The Mathematics of the Longitude - Department of Mathematics

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$17. C. T. Chong and Liang Yu, A$\Pi^1_1$ uniformization principle ...$

Let M be the true altitude of the Moon (arc MO), M be the apparent altitude of the Moon (arc Om), S be the true altitude of the Sun or star (arc SH), s be the observed altitude of the Sun or star (arc sH), D be the true lunar distance (arc SM ), d be the observed lunar distance (arc sm ). Due to the parallax of the Moon, the observed altitude of the Moon is smaller than the true altitude of the Moon as shown in Figure 8.3. And due to refraction, observed altitude of the Sun or star is bigger than the true altitude of the Sun or star. Using triangle ZSM and the law of cosines for sides, we have cos( SM) − cos( ZS) cos( ZM) cos( Z) = , sin( ZS) sin( ZM) cos( SM) − sin( HS) sin( OM) cos( Z) = . (1) cos( HS) cos( OM) Using triangle Zsm and the law of cosines for sides, we have cos( sm) − cos( Zs) cos( Zm) cos( Z) = , sin( Zs) sin( Zm) cos( sm) − sin( Hs) sin( Om) cos( Z) = . (2) cos( Hs) cos( Om)
Equating (1) and (2), we have cos( D) − sin( S) sin( M) cos( d) − sin( s) sin( m) = , cos( S) cos( M) cos( s) cos( m) cos( D) − sin( S) sin( M) cos( d) − sin( s) sin( m) 1+ = 1+ , cos( S) cos( M) cos( s) cos( m) cos( S) cos( M) + cos( D) − sin( S) sin( M) cos( s) cos( m) + cos( d) −sin( s) sin( m) = , cos( S) cos( M) cos( s) cos( m) cos( D) + cos( M + S) cos( d) + cos( m+ s) = , cos( S) cos( M) cos( s) cos( m) ( D ( ) ) 2 M + S 1− 2sin 2 + cos 2 2 cos( S) cos( M) ( D ( M S) ) = + ( ) ( ) { 2 } −1 2cos m+ s+ d + − { 2} cos m s d { 2} { } 2 2 sin 2 cos 2 = cos( s) cos( m) ( ) ( ) cos( S) cos( M) cos m+ s+ d { 2} cos m+ s−d 2 − cos( s) cos( m) Let ( ) ( ) cos( S) cos( M) cos m+ s+ d { } cos m+ s−d 2 2 2 cos ( θ ) = cos( s) cos( m) 2 ( ) Then, ( D 2 M S ) = + 2 { } − ( θ ) sin 2 cos 2 cos 1 = { + ( + ) } − { + } 2 1 1 cos M S 1 cos2θ 2 1 = { cos( M + S) −cos( 2θ ) } 2 ( ) ( ) = sin M + S { + } sin − M + S 2 θ θ 2 { } { } { } . (3) , .
Page 1 and 2: The Mathematics of the Longitude Wo
Page 3 and 4: Chapter 6 Lunar Eclipse Method ....
Page 5 and 6: Acknowledgements Firstly, I would l
Page 7 and 8: otates 15 degrees per hour, if you
Page 9 and 10: Chapter 1 Introduction Throughout h
Page 11 and 12: Chapter 2 Terrestrial, Celestial an
Page 13 and 14: circle. Any other parallel of latit
Page 15 and 16: The ecliptic and celestial equator
Page 17 and 18: observer positioned in the GP of a
Page 19 and 20: Figure 2.11 None of the above horiz
Page 21 and 22: 3.2 Greenwich Apparent Time The tim
Page 23 and 24: Chapter 4 Spherical Trigonometry 4.
Page 25 and 26: These formulas allow us to calculat
Page 27 and 28: Figure 5.2 A man using the cross-st
Page 29 and 30: To find the altitude of a celestial
Page 31 and 32: vernier will lie between the zero a
Page 33 and 34: Figure 5.8 The visible sea horizon
Page 35 and 36: (e) Semi-diameter Semi-diameter, SD
Page 37 and 38: Chapter 6 Lunar Eclipse Method 6.1
Page 39 and 40: 6.3 Lunar Eclipse Method Being lost
Page 41 and 42: Chapter 7 Eclipses of Jupiter's Sat
Page 43 and 44: Figure 7.2 Giovanni Domenico Cassin
Page 45 and 46: Even with the availability of instr
Page 47: 8.3 Clearing the Lunar Distance Whe
Page 51 and 52: (M+S)/2 22°44'19.5" log cos θ = 1
Page 53 and 54: 2 Now, sin ( θ ) = = = = Similarly
Page 55 and 56: C = M − m= 49′ 05′′ c= s−
Page 57 and 58: Figure 9.2 H4, Harrison's fourth ma
Page 59 and 60: T + T ieT .. = Transit 1 2 2 Figure
Page 61 and 62: somewhere on a line, actually a pos
Page 63 and 64: (2) Find the altitude H of a select
Page 65 and 66: Chapter 12 The Intercept Method 12.
Page 67 and 68: intercept method to obtain ship cur
Page 69 and 70: (8) Take the map and draw a suitabl
Page 71 and 72: Chapter 13 Prime Vertical Method Be
Page 73 and 74: Bibliography [1] Charles H. Cotter,

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