arc by perpendiculars imagined to be let fall on the latter ; and the lengths of the arcs and of the perpendiculars may be BBS 374 GEODESY. CHAP. XVIII. computed as before. The difficulty of obtaining the longi- tudes of places with precision is an objection to the em- ployment of this method in the survey of a country ; and the same objection exists to the measurement of an arc on a parallel of terrestrial latitude. The measured length of an arc on a perpendicular to a meridian, and on a parallel of latitude, have, however, been used in conjunction with the measured arc of the meridian at the same place, as means of determining the figure of the earth. In a triangulation carried out from east to west, or in the contrary direction, the sides of the triangles may be computed as arcs of great circles of the sphere : then with these sides and the in- cluded angles, the distances AB, AC, AD, &c. of the se- veral stations may be ob- tained by spherical trigo- nometry ; and from the last of these as M, letting fall Mp perpendicularly on the meridian of A, the arcs Ap, Mp may be computed in the right angled spherical triangle ApM. There must subsequently be obtained the distance from M to q on the arc of a parallel circle, as Mq, drawn through M, and the distance from p to q on the meridian. It has been shown in art. 71. that pq in seconds is approxi- mately equal to ^ p 2 sin. 2 PM sin. 1" (fig. to that article), the radius of the sphere being unity. Now if the arc Mp, com- puted as above mentioned, were in feet, and MC the semi- diameter of the earth be also expressed in feet ; since is M C equal to the measure of the angle MC/? at the centre, and that MC sin. M.cp MC' sin. MC'q, each member being equal to MN ; also, since MC'= MC sin. PM, we have, considering Mp as an arc of small extent, and putting Mp in feet for MC sin. MCp, also for sin. MC'q putting its equivalent sin. p or p sin. 1", MC sin. PM = p sin. I". Therefore pq sin. 1" (or pq in arc, rad. = 1) = * PM or -:MC 2 sin. 2 PM ' putting 2 sin. PM cos. PM for sin. 2 PM (PL Trigon., art. 35.), CHAP. XVIII. LENGTH OF A DEGREE DETERMINED. 375 M 2? 2 and / for the latitude of M, pq (in arc, rad. = 1) = ^ $ tan l\ > MC and, in feet, ^P 1 + j p q - - tan. /. f 2 2 MC In the same article it has been shown that the difference between M^ and M.p (in arc, rad. = 1) is approximatively equal to jr M/? 3 sin. 3 V tan. 2 I. Now if the arc M/> were in feet, and MC the semidiameter of the earth be also expressed in feet, Mj9 sin. \" in the last expression, in which Mp is sup- posed to be in seconds, would be equivalent to - , and that MC TVT T)"^ expression would become \ 3 tan. 2 /; therefore the differ- M C' ence between MJP and M^ in feet is, when ~M.p and MC are in feet, equal to ^ ^tan. 2 /, by which quantity wq exceeds M C" Mp. If, after the several distances AB, BC, CD, &c. have been computed in the triangulation, the latitudes of the stations and the bearings of the station lines from the terrestrial me- ridian passing through one extremity of each be observed or computed ; the lengths of the several arcs of parallel circles, as B, cc, DC?, &c., drawn from each station to the meridian passing through the next may be calculated and -subsequently reduced to the corresponding arcs qh, hk. Sue. on the parallel of terrestrial latitude Mq, which passes through any one, as M, of the stations. The sum of all such arcs will be the value of that whose length it may have been proposed to obtain. Whether the chain of triangles extend in length eastward and westward, or in the direction of the meridian, the value of pg must be subtracted from the computed value of Ap in order to obtain the length of the meridional arc com- prehended between A and the parallel of latitude passing through M : then the latitudes of A and M being determined by computation or found by celestial observations, the difference between the latitudes of A and M will become known, and such difference compared with the measured length of Aq will, by proportion, give the length of a degree of latitude at or near A. In like manner the difference between the longitudes of A and M, obtained by celestial observations, by chrono- meters or otherwise, if compared with the measured lengths of Mp and M q, will, by proportion, give the lengths near A of an arc of one degree on a great circle perpendicular to the meridian and on a parallel of terrestrial latitude. B B 4 376 GEODESY. CHAP. XVIII. 409. The usual method of finding the latitude of a station as A or M for geodetical purposes is similar to that which has been described in the chapter on Nautical Astronomy (art. 334.), some fixed star which culminates very near the zenith being employed, in order to avoid as much as possible the error arising from refraction ; and the altitude or zenith dis- tance being observed with a zenith sector (art. 107.). On the continent, however, lately, the latitudes of stations have been obtained from observed transits of stars at the prime vertical on the eastern and western sides of the meridian ; and the following is an explanation of the process which may be used. The observer should be provided with a transit telescope which is capable of being moved in azimuth, or with an altitude and azimuth circle : that which is called the horizontal axis should be accurately levelled, and the telescope should be brought as nearly as possible at right angles to the me- ridian. This position may be obtained by first bringing the telescope correctly to the meridian by the methods explained in arts. 94, 95. ; and then turning it 90 degrees in azimuth by the divisions on the horizontal circle. Let WNE represent the horizon of the observer, z his zenith, and P the pole of the equator ; also let NZN' represent the meridian, WZE the prime vertical, dss'd' part of the star's parallel of declination, and let s and s' be the places of the star at the times of observation. Imagine hour circles to be drawn through P and s, P and s' ; then P s, PS', each of which is the star's polar distance, are known from the Nautical Almanac, and if the times of the transits be taken from a clock showing mean solar time, the interval must be con- verted into sidereal time by the table of time equivalents, or by applying the " acceleration : " the sidereal interval being multiplied by 15 gives the angle SPS'. From the equality of the polar distances this angle is bisected by the meridian, and the angles at z are right angles ; therefore, the effects of re- fraction being disregarded, we have in the right angled triangle PZS (art. 62. (/') ) rad. cos. ZPS = cotan. PS tan. PZ, and PZ is the required colatitude of the station. But the true value of the angle ZPS is diminished by 'a small quantity depending on the change produced by re- fraction in the star's zenith distance, and a formula for do- CHAP. XVIII. TRANSITS AT THE PRIME VERTICAL. 377 termining the amount of the diminution may be thus inves- tigated. In the right-angled triangle ZPS we have (art. 60. (*)) sin. z S = sin. ZPS sin. p s : Differentiating this equation, considering PS as constant, cos. zs dzs cos. ZPS C?ZPS sin. PS. But, from the equation, sin. PS , and this value of sm. ZPS sin. PS being substituted in the differential equation, there is obtained cos. zs 7 cos. ZPS - d.zs t sin. zs sin. ZPS or cotan. zs dzs = cotan. ZPS , cotan. zs , tan. ZPS . whence C?ZPS = ~ dzs, or = dz s. cotan. ZPS tan. zs And if for dzs we put the value of refraction corresponding to the star's altitude, the resulting value of C/ZPS may be added to that of the angle ZPS which was determined from the half interval between the observations in order to obtain the value of ZPS which should be employed in the above formula for PZ. Ex. At Sandhurst, November 23. 1843, the interval between the transits of a Persei at the prime vertical was found to be, in sidereal time, 2 ho. 52' 22": consequently half the hour angle = 21 32' 45". The star's north polar distance from the Nautical Almanac was 40 41 ' 50".5, its apparent zenith distance by observation was 13 51' (nearly) ; and consequently its refraction = 14".2 ( dzs). Let this be represented in the figure by s s, and the corresponding variation of the hour angle by SPA' : then, by the formula above, log. dzs(l 4".2) - + log. tan. ZPS (21 32' 45") + log. tan. zs (13 51') co. ar. =log. JZPS= 22". zps = 21 32' 45" 1.152288 9.596415 0.608097 In the triangle ZPS. log. cos. ZPS log cotan. PS = log. tan. PZ = 38 39' 28", the quired. - 9.9685218 - 0.0654716 1.356800 74 (=SP) - 9. 9030502 colatitude re- ZPS = 21 33 7. 74 If the time at the station should be well known, the sidereal time corresponding to the middle of the interval between the times of observation may be computed ; and this should agree with the sidereal time at which the star is on the meridian, that is, with the star's right ascension. If such agreement be not found to subsist, the error must arise from the transit telescope not being precisely in the prime vertical : let it be supposed that the telescope is in the plane of the vertical 378 GEODESY. CHAP. XVIII. circle W'E' ; then the places of the star at the times of ob- servation will be p and q, and PZ let fall perpendicularly on W'E' will denote the colatitude obtained from the above for- mula. The true colatitude PZ may then be found in the triangle PZZ ; for PZ has been obtained as above, the angle at z is a right angle, and the angle ZPZ is equal to the dif- ference between the star's right ascension and the sidereal time at the middle between the observations : therefore (art. 62. (/')) cos. ZPZ = tan. PZ cotan. PZ ; . COS. ZPZ . , . , hence cotan. p z = ; or, in this last expression, putting for tan. p z its value - before found, we have cotan. cotan. PS cos. ZPZ cotan. PS cos. ZPS tan. PS pz = , or tan. PZ = - . cos. ZPS cos. ZPZ The result immediately deduced from the observation must be re.duced (art. 152.) to the geocentric latitude. 410. In the progress of a geodetical survey, it becomes ne- cessary, frequently, to determine by computation the difference between the latitudes and the longi- tudes of two stations as A and B, when there are given the computed or measured arc AB, the la- titude and longitude of one station as A, and the azimuthal angle PAB: and if the earth be con- sidered as a sphere, the following processes may be employed for the purpose. Let P be the pole of the earth, and let fall p>t perpendicularly on the meridional arc P A : then, by the usual rules of spherical trigonometry, we have I. (art. 62. (/') ) rad. cos. PAB = cotan. AB tan. A; whence At and consequently Pt are found. II. (art. 60. 3 Cor.) cos. AB cos. Pt cos. At cos. PB ; whence P B the colatitude of B is found. ill. (art. 61.) sin. PB : sin. A :: sin. AB : sin. p ; and the angle APB is equal to the difference between the longitudes of A and B ; and IV. sin. p B : sin. A : : sin. A p : sin. A B P ( the azimuthal angle at B). But avoiding the direct processes of spherical trigonometry, Delambre has investigated formulae by which, the geodetical arc between the stations being given with the observed la- titude and the azimuth, at one station, the differences of CHAP. XVIII. FORMULA FOB DIFFERENCE OF LATITUDE. 379 latitude and longitude between the stations, and the azimuth at the other station may be conveniently found. Thus, as above, for the difference of latitude, let P be the pole of the earth, A a station whose latitude has been observed, and PAB the observed azimuth of B ; it is required to find the side PB, and subsequently the angles at P and B. In the triangle PAB we have (art. 60. (a), (), or (c)) cos. PB cos. A sin. PA sin. AB + cos. PA cos. AB. Now, let I be the known latitude of A; dl the difference between the latitudes of A and B ( = A ra if BWZ be part of a parallel of terrestrial latitude passing through B) ; thfcn I 4- dl is the latitude of B, and the equation becomes sin. (l+dt) = cos. A cos. I sin. AB + sin. I cos. AB, or (PL Trigo., art. 32.) sin. Zcos. dl + cos. I sin. dl cos. Acos. I sin. AB-J-sin. / cos. AB, or again (PL Trigo., arts. 36, 35.) sin. / (1 2 sin. 2 dt) + 2 cos. / cos. \dl sin. \dl cos. A cos. / sin. AB + sin. / (1 2 sin. 2 \ AB) ; whence 2 sin. / sin. 2 ~dl+2 cos. / cos. \dl sin. \dl zr cos. A cos. I sin. AB 2 sin. / sin. 2 ^AB, or dividing by 2 cos. /, tan. I sin. 2 \dl + cos. \ dl sin. \dl = % (cos. A sin. AB 2 tan. / sin. 2 ^AB). Let the second member be represented by p ; also for tan. / put 2 = - i cos. 2 A sin.' 2 AB tan. 7+2 cos. A sin. AB sin. 2 |ABtan. 2 7, and ^ p^ = | cos. 3 A sin. 3 AB, rejecting the remaining terms, which contain powers of AB higher than the third. Now (PI. Trigo., art. 46.) sin. AB = AB AB 3 ; there- fore the first term in the value of 2p becomes AB cos. A AB 3 cos. A ; also the equivalent of p 3 may be put in the form jr A B 3 cos. 2 A cos. A. This being added to the second term just mentioned, the three terms are equivalent to AB COS. A AB 3 sin. 2 A COS. A. Again, putting | AB for its sine, the second term in the value of 2p becomes J AB 2 tan. 7; and the first term in the equivalent of 2 qp 1 becomes -f \ A B 2 cos. 2 A tan. 7. These terms being added together, produce the term i AB 2 sin. 2 A tan. I Thus we obtain for dl or Am (the required difference of latitude) AB cos. A \ AB 2 sin. 9 A tan. I | AB 3 sin. 2 A cos. A (1 + 3 tan. 2 /) &c. Here AB and dl are supposed to be expressed in terms of radius (= 1) ; if A B be given in seconds, we shall have, after dividing by sin. 1", dl (in seconds) = AB cos. A AB 2 sin. V sin. 2 A tan. I | AB 3 sin. ' 1" sin. 2 A cos. A (1 + 3 tan. 2 Z) &c. This formula is to be used when the angle PAB is acute : if that angle be obtuse, as the angle FAB', its cosine being negative, the formula would become dl (in seconds) AB cos. A AB 2 sin. I" sin. 2 A tan. 7 + | AB 3 sin. 2 V sin. 2 A cos. A (1 + 3 tan. 2 7) &c. If in either of these expressions we write for 7 the term I' (77, in which case I' represents the latitude of B ; then since, as a near approximation, tan. (/'- r77) = tan. I - tan. dl, the term \ AB 2 sin. l^sin. 2 A tan. 7 becomes - \ AB 2 sin. 1" sin. 2 A tan. I' + \ AB 12 sin. 1" sin.' 2 A tan. dl\ CHAP. XVIII. COMPRESSION AT THE POLES. 381 and for dl putting AB sin. I" cos. A, its approximate value, we get \ AB 2 sin. \" sin. 2 A tan. I' + \ AB 3 sin.- 1" sin. 2 A cos. A. If to the last of these terms we add the term -I AB 3 sin. 2 V sin. 2 A cos. A, the first expression for dl above becomes, neg- lecting the terms containing tan. 2 /, AB cos. A ~ AB 2 sin. 1" sin. 2 A tan. I' + ^ AB 3 sin. 2 1" sin. 2 A cos. A ; in which all the terms are negative when the angle at A is obtuse. This formula corresponds to that which denotes the value of M/> in art. 360. ; and from it we should have p A = PB + AB cos. A \ AB 2 sin. V sin. 2 A tan. l r + c. conformably to the value of PZ in art. 352. The angle at P, or the difference of longitude between the stations A and B, and also the azimuthal angle at B, will be most conveniently found by the direct formula in. and IV. above given. Delambre observes that the meridional arcs computed on the hypothesis that the earth is a sphere do not differ sensibly from those which would be obtained on the spheroidal hypo- thesis ; because the sums of the spheroidal angles of the triangles are, without any appreciable error, equal to those of the spherical angles, and the chords of the sides are the same. Thus we obtain correctly the sides of the triangles and the arcs between the parallels of latitude whatever be the figure of the earth (provided it do not differ much from a sphere). He adds that the consideration of the spheroidal figure of the earth is of little importance in the calculations, except for the purpose of changing into seconds the terrestrial arcs which have been measured or computed in feet. This may be done by dividing such arc by the radius of curvature at the place, instead of dividing it by what may be assumed as the mean radius of the earth. 411. The length of a degree of latitude being found, from the results of admeasurement, to be greater near one of the poles of the earth than near the equator, it follows that the earth is compressed in the former re- gion, or that the polar semi-axis is shorter than a semidiameter of the equator. For, let A a and B b represent the lengths of two such terrestrial arcs, supposed to be on the same meridian, and 382 GEODESY. CHAP. XVIII. imagine AD, #D, BE, JE to be normals drawn to the earth's surface ; the angles at D and E will each be equal to one degree: and since A#, B# may be considered without sensible error as portions of circles, the sectors A Da, BEZ> will be similar, and the lines AD, BE, which may be considered as radii of curvature, will be to one another in the same ratio as the arcs. Therefore Aa, which is the nearest to the pole P, being longer than B, AD will be longer than BE, or the sur- face of the earth about A will be less convex than the surface about B, which is the nearest to the equator. 412. Adopting now the hypothesis that the earth is a sphe- roid of revolution, such that all the terrestrial meridians are ellipses whose transverse axes are in the plane of the equator, and the minor axes coincident with that of the earth's rota- tion; the principal investigations relating to the values of terrestrial arcs will be contained in the following propo- sitions. PEOP. I. Every section of a spheroid of revolution, when made by a plane oblique to the equator, or to the axis of rotation, is an ellipse. Let XYC, XPC and PCY be three rectangular co-ordinate planes formed by cutting a spheroid in the plane of the equator (XYC) and in those of two meridians at right angles to one another ; and let a plane cut the spheroidal surface in AB and the plane XPC in AR, which for the present purpose may be supposed to be a normal to the spheroid at A; it is required to prove that the section is an ellipse. Let CD ( x), DE ( y) and E B ( = z) be rectangular co-ordinates of any point B in the curve line AB; and let RF( = #'), FB (=y) be rect- angular co-ordinates of B with respect to A R : also let I represent the angle ARX, and 6 normal or vertical plane ABR to the co-ordinate plane XPC. Imagine a perpendicular to be let fall from B on the plane XPC, it will meet the plane in G ; and join G, F : draw also FL in the plane XPC perpendicular, and GK parallel to xc. Then we shall have, BF and FG being perpendicular to A R so that the angle B F G is the inclination of the plane A B R to the plane XPC, y ( BG or ED, or :=: BF sin. BFG) y' sin. ; CHAP. XVIII. SPHEROIDAL TRIANGLES. 383 also, DL or GK being equal to GF sin. GFK or GF sin. /, and GF to BF COS. BFG Or BF COS. } #(:=CDOrCR-h RL + DL) CR + X f COS. I + y' COS. 6 sin. I ', ( = G D or F L F K) = #' sin. / y' cos. 6 cos. /. Substituting these values of x, y and z in the general equation AZ 2 4- B (x 1 -f y Q ) z= C of a spheroid, the latter will manifestly become of the form w/ 2 + px'y' + qx' + ry' s, which is the equation to a line of the second order : the va- riables in it being the assumed co-ordinates of B, it follows that the equation appertains to the curve line AB ; and since the curve, being a plane section of a solid, returns into itself, it is an ellipse. PROP. II. 413. When the sides of a terrestrial triangle do not much exceed in extent any of those which are formed in a geodetical survey, the excess of the three angles above two right angles is, without sensible error, the same whether the earth be con- sidered as a spheroid or a sphere. This may be proved by comparing the angles of a spheroidal, with those of a spherical triangle, when the angular points of both have the same latitudes and equal differences of longitude* agreeably to the method pursued by Mr. Dalby in the " Phi- losophical Transactions" for 1790. Let PRM be a spheroidal triangle of which one point as p is the pole of the earth, and let prm be a corresponding triangle on the surface of a sphere: let also ABC be the plane of the equator. Imagine the normals MD and RE to be drawn cutting AC and BC in g and h ; then the angles M^/A R^B, which express the sphe- roidal latitudes of M and R, will be equal to the angles mca, rcb respectively ; and we shall have MD parallel to me and RE parallel to re. Imagine also the planes DMR', ERM' to pass through MD and RE pa- rallel to the plane r?^c ; then R r D will be parallel to re, and 384 GEODESY. CHAP. XVIII. consequently to HE; and M'E will be parallel to me, conse- quently to MD: also the angles PR'M and PM'R are, respec- tively equal to the spherical angles p r m and pmr. But since, within the limits of the terrestrial triangles, the deviation of the produced normal ER from the plane MDR conceived to pass through the normal MD, is imperceptible from the station M, we may consider the planes MDR and MER as coincident; and the supposed vertical plane passing through MR to cut the parallel planes MR'D, RM'E under equal angles of incli- nation, so that the angle RMR' may without sensible error be considered equal to MRM': then the spheroidal azimuthal angle PRM will be as much less than the spherical angle PR'M ( prm) as the spheroidal azimuthal angle PMR ex- ceeds the spherical angle PM'R ( pmr). Consequently the sum of the spheroidal angles at R and M will, without sensible error, be equal to the sum of the spherical angles prm, pmr. The like will be true for another triangle as PNM, in which we shall have the sum of the angles PMN, PNM equal to the sum of the angles PMN', PN'M ; therefore, describing the arcs R N, R'N', we have the angles PRM + PMR = PR'M + PMR', and PNM+PMN PN'M + PMN': then by addition, PRM + PNM + RMN = PR'M -f P N'M + R'M N'. Again in the triangles PRN, PR'N', we have as before PRN + PNR =r PR'N' + PN'R'; and subtracting this equation from the preceding, we have NRM -f- RNM + RMN N x R r M + R'N X M + R'MN'. That is, the sum of the angles in the spheroidal triangle RMN is^without sensible error equal to that of the angles in the triangle R'MN', which are those of a spherical triangle rmn, whose angular points correspond, in latitude and longitude, with the points R X ,M,N'. By a different investigation, Legendre has ascertained that the difference between the spherical and the spheroidal excess of the angles of a triangle, above two right angles, in the greatest triangle ever formed on the surface of the earth, does not amount to ^ of a second ; therefore, in computing the sides of the terrestrial triangles, the latter may always be considered as appertaining to the surface of a sphere. CHAP. XVIII. RADII OF CURVATURE. 385 PROP. III. 414. To find the radius of curvature at a certain point A (fig. to art. 412.) in the periphery of a vertical section of a spheroid. It has been already proved that the general equation for the curve line produced by such a section is m^ + ny" - px'y' + qx + ry f = s . . . . (i), and comparing the co-efficients of the variables x f and y f with those of the same variables in the equation az~ + bx 1 + by 1 = cPW- of a spheroid, when for x, y, z, are substituted their equivalents (art. 412.) it will be seen that m = a 9 sin. 8 1 + 6* cos. 8 / n = fe 9 + (a a -6 2 ) cos. 8 I cos. 8 p = 2 (a 2 6 2 ) sin. I cos. I cos. 6 q = 2 R c 6 2 cos. Z r = 2 R c 6 s sin. Z cos. = (a 8 -RC 8 )6 8 . also a, representing half the equatorial diameter, b half the polar diameter of the spheroid, and a 2 b* or a 1 e 1 represent- ing the square of the excentricity of the ellipse X AP, we have, by conic sections, RC rz , and AR zr -, - ~ -. s^n 7^ - s ; (1 e" 2 sin. 2 /)* a (1 e 2 sin. 2 Now x f and j/ 7 being the co-ordinates of any point in a curve, the usual formula for the radius of curvature is, when dy f is considered as constant, m _ i 1 + (I)T. ~ 372 / ' and differentiating the equation (i) twice successively, con- sidering dy f as constant ; also, since the radius is required for the point A where y ! 0, making y' equal to zero after each differentiation, we have dtf _ r -px f dy' ~ q + 2mx f * dy'*~ But, from the preceding values of p and r, and since, at the point A, x f AR, it will be found that rpxf, or r px' 0; therefore the radius R of curvature at the point A becomes c c 386 GEODESY. CHAP. XVIII. equal to - ^ ; in which replacing ra, n, q and x* by their values above, we obtain _ f V + (a* - 2 ) cos. 2 / cos. 2 0}' When the ellipse AB coincides with the plane of a terrestrial meridian as XAP, we have 6 ; and designating the radius of curvature in this case by R' (which is then in the direction of AR) we have Z> 2 a (1 - * 2 ) R' = - } or = - 3 : a (1 e 1 sin. a /)' (1 - e 1 sin.. 2 IJ again, if the said ellipse be perpendicular to the plane of a meridian XAP, we have 6 90 degrees ; and designating the radius of curvature in that case by R" (which is then also in the direction of AR) we have R" = The above formulae may be transformed into others, equiva- lent to them, in which e, the compression ( ~ - J is em- ployed instead of e, the excentricity. Thus, by the nature of the ellipse we have - - = e 1 ; now since the difference a 2 between a and b is very small, if we put 2 a for a + b and multiply the above value of the compression by - -(ml) * (t. cP U 1 we shall have a =: s; then comparing this equation with that for e 2 , we obtain \ e 1 =. e, or e 2 2s. Again, let APQ represent a meridian of the terrestrial spheroid, AQ a diameter of the equator, p one of its poles, and F the focus. Imagine PF' to be equal to the semi-transverse axis, and F'c' drawn perpendicularly to p c, to be equal to the difference between the semi-transverse and semi-conjugate axes: also let the angle F'PC' be represented by I. Then, when PF', or PF, or AC, is equal to unity, F'C' ( = e) becomes equal to sin. I : and because the difference between the semi-axes is small, P c ( = 1 e) is sometimes put for p c', and is represented by cos. I ; or CHAP. XVIII. RATIO OF THE EARTH'S AXES. 387 1 cos. I is put for s. But 1 cos. I is (PL Trigo., art. 36.) equal to 2 sin. 2 \ I ; therefore s is equivalent to 2 sin. 2 1 1 : or putting J sin. I for sin. \ I because the angle I is very small, i sin. 2 1 = sin. 2 Ji ; whence s may be represented by | sin. 2 I ; or 2 e, that is, e 1 by sin. 2 I. PROP. IV. 415. The lengths of two meridional arcs as A, B& (fig. to art. 411.), one near the pole and the other near the equator, and each subtending one degree, being given by admeasure- ment ; to find the ratio which the earth's axes have to each other, the meridian being an ellipse. Imagine the normals D, AD, Z>E, BE to be drawn from the extremities of the arcs till they meet in D and E respectively ; and let AD, BE intersect CQ, the equatorial semidiameter, in N andN': also let the angles ANQ, BNQ (the geographical latitudes of the points A and B) be represented by I and I' respectively. Then, by the similarity of the sectors AD a and BE, repre- senting AD and BE, the radii of curvature, by R and R', we have R : R' :: Aa : B, or R . arc B R'. arc A (A). But (art 414.) B= a \~ */ ,; or, developing or (1 e 1 sin. 2 /) 5 by the binomial theorem and neglecting powers of e above the second, B = or (1 - e 1 ) (1 + le 1 sin. 2 /) = a (1 - e 1 + ^e 1 sin. 2 7). In like manner R' =. a (1 e 1 + f e 1 sin. 2 7') : *. , therefore the above equation (A) becomes B (i _ e <> + I e i gin. 2 T) A (1 - e 1 + f e" sin. 2 Z r ), and from this by transposition we obtain A B __ - f (A sin. 2 V - B sin. 2 /)" Thuse 2 is found: tt '2 _ 2 but e 1 = --- ^ 5 whence a 1 (1 e 2 ) = b ', and a (1 ^-e^b : developing the radical as far as two terms, we get 1 | 2 = -, in which substituting the numerical value of e 1 above, we have the required ratio of the semi-axes a and b. The numerical value of e being substituted in the above c o 2 388 GEODESY. CHAP. XVIII. equation R . arc B = R'. arc A, there might from thence be obtained the value of a ; and from the values of e and a, that of b might be found. Thus the equatorial and polar semi- diameters of the earth would be completely determined. PROP. Y. 416. To investigate the law according to which the lengths of the degrees of terrestrial latitude increase, from the equator towards either pole. It has been shown in the preceding article, that, in the plane of the meridian, R a(l-e 2 +fe 2 sin. 2 /). But, at the equator, where I 0, the radius of curvature becomes equal to a(l e 2 ); therefore, by subtraction, the increment of the radius, for any latitude I, above its value at the equator, is equal to f ae 1 sin.- 1: or the increments of the radii vary with the square of the sine of the geographical latitude of the station. Now the lengths of the degrees of latitude have been shown (art. 415. (A)) to vary with the radii of curvature in the direction of the meridian of any station ; therefore, by pro- portion, the increments of the degrees of latitude vary with the increments of the radius ; that is, with the square of the sine of the latitude. PROP. VI. 417. To determine the radius of any parallel of terrestrial latitude, and find the length of an arc of its circumference between two points whose difference of longitude is given. Let P^A be a quarter of the ellip- tical meridian passing through q ; and let q R be the direction of a normal at q. Let AC ( a) be a radius of the equator, and PC ( = &) be the polar semi-axis : let also q Ji ( = s c or x) be a radius of the parallel M q of latitude ; and let a e be the excentricity of the elliptical meridian. Then, by the nature of the ellipse, we have CHAP. XVIII. PARALLEL ON A SPHEROID. 389 now let / be the latitude of q, or the value of the angle A R q ; then s 2 ^ (1 e 2 sin. 2 7)3 Now the angle sm~l iae? cos.l sin' 2 Z} 3600 sin. I' 7 ; c c 3 390 GEODESY. CHAP. XVIII. whence A -A' -a (1 cos. T) 3600 sin. V _ (fa sin. 2 laa cos. / sin. 2 /) 3600 sin. 1" , A A 7 a (1 - cos. Z) 3600 sin. 1" _ 2> >r (a sin. 2 l+*a sin. 2 l-a-\a cos. / sin. 2 T) 3600 sin. 1" ~ A _ A /_2 a sin. 2 1/3600 sin. 1" or (a sin. 2 / sin. 2 %la cos. 2 /) 3600 sin. 1" neglecting e 2 in the above value of A we may, as an ap- proximation sufficiently near the truth, put ; =-^ for a ; ouUU sin. I when we shall have a _ AA' 2 A sin. 2 \l A cos. / A? "" A sin. 2 / sin. 2 | / A cos. 2 I ~ A sin. 2 / sin. 2 \ I A cos. 2 V The value of e being thus found, the ratio of the axes may be obtained as in Prop. IV. (art. 415.) : and in a similar man- ner, the length of a degree perpendicular to the meridian being found by multiplying the value of R" (art. 414.) into 3600 sin. 1", may the ratio be obtained from the measured length of a degree of latitude combined with that of a degree on an ellipse perpendicular to the meridian. PROP. VIII. 419. To find the distance in feet, on an elliptical meridian, between the foot of a vertical arc, let fall perpendicularly on the meridian from a station near it, and the intersection of a parallel of latitude passing through the station. Also, to find the difference in feet between the lengths of the vertical arc and the corresponding portion of the parallel circle. Let P (fig. to Prop. VI.) be the pole of the world, c its centre, M the station ; and let M p, M q be the two arcs, as in art. 410. Since normals to the elliptical meridians PA, PB, at the points M and q, may, without sensible error, be con- sidered as meeting in one point Q on the axis PC ; it is evi- dent that the values of cos. P and of 1 cos. P, or 2 sin. 2 ^p will be the same whether the earth be a spheroid or a sphere, and consequently that they will coincide with the values TVT "1C given in art. 7 1 . Now M Q is equal to - (PL Trigon., COS. X M Q art. 56.), and XMQ is equal to MTB, the geographical latitude of M ; therefore, putting / for the latitude of M, and sub- . . . a cos. / , ._ ,. stituting tor MX or #x its value -7= ^ -. ^-=r-i (art. 417.), (1 e 1 sm. 2 iy CHAP. XVIII. SPHEROIDAL ARCS. 391 we have M Q equal to ^ ^ ; ^J^ : consequently, in art. ( JL **^ ^^ S1X1* L ) a 408., putting for the semidiameter MC the length of the normal M Q, there will be obtained p q (in feet) = -~ tan. Z (le 1 sin. 2 Z)*; or, extracting the root as far as A a two terms, pq (in feet) J-* tan. Z (1 Je 2 sin. 2 Z). Again, considering M.p as a circular arc of which MQ is, without sensible error, equal to the radius of curvature, it is evident that the difference between M q and M p may be ob- tained from its value in art. 408., on substituting for M c the above value of M Q ; and it follows that on a spheroid, M q Mp (in feet) = ^ tan. 2 Z (1 V sin. 2 Z). The value of M q being obtained for any given parallel of spheroidal latitude, it may be reduced to a corresponding arc, as M / q f , having equal angular extent in longitude, by the pro- portion MX : M'X' :: M.q : m'g f . PROP. IX. 420. To investigate an expression for the length of a meridional arc on the terrestrial spheroid; having, by ob- servation, the latitudes of the extreme points, with assumed values of the equatorial radius and the excentricity of the meridian. Let I and Z' represent the observed latitudes of the two ex- treme stations, a the radius of the equator, and a e the excen- tricity ; then (arts. 414, 415.) R, the radius of curvature at a point whose latitude is l y ~~ a (1 e 1 4- f e 2 sin.' 2 Z), or putting for sin. 2 / its equivalent ~ ( 1 cos. 2 Z), e 1 Rrr (1 f dm^ whence a (1 | e 2 cos. 2 Z) dl = c c 4 392 GEODESY. CHAP. XVIII. Integrating this equation between / and /', corresponding to m and mf, a {//_ j_|! (//__ /) f e 2 (sin. 2 I'- sin. 2 /)} - m' m ; or putting for e 1 its equivalent 2 s (art. 414.) and for sin. 2 /' sin. 2 / its equivalent (PL Trigon., art. 41.) we have C 5 cos. (/'-/) sin. (l'~T)\, lf A l 6 -e- -/' Z = m'-wi. If /' 7 be expressed in seconds, the factor and the denomi- nator must, each, be multiplied by sin. V. If m' m be considered as representing the length of the meridional arc, obtained in measures of length from the trian- gulation, this equation may be used as a test of the correct- ness of the assumed values of a and e. 421. In the following table are contained a few determin- ations of the lengths of a degree of latitude in different regions of the earth, from which the fact of a gradual but irregular increase of such lengths in proceeding from the equator towards either pole is manifest. Country. Lat. of the Middle Point in the Arc. Length of a Degree of Lat. in English Feet. Peru India Cape of Good Hope - France England Russia o / // 1 31 OS. 12 32 21 N. 33 18 30 S. 44 51 2 N". 52 35 45 N. 58 17 37 N. 362804 362988 364716 364540 364968 365350 The value of the earth's ellipticity, deduced from the measured lengths of meridional arcs, is liable to some uncer- tainty within, however, very narrow limits ; and the ratio of the equatorial to the polar diameter is supposed to be nearly as 301 to 300. In the appendix to the Nautical Almanac for 1836 it is assumed to be as 305 to 304; whence s (the compression) would be equal to 0.00247. 422. The fundamental base line is unavoidably small when compared with the distances between the stations which form the angular points of most of the primary triangles in a geodetical survey : and since, when the three angles of each triangle are actually observed, the most favourable condition is that the triangles be as nearly as possible equilateral, it follows that the distances between the stations, or the sides of the triangles, should increase gradually as the stations are more remote from the base line, till those sides become of any CHAP. XVIII. SECONDARY STATIONS. 393 magnitude which may be consistent with the features of the country or the power of distinguishing the objects which serve as marks. 423. The stations whose positions have been determined by the means already described, are so many fixed points from which must commence the operations for interpolating the other remarkable objects within the region or tract of country ; and, for the measurement of the angles in the secondary triangles, there may be employed such an azimuth circle as has been described in art. 104., while, for triangles of the smallest class, a good theodolite of the kind employed in common surveying will suffice. A process similar to that which has been already described, may be followed in fixing the positions of the secondary stations ; and where such a process is practicable, no other should be employed, the most accurate method of surveying being that which consists in observing all the angles of the triangles formed by every three objects. Should circum- stances, however, prevent this method from being followed, or permit it to be only partially put in practice, the verifica- tion obtained by observing the third angle of each triangle must be omitted ; and it must suffice to obtain, with the theodolite, the angles, at two stations already determined, be- tween the line joining those stations, and others drawn from them to the station whose position it is required to find. Frequently, also, it will be convenient, in a triangle, to make use of two sides already oomputed, and the angle between them, obtained by observation, to compute the third side and the two angles adjacent to it. When a side of some primary triangle, on any line deter- mined with the requisite precision, is used as the base of a secondary triangle, or as a base common to two such triangles, and the angles contained between the sides of the triangles have been observed ; the rules of plane trigonometry will, in general, suffice for the computation of the lengths of the sides. The stations may also be laid down on paper, if necessary, by a graphical construction, a proper scale being chosen from which the given length of the base line is to be taken. If at any two stations already determined, there be taken, by means of a surveying compass, or the compass of the theodolite, the bearings of any object whose position is required, from the true or magnetic meridian of those stations ; the intersection of lines drawn from the stations on the plan, and making, with lines representing those meridians, angles equal to the observed bearings, will give the position of the object with more or less accuracy, according to the delicacy of the needle, 394 GEODESY. CHAP. XVIII. and the precision with which the directions of the meridians have been previously determined. It would be proper to observe at three or more known stations the angles contained between the lines imagined to join them, and other lines supposed to be drawn from them to each of the objects whose positions are required ; in order that, by the concurrence, in one point, of the intersections of all the lines tending towards each object, the correctness of the operations might be proved. Such concurrence is, how- ever, scarcely to be expected, and the mean point among the intersections must be assumed as the true place of the object. 424. It occurs frequently, in the secondary operations of a survey, particularly in those which take place on a sea,-coast, that the position of an object, or the position and length of a line joining two remarkable objects, are to be determined when, from local impediments, it would be inconvenient to convey the instrument to any known stations from whence the objects might be visible ; and, in order to meet such cases, the follow- ing propositions are introduced. As a graphical construction alone may sometimes suffice, there are given, with the formula for the computations, the processes of determining the positions and distances by a scale. PROB. I. To determine the positions of two objects, and the distance between them, when there have been observed at those objects, the angles contained between the line joining them, and lines imagined to be drawn from them to two stations whose dis- tance from each other is known. Let P and Q be the two objects whose positions are re- quired, A and B the stations whose distance AB from each other is known ; then QPB, QPA, PQA, PQB will be the observed angles. On paper draw any line as p q, and at p and q lay down with a protractor, A ' or otherwise, angles equal to those which were observed at P and Q ; then the intersections of the lines containing the angles will determine the points A' and b. Produce the line A f b if neces- sary, and, with any convenient scale, make A'B' on that line equal to the given distance from A to B: from B' draw B'Q' parallel to bq till it meets A' q, produced, if necessary, in Q/; CHAP. XVIII. SECONDARY STATIONS. 395 and from Q' draw Q'P' parallel to pq till it meets Afp, pro- duced, if necessary, in p'. The figures A! pqb and A'P'Q'B' are (Euc. 18. 6.) similar to the figure APQB formed by lines imagined to join the objects on the ground: therefore P'Q', A'P', &c., being measured on the scale from whence A'B' was taken, will give the values of the corresponding distances be- tween the objects. The processes to be employed in computing the distances are almost obvious. Thus, let any number, as 10 or 100, re- present pq. Then, in the triangle Afpq there are known the angles Afp q ( = A P Q), A! qp ( = A Q p), with the side p q ; from whence (PI. Trigon., art. 57.) A'p may be found. In the triangle pqb there are known the angles bpq (= BPQ), pqb ( PQB), with pq\ to find pb. In the triangle Afpb there are known the angle Afpb (= APB) and the sides Afp^pb', from which A. f b may be found. Again, from the similarity of the figures, Afb : A'B' ( = AB) :: pq : P'Q' ( PQ); and by like proportions any other of the required distances may be found. The construction and formulas of computation will, mani- festly, be similar to those which have been stated, whatever be the positions of the stations p and Q with respect to A and B. PROB. II. 425. To determine the position of an object, when there have been observed the angles contained between lines imagined to be drawn from it to three stations whose dis- tances from each other have been previously determined. Let A, B, c be the three given stations, and P the object whose place is to be determined; then A PC, CPB, or APB will re- present the observed angles. With the three given distances AB, AC, BC, lay on paper, by any convenient scale, the triangle A'B'C', and on the side A'B' make the angles B'A'D, A'B'D, re- spectively, equal to the 396 GEODE&Y. CHAP. XVIII. observed angles CPB, APC; through D and c' draw an inde- finite line, and from A? a line to meet it, suppose in p', making the angle B'A'P' equal to B'DC': the point p' will be the station. For if the circumference of a circle be imagined to pass through A', D, and B', since the angle B'A'P' is by con- struction equal to B'DP', those angles will be in the same seg- ment (Euc. 21. 3.), and the circumference will also pass through P'; therefore the angle B'P'C' will be equal to B'A'D, and the angle A'P'C' to A'B'D. But the angles B'A'D, A'B'D are by construction equal to B PC, APC ; therefore the latter angles are, respectively, equal to B'P'C' and A'P'C': thus, the angle_s at p 7 , on the paper, are equal to the corresponding angles which were observed at P ; and p' represents the object p : therefore the lines A'P', c'p', B'P', measured on the scale, will give the required distances in numbers. The like construction might be used if A, c, and B were in one straight line. If the point D should coincide with c' the case would evidently fail; and the determination of P' will be less accurate as D falls nearer to c'. The formulas for computation may be briefly stated thus : In the triangle A'B'C' the three sides are given ; therefore one of the angles, as B'A'C', may be found (PL Trigon., art. 57.). In the triangle B'A'D all the angles are known, and the side A' B' ; therefore the side A' D may be computed. In the triangle DA'C', the sides A'D, A'C', and the angle C'A'D are known ; therefore the angles A' DC', A'C'D may be found. In the triangle A'C'P' all the angles and the side A'C' are known; therefore the sides A'P', C'P' may be computed. Lastly, in the triangle A' B'P' there are known the side A'B' and all the angles (for A' B'P' is equal to the computed angle A' DC'); therefore P'B' may be obtained. By this proposition the Observatory of the Royal Military College at Sandhurst was connected with three stations whose positions are given in the account of the Trigono- metrical Survey of England. The stations are Norris's Obelisk, which may be represented by A; Yately Church Steeple, represented by B, and the middle of a Tumulus near Hertford Bridge represented by C ; P representing the centre of the dome in the Observatory. With an altitude and azimuth instrument whose circles are, each, twenty inches in diameter, the following angles were taken: APC = 134 32' 4", APB - 171 37' 16"; whence CPB =37 5' 12"; and from data furnished by the Trigonometrical Survey, there CHAP. XVIII. SECONDARY STATIONS. 397 were obtained AB = 20252 feet, AC = 18086 feet, and BC = 8243.66 feet : with these data and the observed angles there were found by computation, as above, AP 6985.2 feet, CP = 12488 feet, and BP ~ 13315.3 feet. There was, at the same time, observed the bearing of Norris's Obelisk from the meridian of the Observatory, which was found to be S. 86 IT 6" E., with which, and the computed distance AP, it was found by the formulae in art. 410; that the difference between the latitudes of the Obelisk and the centre of the dome is 8". 19, and the difference, in time, between the lon- gitudes is 7".28 ; the Observatory being northward and west- ward of the Obelisk. Hence, from the latitude and longitude of the latter in the Trigonometrical Survey, it is ascer- tained that the latitude of the Observatory is 51 20' 32".99, and its longitude, in time, is 3' 3". 7 8 westward of Green- wich. If the object whose position is required were, as at p, within the triangle formed by the three given stations A, B and c, the observed angles being then Ape, C/?B or A/?B, the construction would be similar to that which has been given, except that the angles B'A'D', A'B'D' must in that case be made, respectively, equal to the supplements of B^>c and ApG) and the angle B'A'JP' equal to B'D'C'; for then, as in the former, the circumference of a circle supposed to pass through A?, B' and D' would also pass through p', and this point would represent the object. The formula of compu- tation would be precisely the same as in the other case. When many points are to be determined in circumstances corresponding to those which are stated in this proposition, it is found convenient to obtain their positions mechanically by means of an instrument called a station pointer. This consists of a graduated circle about the centre of which turn three arms extending beyond the circumference, and having a cham- fered edge of each in the direction of a line drawn through the centre : by means of the graduations these arms can be set so as to make with one another angles equal to those which have been observed, as APC and CPB; and then, moving the whole instrument on the paper till the cham- fered edges of the arms pass through the three points A', B' and c', the centre of the instrument will coincide with p', and consequently will indicate on the paper the required po- sition of the object. The position of an object, as P, with respect to two given stations as A and B, may be found by the method of cross- bearings in art. 364. 398 GEODESY. CHAP. XVIII. PKOB. III. 426. To determine the positions of two objects with re- spect to three stations whose mutual distances are known, when there have been observed, at the place of each object, the angles contained between a line supposed to join the ob- jects and other lines imagined to be drawn from them to two of the three given stations, some one of these being invisible from each object. Let P and Q be the two objects whose positions are A r required; A, B and C the \ three stations, and let APB, v"" APQ or BPQ, PQB, PQC or BQC be the four angles which have been observed. With any scale lay on paper a triangle A'B'C' having sides equal to the given distances AB, BC, AC, then on A'B' describe (Euc. 33. 3.) a segment A'P'B' of a circle, which may contain an angle equal to APB, and on B'c' describe a segment B'Q'C' which may contain an angle equal to BQC: it is manifest that the representations of P and Q will be somewhere on the circumferences of those seg- ments. Now, in order to discover readily what should be the next step in the process, imagine P 7 and Q' to be the places of the two objects, and imagine lines to be drawn as in the figure : then wherever p' and Q' be situated, the angles A'P'B' and B'Q'C' will be equal to the corresponding angles APB and BQC ; also the angle B'A'X will be equal to B'P'Q', and B'C'Y to B'Q'P'. Therefore, if the angles B'A'X, B'C'Y be made re- spectively equal to BPQ and BQP, a line drawn through x and Y, and produced if necessary (as in the figure), will cut the circumferences of the circles in p' and Q'; and these points will represent the required places of the two objects, since it is manifest that the angles at P' and Q' will be equal to the observed angles at P and Q. The lines P'Q', A'P', &c., being measured on the scale, will give the required distances in numbers. The points P and Q may be on opposite sides of the triangle ABC, as at p and q, or one of them may be within and the other on the exterior of the triangle, but the construction will be similar to that which has been given. It is evident that CHAP. XVIII. SECONDARY STATIONS. 399 the case would fail if the points x and Y should coincide with one another. The formulas for the computation may be briefly stated thus. In the triangle A'B'C' the three sides are known, therefore the angle A'B'C' may be found (PL Trigo., art. 57.). In the triangle A'B'X there are given A'B', and the angles B'A'X (= BPQ) and A'XB' (= APB); therefore the side B'X may be found. In like manner, in the triangle C'B'Y, there are given the side B'C' with the angles B'C'Y (= BQP) and B / YC / (=:BQC); therefore the side B'Y can be found. By subtracting (in the figure) the angle A'B'C' from the sum of the angles A'B'X and C'B'Y, the angle x B'Y will be obtained ; and then, in the triangle XB'Y, the sides B'X, B'Y and the contained angle will be known, and consequently the angles B'XP' and B'YQ' may be computed. In the triangle B'P'X there are known all the angles and the side B'X; whence B'P' may be found ; in like manner may B' Q be found in the triangle B'Q'Y. In the triangle B'P'Q' all the angles and the side B'P' are known ; therefore P'Q' may be found; and lastly, the distances A'P', C'Q' may be found if required in the triangles A' B'P' and C'B'Q'.* 427. A geodetical survey of a country is always accom- panied by determinations of the heights of its principal mountains; for which purpose, in general, the angular ele- vations of their summits are observed by means of the vertical circle belonging to the azimuth instrument or theodolite ; with which, after due allowance for the effects of refraction, and the curvature of the earth, the required altitudes may be computed by the rules of plane trigonometry. The allowance for terrestrial refraction is to be determined by the method explained in art. 165.: and the correction of an angle of elevation on account of the deflection of the earth's surface, supposed to be spherical, from a plane touch- ing the sphere at the place of the observer may be investigated in the following manner. Let C be the centre of the earth, A the place of the observer's eye, and B the summit of a mountain : let CA audcB be radii of the earth, and imagine C D to be equal to C A, so that A and D may be level points ; also let AH be perpendicular to AC, so as to represent a horizontal line at A: then the angle BAH, after allowance * Much useful information respecting the processes to be employed for " filling-in " a geodetical triangulation will be found in the works on tri- gonometrical surveying by Capt. Frome, Royal Engineers (Weale, 1840) ; Major Jackson, Hon. E.I.C., Seminary, Addiscombe (Allen and Co., 1838); and Mr. G.D.Burr, Royal Military College, Sandhurst (Murray, 1829). 400 GEODESY. CHAP. XVIII. for refraction, is the elevation of B above the horizon of A. Draw the straight line AD ; then the triangle ACD will be isosceles, and the angle CAD toge- ther with the half of ACD is equal to a right angle : but the angle C A H is a right angle ; therefore HAD is equal to half the angle at c. This angle is known, since the radius of the earth is known, and the horizontal distance of A from B, which may be considered as equal to the arc A D, is supposed to be given by former operations. When B is, as in the figure, elevated above AH, the angle HAD must evidently be added to BAH in order to have the corrected elevation DAB: but if B were, as at b, below AH, the angle HA&, of depression, should be subtracted from HAD or from half the angle at c. Then, with AD as a base, the angle ADB as a right angle, and the correct elevation DAB or DA b, the height BD or D might be computed. The mean length of a degree on the earth's surface being 364547 feet, when AD is equal to 1 mile, or 5280 feet, the angle ACD = 52". 14; and then the deflexion DH is equal to 8.0076 inches. Now (2CD + DH) . DH AH* (Euc. 36.3); but DH being very small compared with 2 CD, the factor 2 c D + D H may be considered as constant ; therefore D H, the other factor, may be said to vary with the square of A H, or very nearly with the square of AD. Thus, for any given distance AD, not exceeding a few miles, we have (5280 feet) 2 : (AD, in feet) 2 :: 8.0076 in. : DH, and the last term is the deflexion, in inches, corresponding to the given distance. Hence, if BH were computed in the triangle AHB, the angle at H being by supposition a right angle and, AH being considered as the given distance, the value of D H found from the proportion might be added to B H in order to have the correct height B D. 428. The determination of the heights of mountains by trigonometrical processes, is, with good instruments, sus- ceptible of a high degree of accuracy ; but the operations being laborious the relative heights of the remarkable sum- mits in a mountainous country are more generally ascertained by means of the observed heights of the mercurial column in a barometer at the several stations. When Colonel Mudge and Mr. Featherstonehaugh determined, in North America, the levels along the axis of elevation, from the head of the Penobscot and St. John's rivers to the Bay of Chaleurs, they CHAP. XVIII. HEIGHTS, BY THE BAROMETER. 401 were provided with several barometers, one of which was kept constantly at the Great Falls of St. John, in a building to which the air had free access : the man who had it in charge made the observations at 8 A. M., at noon, and at 4 P. M. daily, and the travelling party always endeavoured to make their observations as nearly as possible simultaneously with those at the Falls. The instruments employed for such purposes are, in general, similar to the ordinary barometer but more delicate ; occasionally, however, a Siphon Barometer, of a kind invented by M. Gay Lussac, and now frequently made in this country, has been used. A formula for deter- mining the relative heights of stations, from the heights of the mercurial columns supported by the atmosphere at the stations, may be thus investigated. 429. The particles of the atmosphere which surrounds the earth gravitate towards the surface of the latter, but, being of an elastic nature, they exert an expansive force in every direction ; and, when the atmosphere is tranquil, that part of the force which, at any point, acts from below upwards is in equilibrio with the gravity or weight of the column of air which is vertically above that point. Now, in any small volume of air within which the density may be considered as uniform, that density is proportional to the external force which, acting against it, tends to compress it : and this force of compression on every part of the surface of the volume is equal to the weight of the column of air which is incumbent on that part ; it follows therefore that, since the weight of a volume of air of uniform density is proportional to the density, the weights of small portions of air, equal in volume, in every part of the atmosphere (gravity being supposed constant), will be the same fractional parts of the weights of the atmospherical columns above them. This law being admitted, let AB be part of the surface of the earth, AZ the indefinite height of a slender cylindrical column of the atmosphere, and let this be divided into strata of equal thickness A, ab,bc, &c. Then, taking w to represent the weight of the column z/J -w may represent the weight of the small column ef ^ b and w + w, or w. will express the weight of the n n column ze. Again, by the law above mentioned, -( - W] will represent the weight of the small column de, ancl D D 402 GEODESY. CHAP. XVIII. ^J: w + i (5+l w \ or ( ) 2 w wi n be the weight of id. n n\ n I \ n ) Continuing in this manner it will be found that the weights of the columns z ze, zd, &c., and also of the columns ef, ed, dc, &c., will form a geometrical progression whose common ratio is : that is, the weights or pressures at f, e, d, &c. ft will form a geometrical progression, while the depths fe, fd, fc, &c., form an arithmetical progression. 430. But, by the nature of logarithms, if a series of natural numbers be in a geometrical progression, any series of numbers in an arithmetical progression will be logarithms of those natural numbers ; therefore, if there were a kind of logarithms adapted to the relation between the densities of the air and the depths of the strata, on finding the densities of the air at any two places as between a and b and between c and d, such logarithms of those densities would express the depths fa and/W, and the difference between the logarithms would be equal to ad, which is the height of d above . Thus a d =. log. dens, at a log. dens, at d, or , , dens, at a ad \QQ. , j. dens, at d There are no such logarithms, but, from the general pro- perties of logarithms, the formula may be adapted to those of the ordinary kind. Thus, the weight of an atmospherical column, as Af, is equal to the sum of the increasing weights of the series of strata from f down to A, and may be re- presented by the product of the greatest term (the weight of the air in Act) by some constant number M, which is therefore the modulus * of the system of logarithms whose terms are fe,fd,fc, &c., to /A. Now M may represent the height of a homogeneous atmosphere whose uniform density is equal to that of the stratum Act, and whose weight or pressure on A, is equal to that of the real atmosphere ; and since such column, at a temperature expressed by 31 (Fahrenheit's * From the series 1,2 (art. 213. Elem. of Algebra), we have nuw = log. (l+w) where nuw may represent /A, and (1 + w} n the weight of the stratum in A a : let the first member be represented by x and (1 4-w) by y. Then, by subtraction, in the series 2, the increment dx of HM w is M w ; and after developing (1 -f w} n , (1 + M>)+I by the binomial theorem, we have, by subtraction, the value of dy, the increment of y: from the values of these increments it will be found that y dx = M dy. But the integral of ydx expresses the sum of the weights of all the strata ; therefore the integral of Mefy, that is, My, or the 'product of the greatest term (the weight of A a) by the modulus M of the system, is equal to that sum or to the weight of the column A/. CHAP. XVIII. HEIGHTS, BY THE BAROMETER. 403 thermometer), would hold in equilibrio at the level of the sea a column of mercury equal in height to 30 inches, it follows (the heights of two columns of homogeneous fluids, equal in weight, being inversely proportional to their densities) that the height of the column of homogeneous atmosphere at that temperature would be 4343 fathoms, and this may be con- sidered as the value of M. The modulus of the common logarithms is 0.4343 ; and since, in different systems, the logarithms of the same natural number are to one another in the same proportion as their moduli, we have , dens, at a lrmArt , dens, at a 0.4343 : 4343 :: com. log. , r- , ' 10000 com. log. -, 3 dens, at d & dens, and the last term is equivalent to atmospheric log;, -= - - - ,. 3 dens, at d But the heights of the columns of mercury in a barometer, at any two stations, a and d, are to one another in the same ratio as the densities of the atmosphere at those stations, or as the weights of the columns of air above them ; therefore the height of d above a may be expressed, in fathoms, by the , barom. at a formula 10000 common loo;. , ; 7 , 5 barom. at d or by its equivalent, 10000 (log. barom. at a log. barom. at d). Thus the height of the column of mercury in a barometer being observed at any two stations, as a and d, as nearly as possible at the same time, there may be obtained the relative heights of the two stations. In using the formula it is evident that the heights of the mercurial columns at the two stations ought to be reduced to those which would have been observed if the temperature of the air and of the mercury were 3 1. In order to make such reduction, since the mean expansion of a column of mercury is a part expressed by .000111 of the length of the column for an increment of temperature equal to one degree of Fah- renheit's thermometer, if this number be multiplied by the difference between the temperature at each station and 31 (that temperature being expressed by the thermometer at- tached to the instrument), the product will be the expansion of the column (in parts of its length) for that difference. Therefore, multiplying this product into the observed height of the column, in inches, at each station, we have the ex- pansion in inches ; and this last product being subtracted from the observed height, if the temperature at the station be greater than 31, or added if less, there remains the corrected height of the column. The logarithms of these corrected heights, at the two stations, being used in the last formula D D 2 404 GEODESY. CHAP. XVIII. above, the resulting value of ad will be a first approximation to the required height. By experiments it has been found that the relative height thus obtained varies by ^ 7 of its value for each degree of the thermometer in the difference between 31 and the mean of the temperature of the air at the two stations : consequently, if d be the difference between 31 and the mean of two detached thermometers, one at each station, the correction on account of the temperature of the air will be expressed by the formula -^-z d, where ad is the first approximate value of the height. This correction is to be added to that approximate height when the mean of the detached thermometers is greater than 31, and subtracted when less. The result is very near the truth when the height of one station above the other does not exceed 5000 or 6000 feet and when the difference of temperature does not exceed 15 or 20 degrees : in other cases, more accurate formulas must be em- ployed, and that which is given by Poisson in his " Traite de Mecanique " (second ed. No. 628.), when the measures are reduced to English yards, and the temperatures to those which would be indicated by Fahrenheit's thermometer, is H = A { log. barom. at a log. k ] ; in which H is, in yards, the required height of one station above the other, as d above . 20053.95 / l ,t + t f 64x = 1 - .002588 cos. 2X ( ' 900 /' k = height of barom. at d, X (l + ~ ). t and f are the temperatures of the air, by detached ther- mometers at the two stations. T and T' are the temperatures of the mercury, by attached thermometers. A is the common latitude of the stations, or a mean of the latitudes of both if the stations be distant from each other in latitude. 431. The siphon barometer is a glass tube formed nearly as in the annexed figure, and con- taining mercury; it is hermetically sealed at both extremities, and has at A a very fine perforation, which allows a communication with the external air without suffering the mercury to escape. The atmosphere pressing on the mercury at N, balances the weight of a column of that fluid whose upper extremity may be at M. There is CHAP. XVIII. HEIGHTS, BY THE BAROMETER. 405 a sliding vernier at each extremity of the column ; and, the zero of the scale of inches being below N, the difference between the readings at M and N on the scale is the required height of the column of mercury. In other mountain barometers the tube is straight, and its lower extremity, which is open, enters into a cis- tern A B containing mercury : the bottom of the cistern is of leather, and by means of a screw at c, that mercury can be raised or lowered till its upper surface passes through an imaginary line, on which, as at N, is the zero of the scale of inches ; M being the upper extremity of the column of mercury in the tube, the height MN is read by means of a ver- nier at M. The external air presses on the flexible bottom of the cistern, and this causing the surface of the mercury at N to rise, or allowing it to fall, the corresponding variations in the elasticity of the air in the part AN of the cistern, produce the same effect on the height of the mercurial column M N, as would be produced by the external air if it acted directly on the surface at N. The barometer invented by Sir H. Englefield has no screw for regulating the surface of the mercury in the cistern with respect to the zero of the scale of inches ; and the atmosphere, entering through the pores of the box-wood of which the cistern is formed, presses directly at N on the surface of the mercury ; there can, consequently, be only one state of the atmosphere in which the surface is coincident with that zero. The exact number of inches and decimals, on the scale, at which the extremity M of the column of mercury stands when the surface at N coincides with the zero is found by the artist, and engraven on the instrument ; and, when the top of the column is at that height (or at the neutral point, as it is called) no correction is necessary on account of the level of the mercury in the cistern. In other cases such correction is determined in the following manner : The ratio between the interior area of a horizontal section through the cistern, and the area of a like section through the bore of the tube, is ascertained by the artist and engraven on the instrument : let this ratio be as 60 to 1 : then the lengths of cylindrical columns, containing equal volumes, being inversely propor- tional to the areas of the transverse sections, the required correction will be ^ of the difference between the height of the neutral point, and that at which the top of the column stands in the tube. This correction must be added to, or subtracted from, the height read on the scale according as the top of the column is above or below the neutral point. D D 3 406 GEODESY. CHAP. XVIII. The difficulty of transporting the usual mountain barometer overland has induced travellers to use, for the purpose of determining the relative heights of stations, the instrument invented by Dr. Wollaston, and called by him a thcrmometrical barometer. This consists of a thermometer, which may be of the usual kind, but very delicate, whose bulb is placed in the steam arising from distilled water in a cylindrical vessel about five inches long, the water being made to boil by an oil or spirit lamp. Now it is known that water boils when the elastic power of the steam produced from it is equal to the incumbent pressure of the atmosphere; and thus, the tem- perature at which, in the open air, the water boils, will de- pend upon the weight of the atmospherical column above it. Therefore, since this weight becomes less as the station is more elevated, it is evident that water will boil at a lower temperature on a mountain than on a plain at its foot ; and, for the purpose of determining the height of the mountain, it is only necessary to find an expression for the elastic power of steam, at a given temperature under the pressure of the atmosphere, in terms of the height of an equivalent column of mercury in a barometer. From the experiments of De Luc, M. Dubuat has obtained a formula equivalent to or log. h = 3.84 (log. t 1.876105), or log. h = 3.84 log. t 7.204, where h is the height of the ordinary barometrical column expressed in English inches ; t is the temperature at which the water boils in the open air, expressed by degrees of Fah- renheit's thermometer, and reckoned from the freezing point (32) as zero. The value of h being thus obtained for each of the two stations, the height of one station above the other may be found by the formula for the usual mountain baro- meter. 432. The processes of a trigonometrical survey for deter- mining the figure of the earth are too extensive to be fre- quently put in practice; they also require the combined efforts of many persons, and they involve expenses which are beyond the resources of private individuals. On the other hand, the variations of terrestrial gravity at different places on the earth's surface are capable of being determined with great precision by comparing together the observed times of the vibrations of pendulums at the places : and as these va- riations, though in part due to a want of homogeneity in the CHAP. XVIII. EMPLOYMENT OF PENDULUMS. 407 mass of the earth, depend chiefly on the deviation of its figure from that of an exact sphere ; a series of well-conducted ex- periments on the vibrations of pendulums at stations remote from each other, which may be made by two or three persons at a comparatively small expense, afford the most convenient means of ascertaining the form of the earth. 433. The pendulums which are employed for this purpose are generally of the kind called invariable ; that is, they are not provided with a screw by which their lengths may be in- creased or diminished like the pendulums applied to ordinary clocks ; so that their lengths can only vary by the expansion or contraction of the metal in consequence of the variations of temperature to which they may be subject. The effects arising from the variations of temperature and from all the other circumstances which interfere with the action of gravity upon them, are determined by theory and applied as cor- rections to the observed times of the vibrations. These pendulums are employed in two ways : they may be attached to the machinery of a clock for the purpose of con- tinuing the oscillations and registering their number, or they may be unconnected with any maintaining power, and left to vibrate by the action of gravity till the resistance of the air and the friction on the point of suspension bring them to a state of rest. The late Captain Kater, availing himself of that property of vibrating bodies by which the centres of suspension and oscillation are convertible, constructed pen- dulums which admit of being made to vibrate upon either of those centres at pleasure ; by this construction the effective length of the pendulum (which is the distance between those centres) is easily found, by measurement, when the places of the centres have been determined by the experimental number of vibrations made upon each being equal, in equal times ; and such invariable pendulums are now generally employed by the English philosophers for the purposes of experiment. They are furnished with steel pivots or axles (called knife edges) at the two places which are to be made alternately the centres of suspension and oscillation ; and these rest upon the upper edge of a prism of agate or wootz, so that the pen- dulums may vibrate with as little friction as possible. 434. In making the experiments with a detached pendulum, the latter is placed in front of a clock regulated by mean solar, or sidereal time, but quite unconnected with its motion. On the pendulum of the clock is placed a disk of white paper opposite to a vertical wire crossing an opening in the rod of the detached pendulum. A telescope is fixed a few feet in front of the pendulums, so that, when these are put in D D 4 408 GEODESY. CHAP. XVIII. motion, the disk of paper may be seen to pass over the field of view. Now, since the detached pendulum and that of the clock have not exactly the same velocity, if we suppose the vertical axis of the former to have been originally in coin- cidence with the centre of the disk, these will separate from each other by the excess of the velocity of one above that of the other : but after a certain number of oscillations, they will again coincide, moving in opposite directions; and then the detached pendulum may be said to have gained or lost one oscillation. For example, if 30 vibrations of the clock pendulum had been observed in the interval between two consecutive coincidences, it is evident that the invariable pen- dulum must have made either 29 or 31 vibrations. After being in coincidence the pendulums separate as before, and again, subsequently, they coincide ; and so on. The number of oscillations made by the pendulum of the clock between two, three, or more coincidences is counted, and the times of the several coincidences are shown by the clock, when they take place : then the number of oscillations made by the clock pendulum in the time of any number of coincidences (suppose n) is to the number of oscillations of the detached pendulum in the same time (which number in this case will exceed or fall short of the former number by n), as 86400 seconds (the number of oscillations made by the clock in a mean solar, or sidereal day, according as the clock is regulated by solar or sidereal time) are to the number of oscillations made by the detached pendulum in the same time, at the station. With respect to the invariable and attached pendulum, the number of vibrations performed by it in a mean solar, or sidereal, day is ascertained by observing the times indicated on the dial of the clock to which it is attached, at the end of equal intervals of time, as 12 or 24 hours. And in both cases it is most convenient to determine the measure of time by the transits of stars. Captain (now Colonel) Sabine's experiments in his two voyages during the years 1822, 1823, were made with pen- dulums whose lengths were invariable except in respect of temperature. In the first voyage the pendulum was detached from any clock-work, so that after a certain number of os- cillations it rested : but in the second voyage the pendulum was attached to a clock, and its oscillations were continued by the maintaining power of the clock. The uniformity of the maintaining power was inferred from that of the extent of the arc of vibration. 435. The following propositions contain the principal sub- jects relating to the corrections which are required for the CHAP. XVIII. REDUCTIONS OF VIBRATIONS. 409 purpose of determining the intensities of gravity in different places, and the ellipticity of the earth, by the vibrations of pendulums. PROP. I. To reduce the number of vibrations made on a given arc, which is supposed to be constant, to the number which in an equal time would be made in an arc of infinitely small extent. If t be the time of performing a vibration in an infinitely small arc, and t' the time of vibration in an arc of given extent ( = 2 ), we have by Mechanics, as a near approxi- mation, Now, let N be the number of vibrations made in an infinitely small arc by a simple pendulum in the time T, and let N' be the number of vibrations made in the same time T by a pen- dulum of equal length, and vibrating in an arc equal to 2 a ; then t and t' being the times of making one vibration in each case, we have T T * = -, and*' =-,: N N' these values of t and t f being substituted in the preceding equation, we get . /, sin. 2 \ N = N/ 1 1 + riff- Therefore, if N', the number of oscillations made in an arc equal to 2 a during a given time, as a sidereal day, be given, we may obtain the number which the same pendulum would have made in an infinitely small arc, in an equal time. Note. If, as usual, the arcs continually diminish, the mean of the extents observed at the commencement and end of the time may be considered as equal to 2 a, half of which is the arc a in the formula. PROP. II. 436. To find the correction of the length of a pendulum, or of the number of vibrations made in a given time, on ac- count of the buoyancy of the air. The loss of weight to which a pendulum is subject when it vibrates in air being equal to the weight of an equal volume 410 GEODESY. CHAP. XVIII. of air ; we have (since the loss of weight is equivalent to a diminution of the action of gravity), putting s for the specific gravity of the pendulum, s that of air, and f for the force of gravity on a pendulum vibrating in air, s : s \:f:df 9 where df represents an increment of the force of gravity ; and this increment must be added to the force of gravity de- duced from the experimental vibrations of the pendulum in air, in order to have the force of gravity on the pendulum in vacuo. But the force of gravity is proportional to the length of the pendulum, when the times of making one vibration are equal ; therefore the above proportion becomes, I being the length of a pendulum vibrating seconds in air, s : s:: l\ dl, and I + dl becomes the length of a pendulum vibrating se- conds in vacuo. * Now if it were required to find the excess of the number of vibrations performed in vacuo over the number performed in air, in an equal time, by the same pendulum; imagine the pendulum, whose length in vacuo is I + dl (as above determined) to be reduced to the length I of the pendulum in air. Then, since, by Mechanics, the lengths are inversely as the squares of the number of vibrations performed in equal times, I: l + dl: : N /2 : N 2 ; where N' is the number of vibrations performed in a given time by the pendulum I -\- dl in vacuo (equal to the observed number of vibrations described by I in air), and N the number of vibrations performed in vacuo by the pendulum / in the same time : and by division, I \dl\\ N' 2 : N 2 N'*; where N 2 N /2 is an augmentation ( =: a) of N /2 , or of the square of the number of vibrations described in air by an equal pendulum in an equal time, and N /2 + a =. N 2 : or again, s : s : : N /2 : , and thus a may be found. From the equation N /2 + a N 2 , we have (N /2 + a)* == N, and developing the first member by the binomial theorem as far as two terms, we have N' + iN'- 1 ^ = N ; or N' + ? ,- = N; consequently -, N 2, N is the increment, which being added to N', gives the value of N. The value of s, the specific gravity of the air, should be determined for the state of the barometer and thermometer at CHAP. XVIII. REDUCTIONS Or VIBRATIONS. 411 the time of the experiment. And it may be observed that no notice has been taken of the retardation caused by the air which is moved with the pendulum as the latter vibrates. PROP. III. 437. To find the correction of the length of a pendulum on account of temperature. Or to reduce the number of os- cillations made in a given time at an observed temperature to the number which would be made in an equal time at a standard temperature. N". B. The variation in the length of the pendulum corre- sponding to an increment of temperature expressed by one degree of the thermometer is supposed to be known from the tables of the expansions of metals by heat, or from experi- ments made on the pendulum itself. Let e express the expansion, in terms of the length of the pendulum, for one degree of Fahrenheit's thermometer ; let t be the standard temperature, and f the observed temperature : then t t' ( rr + b) is the difference between the observed and standard temperatures, and + b e is the expansion or con- traction of the pendulum in terms of its length ; consequently, / being the experimented length of the pendulum (in inches) from the point of suspension to the centre of oscillation at the standard temperature, (1 + be) I is the length of the pen- dulum in inches at the observed temperature ; let this be re- presented by /'. Then, by Mechanics, I ; Z':: n' 2 : ri 1 -, where n' is the observed number of vibrations in a sidereal day, for example, and n is the number of vibrations which would be performed in an equal time if the pendulum were corrected for expansion, or reduced to the length which it would have at the standard temperature. It may be observed, that the expansion of a pendulum is to be ascertained by immersing it successively in fluids of different temperatures, and measuring its lengths by a mi- croscopical apparatus ; the difference between the lengths so measured will be the amount of expansion corresponding to the number of degrees in the difference between the tem- peratures of the fluids. Or the correction for expansion may be found by observing the number of vibrations performed in a sidereal day in apartments brought by artificial means to different temperatures : thus, the corrections being first ap- plied for the magnitude of the arc of vibration and for the 412 GEODESY. CHAP. XVIII. buoyancy of the air, the difference between the numbers of vibrations in a day, divided by the difference of the temper- atures in degrees of the thermometer, gives the number of vibrations, or the parts of a vibration, which, for one day, are due to a difference of temperature expressed by one degree. PROP. IV. 438. To reduce the length of a pendulum at any station to that which it should have in order that it may perform the same number of vibrations in an equal time at the level of the sea. Let R be the semidiameter of the earth at the level of the sea in the latitude of the station ; h the height of the station above the level of the sea ; I the observed length of the se- conds' pendulum, and I' the required length at the level of the sea. Then, the lengths of pendulums vibrating in equal times varying directly as the force of gravity, and the force of gravity varying inversely as the square of the distance of the station from the centre of the earth, we have /':*:: (R + A) 2 : R 2 ; whence /' = / ^fL ; R on developing (R + A) 2 and rejecting powers of h above the first, we get 7/ Note. Captain Kater's allowance for the height of a station is rather less than that which would be given by this rule, because he used a co-efficient depending on the attraction of the matter between the level of the sea and the place of observation. 439. In Captain, now Colonel, Sabine's " Account of Experiments to determine the Figure of the Earth " (1825), there is given (page 351.) from his own experiments, with those of Biot, Kater, and others, a table of the lengths of pendulums at several stations, extending from the equator to Spitzbergen ; and from that table the following values have been taken. Stations. Latitudes. Lengths of Pend. in Inches. Sierra Leone Jamaica New York London Unst Hamnierfest - Spitzbergen - 8 29 28 17 56 7 40 42 43 51 31 8 60 45 26.5 70 4 5 79 49 58 39.01997 39.03510 39.10168 39.13929 39.17164 39.19519 39.21469 CHAP. XV11I. LENGTHS OF SECONDS PENDULUMS. 413 Now it is proved in Mechanics that the length of a seconds pendulum varies directly with the force of gravity, and (Airy's Tracts, Fig. of Earth, art. 63.) that in any place whose latitude is X, the force of gravity is expressed by 1 + n sin. 2 X, where n is the excess of the force of gravity at the pole above the force at the equator, the latter force being represented by unity, and the earth being a spheroid ; there- fore the force of gravity at a station whose latitude is X, is to the force at the equator, as 1 + n sin. 2 X is to 1 : hence, substituting /, the length of a seconds pendulum at the station for the force of gravity there, and A for the length of the like pendulum at the equator, we have l = A + An sin. 2 X. In like manner, for a place whose latitude is X', representing the length of the seconds pendulum there by /', we have J'=A + AW sin. 2 X'. Subtracting the former equation from the latter, we get I / = An (sin. 2 A'- sin. 2 A) : whence An may be found, and, by substitution in either equation, the separate values of A and of n are obtained. The conclusions drawn by Colonel Sabine from all his pendulum experiments, are that A is equal to 39.0152 inches; whence n = 0.005189 : from this last, by Clairaut's theorem -~ ne (Airy's Tracts, Fig. of Earth, art. 62.), the value JO of e, the ellipticity of the earth, may be found. In the formula, m represents ^-gy * ne value at the equator of the centrifugal force, that of gravity there being unity; and from the above data it will be found that e is equal to ^^ 5 or very nearly equal to m : by the measured lengths of degrees of latitude, e is found to be about ^yj. From the above equation for I we have / A = An sin. 2 X ; it follows, therefore, that the increase in the length of a seconds pendulum from the equator towards either pole varies with the square of the sine of the latitude of the station: also, the intensities of gravity at different stations being proportional to the length of the seconds pendulums at the stations, the increase of the force of gravity from the equator towards either pole varies with the square of the sine of the latitude. 440. Experiments for determining the elements of ter- restrial magnetism being now usually carried on in connection with geodetical operations, it will be advantageous to notice 414 GEODESY. CHAP. XVIII. in this place their nature and the manner of performing them. Those elements are of three kinds: the Declination or, as it is generally called, the variation of the magnetized needle ; the Inclination or dip of the needle, and the intensity of the earth's magnetic power. 441. The ordinary variation, or azimuth compass, is well known : the needle is supported near its centre of gravity, and is allowed to traverse horizontally on the point of a vertical pivot made of steel, and the box is furnished with plane sights, which may be placed in the direction of the meridian, or may be turned towards a terrestrial object, as the case may require. Compasses of a superior kind, like that of Colonel Beaufoy, differ from the others in having, instead of plane sights, a small transit telescope, by which the middle line of the box may, by the observed transits of stars, be placed accurately in the direction of the geographical meridian. The usual clipping needle is a bar of steel which, before the magnetic quality is communicated to it, is balanced accurately upon its centre of gravity, where, by a horizontal axis of steel, terminating above and below in what is called a knife-edge, the needle rests, on each side, on the edge of an agate plate. With good needles there is an apparatus con- sisting of screws by which the centre of gravity and centre of motion are rendered coincident ; this adjustment is made before the needle is magnetized by causing it to vibrate on the points of support, and observing that it comes to rest in a horizontal position ; then, after reversing it on its axis, so that the uppermost edge of the needle becomes the lower, the needle is again made to vibrate, and alterations, if necessary, are made till the needle is found to settle horizontally in both situations. The needle thus prepared, and being duly mag- netized, is placed in a plane coinciding with what is called the magnetic meridian (a vertical plane passing through a well- balanced compass needle) ; when the position which it assumes indicates, by means of the graduated circle on whose centre it turns, the absolute inclination or dip of the needle, or the line of direction of the resultant of all the magnetic forces in the earth. On being made to vibrate like a pendulum, the number of oscillations performed in a given time affords one of the means of determining the intensity of terrestrial magnetism in the direction of that resultant. The intensity of terrestrial magnetism in horizontal and vertical directions are resolved parts of the intensity in the CHAP. XVIII. MAGNETIC NEEDLES. 415 direction of the resultant, that is, in the direction assumed by the magnetic axis of the dipping needle. Thus H o being a horizontal line passing through the centre C of the needle's motion, in the plane of the mag- netic meridian, and ns being the direction of the needle when subject to the force of terrestrial mag- netism, so that the angle HCn is the inclination or dip ; on drawing nil perpendicularly to HO, the three lines nc, ch, nh will, re- spectively, represent the absolute intensity and its horizontal and vertical components : hence if f represent the absolute intensity, and d the dip, we shall have f cos. d for the horizontal intensity, and f sin. d for the vertical intensity. 442. The dipping needle is not always so simple as that which has been described : in order to increase the facility of its vibrations, Mayer of Gottingen attached at the centre of gravity of the needle a wire perpendicular to its length, and in the plane of its vibration ; this wire, which carries at its extremity a brass ball, may, by inverting the needle, be either above or below the latter, and the number of vibrations made in a given time may be observed in both positions of the needle. The intention, in separating the centre of motion from that of gravity, is to give the needle a power, resulting from the weight, to overcome the friction of the axis, and allow it, after having vibrated, to return with greater certainty to the same point on the graduated circle than if the centres of gravity and motion were coincident. In using the needle, the dip or inclination should be observed with the axis in one position, and again with the axis reversed, so that the upper edge may become the lower : and a mean of the two readings should be taken. Also, should the situation of the brass ball be such that its centre does not, when the needle is in a horizontal position, lie vertically above or below the centre of gravity of the latter, four such observations should be made, two with the poles of the needle in their existing state, and two others with the poles reversed. The reversion of the poles is effected by the usual method of magnetizing needles. 443. In order to obtain the correct dip from two observ- ations, when the centre of gravity is alternately below and above the point of support; let ZONH be the vertical circle 416 GEODESY. CHAP. XVIII. in the plane of the magnetic meridian : let c N be the actual direction of the needle when a weight w is applied at the end of the wire cw perpendicularly to NC, and let Cn be the direction which the needle ought to assume by the influence of terrestrial magnetism when the centre of gravity of the whole needle coincides with the point of support. Then, H o being a horizontal line, the angle H C n is the true inclination or dip (=d), and HCN ( = cwb) the false dip ( = d'), wb being a vertical line drawn through w to represent the weight of the ball w. Now, an equilibrium must be supposed to exist between the weight at w and the force of magnetism acting at c (the centre of gravity of the arm N c) ; the former force causing the needle to turn about c towards c H, and the latter causing it to turn about C towards C z'. Let the magnetic force acting at c be represented in magnitude and direction by cm parallel to Cn } and let fall mp perpendicularly on CN; also from b let fall be perpendicularly on CW. Then, by Mechanics, the force be being supposed to act at w in a direction parallel to be, and pm to act at c perpendicularly to CN, be x CW mp x Cc . . . . (A): now be ( w sin. cwZ>) w# sin. d f , and mp ( = c m sin. mcp cm sin. NC n) cm sin. (d e?'). When the needle is inverted, so that the weight w may be above it, as at w', let its position be c N 7 : then HCn, being the true dip as before, the angle HCN' (d") will be the false dip ; and the resolution of the forces being similar in both cases, we have b'e f X cw' = m'p' x Cc' . . . . (B) : also b f e f =w f b f sin. d" and m'p' c'm' sin. (d"d). Substituting the values in the equations (A) and (B), and cancelling the terms which by their equality destroy one another in division, there results sin (d - d'} _sin. (d"d) . sin. d 1 sin. d" Developing the numerators by trigonometry, the equation becomes sin. d cos. d 1 cos. d sin. d' __sin. d" cos. d cos. d" sin. d sin. d' sin. d' f ' CHAP. XVIII. FORMULAE FOR CORRECT DIP. .. 417 or sin. d cotan. d' cos. d cos. d cotan. d" sin. d ; or again, dividing by sin. d, and transposing, cotan d' + cotan. d" 2 cotan. d : thus, from the observed values of d' and d", that of d, the true dip, may be found. In the " Transactions " of the Royal Society of Sciences at Gottingen for 1814, the following formula for the true dip d is investigated in the case of four observations being made with the needle and its poles in direct and reversed positions, as above mentioned. Let d' d" be the values of the dip in a direct and a reversed position of the needle, as in the last case ; and, after the poles are reversed, let d'", d lv be values of the dip in a direct and reversed position of the needle : then, putting A for cotan. d f + cotan. d" ', B for cotan. d' cotan. d", C for cotan. d" f + cotan. d lv , and D for cotan. d'" cotan. d lv , there is obtained A.D B.C 2 cotan. d = - + B + D B + D 444. The place of the centre of magnetism in each arm of a balanced magnetized needle may be found precisely as the centre of gravity in any solid body would be found ; and, if the needle be cylindrical or prismatical, that centre would be in the middle of the length of each arm : the centre of oscillation might also be found for a needle as it would be found for a common pendulum ; and by the theory of pendulums, it may be shown that the intensities of magnetical attractions in dif- ferent parts of the earth are inversely as the squares of the times in which a given number of vibrations are made, or directly as the squares of the number of vibrations made in a given time. In making experiments with a needle, the number of vibra- tions made in a given time, and in an arc of a certain extent, must be reduced to the number which would be made in the same time in an arc of infinitely small extent : corrections should also be made for the buoyancy of the air ; and the formulas to be used for these purposes are similar to those which have been already given for the vibrations of pen- dulums by gravity. E E 418 GEODESY. CHAP. XVIII. A variation of temperature affects the magnetism of a needle as well as its length ; and a formula, depending on the length, similar to that which may be employed for a common pendulum, would not be sufficiently precise for the correction of the error arising from that cause. The method employed to find experimentally the coefficient for reducing the time of making a given number of vibrations at a certain temperature to the time in which an equal number would be made at a standard temperature, consists in counting the number of vibrations performed in a given time by the needle when placed in a vessel within which the temperature of the air may be varied at pleasure. For this purpose the apparatus is placed in a vessel of earth, or a trough of wood, with a glass top ; and this vessel is placed within another : between the two, cold water first, and hot water afterwards, is poured, in order to bring the temperatures to any convenient states, which may be indicated by a thermometer ; care is, however, taken not to raise* the temperature by the hot water higher than about 120 (Fahrenheit), lest the magnetism of the needle should thereby be permanently changed. Now, let T, in seconds, be the time in which a needle makes any number (suppose 100) of vibrations in the vessel when the temperature is t, and T' the time of making an equal number of vibrations when the temperature is raised to tf ; then rp/ rn t't : T' T:: 1 : 7^-71 i ~-~~t and this last term is the increase, in seconds, in the time of making that number of vibrations in consequence of an in- crease of temperature expressed by one degree of the ther- mometer. If the value of this fraction be represented by m, then m (r T'), in which T' denotes a standard temperature, suppose 60 (Fahr.), and r the temperature at which an ob- servation was made, will express the increase or diminution due to the observed time of any number of vibrations, with respect to the time of making an equal number at the standard temperature. Consequently, if T be the observed time of making any number of vibrations, and T' the required time of making an equal number of vibrations at the standard temperature, we should have T + m (r T') = T'. The magnetic condition of a needle ought to be ascertained at certain intervals of time, and allowance must be made for any variations which may be detected when observations at CHAP. XVIII. MAGNETOMETERS. 419 considerable intervals of time are compared together. In the " Voyage of the Beagle," it is stated that the time in which the needle performed 300 vibrations had increased in 5 years from 734".45 to 775".8. The state of a needle should not, however, be interfered with during the progress of any course of experiments for determining the intensity of terrestrial magnetism. 445. For the more delicate researches relating to terrestrial magnetism it is now found convenient, instead of the usual variation compass, to employ what is called a Declination Magnetometer ; and, instead of ascertaining the intensity di- rectly by a dipping needle, to employ horizontal force and vertical force magnetometers, by which the intensities in those directions may be determined ; and, from them, the direct in- tensity may be deduced. 446. The Declination Magnetometer is a bar magnet from twelve to fifteen inches long, nearly one inch broad, and a quarter of an inch in thickness ; this is made to rest horizon- tally in a stirrup of gun- metal, which is suspended, from a fixed point above, by fibres of untwisted silk nearly three feet long, and the whole is inclosed in a box to protect it from the agitations of the air. The bar is furnished with two sliding pieces, one of which carries a glass lens, and the other a scale finely graduated on glass, the scale being placed at the focus of the lens. By means of a telescope at a certain distance, the divisions on the scale may be seen through the lens (in the manner of a collimating instrument), and minute changes in the position of the axis of the magnet may therefore be detected by the divisions of the scale, which may be observed in coincidence with a fixed wire in the telescope. In order to have the suspending threads free from torsion at the commencement of the observations, a bar of gun-metal is previously placed in the stirrup, and allowed to remain there till the threads are at rest, when a button, carrying the point of suspension, is turned horizontally so as to bring the bar into the direction of the magnetic meridian ; after which the bar is removed, and the magnet is introduced in its place. It is obvious, however, that though the threads may be free from torsion when the needle lies in the plane of the mag- netic meridian, yet, as soon as by changes in the declination, the needle turns from that position, the threads will become twisted, and the apparent deviation from the mean position of the needle will be less than the true deviation. In order to E E 2 420 GEODESY. CHAP. XVIII. ascertain the correction due to this cause of error, the ratio of the force of torsion to the magnetic force must be found by experiment. For this purpose, NS representing the position of the suspended needle when in the magnetic meridian, let the button of the torsion circle \ be turned upon its centre, vertically above c, \ till its index has described any angle N C A ; (suppose a right angle), then the needle taking a position as n s and resting between the force of torsion acting horizontally from n towards A, and the force of magnetism acting horizon- tally from n towards N, if H represent the former force, and F the latter, we shall have by Mechanics, F : H :: ACW : and by proportion, F + H ACN F + H : F :: ACN : AC?Z; whence - - . The terms in the second member of this equation being given by the experiment, the ratio of F to F -f H, or of TT 1 to 1 H -- , is found. Any change in the declination of the F needle, which may be observed with the instrument, is to the corresponding change, free from the error produced by torsion, TT as 1 is to 1 H ; and hence the observed change multiplied F TT Ly 1 H -- gives the corrected change. F 447. In order to obtain the value of the horizontal intensity of terrestrial magnetism by the instrument, a magnetized bar, called a Deflector, is placed in a horizontal position at right angles to the magnetic meridian, and in a line imagined to be drawn through the centre of the suspended magnet. Its centre is to be placed at two different distances from the latter on this line ; and, in each position, the observer is to take notice of the deflexions produced by its attraction on the declination magnet when the north end of the deflecting bar is turned successively towards the east and towards the west ; half the difference of the deflexions with the north end towards the east, and afterwards towards the west, being considered as the required deflexion at each distance. The experiments are then to be repeated on the other side of the suspended magnet at equal distances from its centre, and a mean of the four deflexions, two on each side at equal distances, is to be taken CHAP. XVIII. COMPONENTS OF MAGNETISM. 421 for the deflexion at each of the two distances. Now it is demonstrated by M. Gauss (Intensitas Vis Magnetics Terre- stris, 1833) that if m represent the momentum of magnetism in the suspended bar, x the horizontal intensity of the earth's magnetism,, R and R' the distances of the centre of the deflecting bar from that of the suspended magnet, and , /-2 T?2\ t \ri K ) therefore the ratio between the horizontal force of terrestrial magnetism, and the magnetism of the bar or needle, can be found from the experiment. Next, the declination magnet being removed, the deflecting bar is to be attached to the suspension threads, and allowed to vibrate horizontally, and the time T of one vibration is to be determined from at least 100 oscillations : then, the momentum K of the bar's inertia being calculated, we have 7T 2 K rax rz * From these two equations m may be eliminated and the value of X may be found. If the value of the inclination or dip (d) be found by observation with the inclination instrument or dip circle in the plane of the magnetic meridian, the most correct method of ascertaining the vertical intensity will be to deduce it * Here mx expresses the momentum of the horizontal force of terres- trial magnetism by which the bar is made to vibrate ; and, K being the momentum of the bar's inertia, ^ X denotes the angular velocity of the K bar ; but, by the nature of the centre of oscillation, in Mechanics, I being the distance of that centre from the point of suspension, and g the force of gravity, ^, or^, or its equivalent "^ (TT= 3.1416), denotes the like angular velocity; therefore wx=^-. The factor (1+ ) is the cor- rection of the square of the time of vibration on account of the torsion of the suspension thread, as above shown. The bar being rectangular, the momentum K of its inertia with respect to a vertical axis passing through its centre of gravity may be shown, by Mechanics, to be equal to T ] T M (a -\-~b ) ; in which expression a is the length, b the breath, and M the mass of the bar. E E 3 422 GEODESY. CHAP. XVIII. from the value x, of the horizontal intensity, found as above. It is manifest that, if ch (fig. to art. 441.) represent X, hn will represent the vertical intensity which, consequently, is equal to x tan. d. Let this be represented by Y ; then the total intensity, represented by R and acting in the direction C n, may be obtained from the formula K := V (x 2 + Y-). 448. That which is called the horizontal-force-magneto- meter, and which is also employed for determining the horizontal intensity of terrestrial magnetism, is a bifilar instrument consisting of a magnetized bar or needle, sus- pended by a slender and very flexible wire passing under a pulley (from the axle of which a stirrup carrying the magnet is suspended) and through two perforations in a small bar above the pulley ; the wire is attached at its extremities to a plate at the top of the apparatus, so that the two halves of its length are parallel to one another when the needle is in the plane of the magnetic meridian. Now, turning the plate at the top of the apparatus till the needle is made to take a position at right angles to the magnetic meridian, resting there in equilibrio between the horizontal component of magnetic attraction by which it is drawn towards that meridian, and the force of torsion by which it is turned from it ; then, the force of torsion being computed, that of the magnetic component in the horizontal direction may be obtained. The same instrument is employed to determine, by the observed extent of the horizontal oscil- lations of the needle about its mean place, at right angles to the magnetic meridian, the ratio between the corresponding variations of the horizontal intensity, and the whole amount of that component. The formula for this purpose, as given in the " Report of the Committee of Physics," published by the Royal Society, 1840, is AF - cotan. v A u, F in which F represents the horizontal intensity, A F its va- riation, v is the angle formed by lines joining the upper ex- tremities and the lower extremities of the suspending wires, and u, expressed in terms of radius, is the observed deviation of the magnet from its mean place. The vertical force magnetometer is used for determining the variations in the vertical component of terrestrial mag- netism. It consists of a magnetic bar or needle about 12 inches long, having a horizontal axle formed into a knife edge and resting upon two agate planes which are supported on pillars of copper. The needle is provided, on each arm, with CHAP. XVIII. MAGNETICAL INTENSITY. 423 a screw ; one of these acts as a weight to keep the needle in a horizontal position, and the other, to render the centre of gravity nearly coincident with the centre of motion. The apparatus has an azimuthal motion by which the needle may be allowed to vibrate in the plane of any verticle circle ; and it is usually placed at right angles to the magnetic meridian. From the observed extent of the vertical vibrations of the needle about its mean place, in a horizontal plane, at right angles to the magnetic meridian, the ratio between the corresponding variations of the vertical intensity and the whole amount of that component may be determined. The formula for this purpose, in the "Report" above mentioned, is, A -p m/2 - -pj- cotan. 6 A 77, in which F represents the vertical intensity, A F its varia- tion, the dip or inclination, T and T' are the times in which the needle vibrates in vertical and horizontal planes respectively, and rj, expressed in parts of radius, is the ob- served deviation of the magnet from its mean place. 449. For the sake of regularity, in the expressions for the numerical values of the intensities of terrestrial magnetism, all measures of length, as well the linear dimensions of the magnetic bars or needles as the distances of the deflecting bars from them, should be given in feet and decimals of a foot ; the mass of a bar should be denoted by its weight in grains ; the times in which vibrations are performed should be ex- pressed in seconds, and angular velocities by the decimals of a foot described in one second on the arc of movement. In such terms, from the formulas for and TTZX above, Lieu- x tenants Lefroy and Kiddle, in 1842, found 3.72 as an ap- proximation to the value of the horizontal intensity at Woolwich (Phil. Trans. 1843, p. II.), at the same time the inclination, or dip, was found to be 69 3' ; therefore X tan. d, the vertical intensity, would have been expressed by 9.72, and ^ the total intensity, by 10.404. cos. d' At present the unit of absolute intensity is taken to re- present the state of that element on the peripheries of two curve lines surrounding the earth, and containing between them a band of irregular breadth crossing the geographical equator. The northern limit of this band was, at one time, supposed to be the line of least intensity ; and, with reference 424 GEODESY. CHAP. XVIII. to this unit, the intensity at London is expressed by the number 1.372. 450. When the results of several observations and ex- periments for the determination of an astronomical or geo- detical element have been obtained, they are next to be combined together, so as to obtain from the whole that value of the element which has the greatest probability of being strictly or nearly true : the principles upon which such com- binations are made have been already explained (arts. 328, 329, &c.) and it is intended now to show by a few examples in what manner the equations of condition are formed from the circumstances of a case. One of the most simple is that in which, at any station, there 'may have been observed the angles subtended by the distances between two objects A and B, between B and another object C, and also between A and C, which may be either the sum or the difference of the other angles, suppose the sum ; and it is required to find the most probable values of the two first angles when, the observations not being equally good, the weights due to the three observed angles are represented by 4, 7, and 9. Let the observed angle between A and B be 4214 / 6".5, between B and c be 3418 / 4 // .25, and between A and c be 7632 / 14 // .5. Now, in order to save trouble, since there is a presumption that the errors can only exist in the seconds, let the degrees and minutes be omitted till the operation is con- cluded, and let x, y, z be the most probable values of the seconds in the different angles: then (6.5 x) 2, (4.25 y) V 7, and (14.5 x y) 3 are the three errors multiplied by the square roots of their respective weights (art. 326.) in order to bring them to the same degree of precision ; and they constitute the first ^members of three equations of condition, the other members being the presumed errors, which may be represented by E 1? E 2 , and E 3 . Multiplying each equation by the coefficient of x in it, with its proper sign, and making the results equal to zero, agreeably to the method of Least Squares (art. 330.), we have 2 ( - 6.5 + x) - and 3 ( 14.5 + x + y} 0, whose sum is 56.5 4-5^+3y:=0. Again, multiplying each by the coefficient of y in it, and making the results equal to zero, we get CHAP. XVIII. EQUATIONS OF CONDITION. 425 A/7. (-4.25 +y).= and 3( 14.5 + x + y) = 0, whose sum is 54.737 + 5.644y + 3x = 0. From these sums we obtain x - 8".05, y - 5"A2 and x + y = 13".47. Prefixing to these numbers the degrees and minutes, the results will be the most probable values of the three angles. If a relatively correct determination of the instant at which a certain star is bisected by the middle wire of a transit te- lescope were required ; it might be obtained by a combination of several equations of the form T t x ay + bz =E, in which T is the true right ascension of a star as given in the Nautical Almanac, t is the observed time of the transit of the same star at the middle wire, x is the error of the sidereal clock supposed to be too fast, y is the error of colli- mation, in time, on the star's parallel of declination (art. 88.) and its coefficient a is the secant of the star's declination ; also z is the azimuthal deviation of the telescope, in time, and its coefficient b is that which is designated n in art. 95. If t in the equations were strictly correct the first member of each would be equivalent to zero, but, on account of the errors of observation, that member must be made equivalent to some small error which is represented by E. Multiplying the left hand member of each equation first by the coefficient of x in it, then by that of y, and after- wards by that of z, and making the sums of the equations separately equal to zero ; there may, from the three equa- tions, be obtained the most probable values of x, y and z ; and from these, by substitution, the most probable time t of the transit. Equations of condition for the length of a meridional arc may be of the form given in art. 420. ; in which repre- senting a by x, s by y, I' I (the coefficient of ) by p and /. , cos. (I 1 - 1) sin. (V - T)\ A , (^ + I >L_ > J (// _ /) by q . a is m > _ m by L, we shall have px + qxy L, or L px qxy=^> (E representing the supposed error arising from incorrectness in the data). From any number of these equations there 426 GEODESY. CHAP. XVIII. may be obtained the most probable values of x and y, that is of a and e. (Fig. of Earth, Ency. Metrop., p. 218.) Equations of condition for the lengths of pendulums may be of the kind Z A -f An sin. 2 X (art. 439.), the values of / being obtained directly from the experiments, and A and n deduced by computation ; and, putting the equations in the form I A A sin. 2 \ n = E (E representing the presumed error) the most probable values of A and n may be ob- tained. (Sabine's "Pendulum Experiments," p. 349, &c.) In some circumstances the determination of the most probable values of elements depends upon several independent relations which it is necessary to satisfy simultaneously, and this subject has been treated by M. Gauss in his " Sup- plementum Theorise Combinationis, &c.," Gottingen 1828. An application of the method has been made by Mr. Galloway in the " Memoirs of the Astronomical Society for 1844," to a triangulation executed in England, when the trigonometrical survey of the country was commenced ; and the following is an outline of the process. Every observed angle is considered as subject to a small error, and the equations of condition are of four different kinds. Those of the first kind are founded on the fact that the three angles of each geodetical triangle are equal to two right angles together with the spherical excess ; and its form is a + x + b-\-y-}-c + z = }. 80 + excess ; in which a, /;, c denote the three observed angles, or rather their several weighted means, and x, y, z are the un- known errors which may exist in the angles respectively. Equations of the second kind are found by making the sum of all the horizontal angles at any one station, when they comprehend all the circuit of the horizon, equal to 360 degrees ; and their form is a + x + b+y + c + z = 360, in which a, #, c represent the angles, and x, y, z the errors. Again, in each triangle let a side be taken which is common to that and the next triangle, in the latter let a side be taken which is common to it and to a third triangle, and so on in a circuit ending with a side which is common to the last and first triangles; then, the sines of these sides being to one another as the sines of the opposite angles, (art. 61.), if the observed angles which are opposite to the sides be represented by , b, c, &c., and the errors by x, y, z, x', y', z r , &c., there may be formed a series of the fractions CHAP. XVIII. EQUATIONS OF CONDITION. 427 sin. (a + #) sin. (b + y) sin. (b + y) sin. (c + z) sin. (b + yY sin. ( + *)' sin. (c+*)' sin. ' which being multiplied together have unity for the equivalent of their product : in this manner there -may be formed equations of a third kind. Equations of a fourth kind may be formed when an observed angle is made up of two or more angles, each of which has been also observed, by making the whole angle equal to the sum of all its components. The values of the errors or corrections in the several equations of condition are to be determined by the method of least squares, and the angles are at the same time to satisfy those equations : thus, not only will the most probable values of the angles be obtained ; but, from one extremity to another of a triangulation, the sides computed with those angles will be the same in whatever order the series of triangles may be taken. THE END. LONDON : Printed by A. SPOTTISVVOODE, New- Street- Square. 39, PATERNOSTER ROW, AUGUST 6, 1844. & jfcelect Catalogue of BOOKS ON EDUCATION, IN ALL BRANCHES OF KNOWLEDGE, PRINTED FOR LONGMAN, BROWN, GREEN, AND LONGMANS. MESSRS. LONGMAN AND Co. have recently published the following important NEW SCHOOL BOOKS : Mrs. Felix Summerly' s Mother's Primer. 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To which is added, a Synopsis of the Greek Metres, by the Rev. J. R. Major, D.D. Head Master of King's College School. London. 2d Edition, revised and corrected by the Rev. F. E. J. Valpy, M.A. formerly Head Master of Reading School. 8vo. 15s. cloth. Giles's Greek and English Lexicon. A Lexicon of the Greek Language, for the use of Colleges and Schools ; containing 1. A Greek-English Lexicon, combining the advantages of an Alphabetical and Derivative Arrangement ; 2. An English-Greek Lexicon, more copious than any that has ever yet appeared. To which is prefixed, a concise Grammar of the Greek Language. By the Rev. J. A. GILES, LL.D. late Fellow of C. C. College, Oxon. 2d Edit, with corrections, 1 thick vol. 8vo. 21s. cloth. V The English-Greek Part separately. 7s. 6d. cloth. " In tw points it excels every other Lexicon of the kind ; namely, in the English-Greek part, and in the Classification of Greek Derivatives under their primitives." MOODY'S ETON GREEK GRAMMAR. 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The New Eton Greek Grammar ; with the Marks of Accent, and the Quantity of the Penult: containing the Eton Greek Grammar in English : the Syntax and Prosody as used at Eton ; also, the Analogy between the Greek and Latin Lan- guages ; Introductory Essays and Lessons : with numerous Additions to the text. The whole being accompanied by Practical and Philosophical Notes. By CLEMENT MOODY , of Magdalene Hall. Oxford ; and Editor of the Eton Latin Grammar in English. 2d Edition, carefully revised, &c. 12mo. 4s. cloth. Valpy's Greek Grammar. The Elements of Greek Grammar: with Notes. By R. VAI.PY, D.D. late Master of Reading School. New Edition, 8vo. 6s. 6d. boards; bound, 7s. 6d. Valpy's Greek Delectus, and Key. Delectus Sententiarum Grsecarum, ad usum Tirpnum accommodatus : cum Notulis et Lexico* Auctore R. VALPY, D.D. Editio Nova, eademque aucta et emend ata, 12mo. 4s. cloth. KEY to the above, being a Literal Translation into English, 12mo. 2s. 6d. sewed. STANDARD EDUCATIONAL WORKS. Valpy's Second Greek Delectus. Second Greek Delectus ; or, New Analecta Minora: intended to be read in Schools between Dr. Valpy's Greek Delectus and the Third Greek Delectus: with English Notes, and a copious Greek nnd English Lexicon. By the Rev. F. E, J. VALrY, M.A. formerly Head Master of TReadmg School. 3d Edition, 8vo. 9s. 6d. bound. Tire Extracts are taken from the following Writers : Hierocles Piilaephatus Plutarch JElian The Septuagint St. Matthew Xenophon Euripides I Sophoctes .Sschylus Aristophanes Herodotus Anacreoa Polypenus Valy's Third Greek Delectus. hird Greek Del Tyrtseus Eton ; Moschus Erycius of Cyziciim Archytas. py The Third Greek Delectus ; or, New Analecta Majora : with English Notes. In Parts. By the Rev. F. E. J. VALPT, M.A. formerly Head Master of Reading School. 8vo 15s. 6d. bound. The Parts may be had separately. PART 1. PROSE. 8vo. 8s. 6d. bound. The Extracts are taken from Herodotus I Isocrates I Demosthenes Xenophon | Plato | Lysias PART 2. 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New Edition, 8vo. 5s. boards. KEI, 3s. boards. Howard's Greek Vocabulary. A Vocabulary, English and Greek ; arranged systematically, to advance th learner in Scientific as well as Verbal Knowledge : with a List of Greek and Latin Affinities, and of Hebrew, Greek, Latin, English, and other Affinities. By NATHANIEL HOWARD. New Edition, corrected, 18mo. 3s. cloth. Howard's Introductory Greek Exercises, and Key. Introductory Greek Exercises to those of Huntingford, Dunbar, Neilson, and others; arranged under Models, to assist the learner: with Exercises on the different Tenses of Verbs, extracted from the Table or Picture of Cebes. By NATHANIEL HOWARD. New Edition, with considerable improvements, 12mo. 5s. 6d. cloth. KEY, 12mo. 2s. 6d. cloth. Dr. Major's Greek Vocabulary. Greek Vocabulary ; or, Exercises on the Declinable Parts of Speech. By tlw Rev. J. R. MAJOR, D.D. Head Master of the King's College School, London. 2d Edition, corrected and enlarged, 12mo. 2s. 6d. cloth. Evans's Greek Copy-Book. rpad>VS Aovol; sive, Calamus Scriptorius: Copies for Writing Greek in Schools. By A. B. EVANS, D.D Head Master of Market- Bos worth Fre Grammar School. 2d Edition, 4to. 5s. cloth. The use of one Copy-Book is sufficient for securing a firm and clear Greek hand. MESSRS. LONGMAN AND CO. S Dr. Major's Guide to the Greek Tragedians. A Guide to the Reading of the Greek Tragedians; being a series of articles on the Greek Drama, Greek Metres, and Canons of Criticism. Collected and arranged by the Rev. J. R. MAJOR, D.D. Head Master of King's College School, London. 2d Edition, enlarged, 8vo. 9s. cloth. In this second edition the work has undergone a careful revision, and many important additions and improvements have been made. Seager's Edition of Bos on the Ellipsis. Bos on the Greek Ellipsis. Abridged and translated into English, from Professor SchsefFer's Edition, with Notes, by the Rev. J. SEAGER, B.A. 8vo. 9s. 6d. bds. Seager's Hermann's Greek Metres. 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Horatius Restitutus ; or, the Books of Horace arranged in Chronological Order , according to the Scheme of Dr. Bentley, from the Text of Gesner, corrected and improved: with a Preliminary Dissertation, very much enlarged, on the Chronology of the Works, on the Localities, and on the Life and Character of that Poet. By JAMES TATE, M.A. 2d Edition, to which is now added, an original Treatise on the Metres of Horace. 8vo. 12s. cloth. Turner's Latin Exercises. Exercises to the Accidence and Grammar; or, an Exemplification of the several Moods and Tenses, and of the principal Rules of Construction: consisting chiefly of Moral Sentences, collected out of the best Roman Authors, and translated into English, to be rendered back into Latin ; with references to ths Latin Syntax, and Notes. By WILLIAM TURNER, M.A. late Master of the Free School at Colchester. New Edition, 12mo. 3s. cloth. STANDARD EDUCATIONAL WORKS. Beza's Latin Testament. 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