Example 1. Fig. 40. Given the Moons apparent altitude 56° 20' Stars apparent altitude Apparent distance Moon's horizontal parallax 49° 48' 59′ 9′′ To find the true distance by projection, draw the lunar line at pleasure AB, and draw the line AC to make an angle with the former equal to the apparent distance; from the point A lay off the sign of the moon's apparent altitude AC, and that of the stars equal to A✶ the star's apparent altitude. Through the point and draw perpendicular lines, which will intersect in D; the line DD is the first correction, which being measured on the line of chords, and calling each degree a mile, will be found equal to 32', and this being on the solar or star side of the lunar line, is subtractive; multiply it by the given horizontal parallax, and the product divide by 62, which gives 30' 32", which subtract from the apparent distance, viz. Apparent distance Second correction subtract 72° 36′ 13" 30 32" True distance 72° 5′ 41′′ As logarithms are constructed to facilitate the calculation of multiplication and division, so that by them multiplication is performed by addition, and division by substraction. EXAMPLE. The first correction 32' prop. log. 7501 Moon's horizontal parallax 59′ 9′′ prop. log. 4833 1.2334 To find the true distance, draw the lines ABC as before, making the angle A equal to 30° 00' the apparent distance, lay off A and A✶ equal to the signs of their apparent altitude. Then, through✶ and D draw perpendicular lines, which will intersect in D, then is the line D equal to 24° 14' on the line of chords, which call 24′ 14′′ the first correction, and being on the contrary side of the star that is on the left side of the lunar line, it is additive; then 24' 14" being multiplied by 60' the given horizontal parallax, and divided by 53' will give 27' 27", the second correction being added to 30° 00' the apparent distance will give the true distance, 30° 27' 27". Which being added to the apparent distance, gives the true 5310 0.8167 distance. Example 3. Fig. 42. Given in the moon's apparent altitude 10° 00' Star's apparent altitude Apparent distance Moon's horizontal parallax, 70° 00' 60° 00' 60' Let ABC be described to an angle at A equal to 60° the apparent distance, and lay off from the line of sines AD and A equal to the moon and star's apparent altitudes respectively. Through and draw perpendicular lines which will meet in D, then is D equal to 56° 00′ on the line of chords, calling each degree on the scale a mile, which is the first correction, then 56 x 60 62 is equal to 54′ 12", the second correction which is subtractive from the apparent distance being on the right side of the lunar line will give the true distance. Example by Proportional Logarithms. 5213 Difference gives prop. log. of second correction 54′ 12′′ To find the correction and apparent distance, 24° 00' 18° 00' 44° 00' 56' 30" Describe ABC as before, making an angle at A equal to the apparent distance, lay off A D and Aequal to the sine of the sun and moon's apparent altitudes respectively; through D and draw perpendicular lines, which in this case will meet in D, therefore it appears there is no correction to be made, so that the apparent distance will be the true distance. Example 5. Fig. 44. The following example in projection may be performed with equal facility and accuracy, by the scale of chords only, when the apparent distance is not more than 90°. Given the moon's zenith distance 80° 00' 60° 00 24° 00' 60' To find the true distance: With the chord of 60o describe the semicircle EBF, make EB the lunar line, and make AC equal to an angle of 24° to equal the apparent distance; lay off 80° the moon's zenith distances, (measured on the line of chords,) both ways from B to X and Y, and draw the line XY, then lay off the star's zenith distance 60° both ways, from C to M and MN; observe where MNXY intersect in D, and DD is the line of correction, which falling on the right side of the lunar line it is subtractive, then multiply the first correction by the given horizontal parallax 60' anddivide by 62 gives the second correction, or by proportional logarithms, as in the preceding rules. 1st Cor. 48' 45" Hor. Par. Prop. Log. of 62' Prop. Log. 5673 60' Prop. Log. 4771 1.0444 0.4629 0.8515 24° 00' 00" Prop. Log. of Second Correction 47' 11" Ap. Dist. True distance 23° 12′ 49" |