Aftronomical Problems. PROBLEM I. The Longitude, or Place of the Sun, in the Ecliptick, being given; To find, 1. The Sun's Right Afcenfion. 2. The Sun's Declination. 3. The Angle of Pofition made by the Interfection of the N this Problem there is given the Sun's or a Star's Place, I in Refpect of the Ecliptick, and his Place, in refpect of the Equinoctial, is required, I. By the Cœleftial Globe. Example. Let the Place of the Sun be in 29 Deg. of Tau rus 8, (that is, 59 Deg. from the beginning of Aries V, which is the nearest Equinoctial Point.) The Globe being in any Pofition, (for in this Problem there is no regard to be had to the Latitude) count the Sun's Place in the Ecliptick upon the Ecliptick Circle, from r; and bring that Point to the Graduated Side of the Brass Meridian. Then, (1.) Will the Brafs Meridian cut the Equinoctial Circle in 56 d. 46 m. counted alfo from ; and that is the Right Afcenfion. And, (2.) The Number of Degrees of the Brass Meridian, comprehended between the Equinoctial and Ecliptick Circles, will be 20 d. which is the Sun's Declination. And, (3.) The Angle made by the Interfection of the Brafs Meridi an, and the Ecliptick Circle in the Point of the Sun's Place, will be 77 d. 23 m. which is the Angle of Pofition, in respect of the Meridian and Ecliptick. II. By Trigonometrical Calculation. Fig. The Globe being in this Pofition, there is reprefented upon the Superficies thereof, Two Right-angled Spherical Triangles, fuch as are expreffed in the Diagrams; and are there noted with OR and R; in which, the Sides and O XXVII. are Arches of the Ecliptick Circle, and is the Sun's Longitude: XXVIII. Hh2 The 198 Fig. The Sides XXVII. is the Sun's R and R are Arches of the Equino&ial, and Right Afcenfion; and the Sides OR are Árches XXVIII. of the Brafs Meridian, and is the Sun's Declination.-Alfo, the Angle at or, is the Angle of the greatest Obliquity of the Ecliptick, (and is equal to the Sun's greateft Declinati on 23 d. 31 m.) The Angle R is a Right Angle, and the Angle is the Angle at the Sun's Pofition, in refpect of the Ecliptick and Meridian Circles. To refolve this Problem Trigonometrically, you are to confider the Quadrant of the Ecliptick, in which the Sun is, which in the Figures are fignified by one of thefe Numbers, 1. 2. 3. 4. Of which, the firft is of the Spring, from the beginning of Aries T, to the beginning of Cancer, &c. Then in the Triangle R or RO. 1. The Angle at y or, 23 d. 31 m. 2. Rr or R: The Sun's Distance from the next Equino&ial Point; to be numbred from ror, unto the Degrees of the Sun's Longitude or Place in the Ecliptick given. And, There is given, There is required, 1. R o, The Sun's Declination for that Sign which he is in; whether North or South. 2. The Arch r R, or f1. The Degrees found by the? 4. The Degrees found muft be Right Afcenfion. 3. The Angle, or the Angle of Pofition, in refpect of the Meridian and Ecliptick Circles. The Canons for Calculation. The Sun being in 29 d. of Taurus 8, which is 59 d. from Aries T. 1. For the Right Afcenfion Y or R. By Cafe III. of R. A. S. T. As the Sine of 90 d. Is to the Co-fine of the greateft Obliquity of the Ecliptick 66 d. 29 m. - So is the Tangent of the Sun's Distance from Yor, 59 d. D. M. The R. Afcen-123 14 236 46 If the Sun's (1 d. of Leo, in Quad. 2. As the Sine of 90 d. 303 303 15 Is to the Sine of the greateft Obliquity of the Ecliptick 23 d.31m. So is the Sine of the Sun's Diftance from Y or 59 d. To the Sine of 20 d. 'the Sun's prefent Declination. Which is North, because he is in a Northern Sign. 3. For the Angle of the Sun's Pofition. By Cafe III. of R. A. S. T. As the Sine of the Sun's prefent Declination 20 d. Is to the Sine of greatest Declination 23 d. 31 m. So is the Sine of the Sun's R. Afcenfion 56 d. 46 m. To the Sine of 77 d. 23 m. The Sun's Angle of Pofition, mude by the Meridian and Ecliptick Circles. PROB. II. The Right Afcenfion, or Declination of the Sun given; To find his Longitude (or Place) in the Ecliptick. THI HIS is the Converfe of the foregoing Problem; for in this, the Sun's Place, in refpect of the Equinoctial Circle, is given: And his Place, in refpect of the Ecliptick Circle, is required: And may be refolved as followeth, Fig. XXVII. XXVIII. I. By Fig. XXVII. XXVIII. I. By the Cœleftial Globe. Example. Let the Sun's Right Afcenfion be 303 d. 14 m. and his Declination 20 d. Southward: And let his Longitude (or Place in the Ecliptick) be required. Count 303 d. 14 m. the Right Afcenfion given, upon the Equinoctial Circle, from the Vernal Equinoctial Point Aries V, and bring that Point to the graduated Side of the Brass Meridian: Then will the Brafs Meridian cut the Ecliptick Circle in 1 d. of Aquarius, and in that Sign and Degree the Sun is at Noon, when his Right Afcenfion is 303 d. 14 m. But to find his Place by his Declination, Count 20 d. (the De clination given) upon the Brass Meridian, downwards towards the South Pole (because the Declination given was Southerly) and turn the Body of the Globe about till 20 d. of the Meridian do cut the Ecliptick Circle; which it will do in Two Points, viz. One in 29 d. of Scorpio m, in the third Quadrant; and the other in d. of Aquarius, in the fourth Quadrant; in both which Points the Sun being, he hath 20 d. of South Declination, and which of thofe Points you feek is determined by the Degrees of Right Afcenfion given, as here 303 d. 14 m. and the Globe being in this Pofition, will fign out the fame Triangles, mentioned in the foregoing Problem. II. By Trigonometrical Calculation. Confider the Quadrant of the Ecliptick in which the Sum is, (which the Degrees of Right Afcenfion given, will determine) and in this Example will be the fourth Quadrant: But the given Declination is indifferent in all the Four Quadrants. There is given, Wherefore, in the Triangle Rv o. 1. The Angle at Yor, 23 d. 31 m. the Sun's greatest Declination. The Side RO, the Sun's prefent Declination 20d. And alfo either of the other Sides Rr. (or R) an Ark of the Equator, to be numbred from the nearest Equino&ial Point, to the Degrees of Right Afcenfion given. And And there is required The Side (or) the Sun's Diftance in Fig. the Ecliptick Circle; from the nearest Equi. XXVII. noctial Point, from which Point the Longi- XXVIII. tude required is to be numbred upon the Eclip tick Circle. So the Right Afcenfion 303 d. 14 m. given, it being in the Fourth Quadrant, will give in the Ecliptick 1 d. of Aquarius ****. The Canons for Calculation. 1. By the Right Afcenfion given. The Right Afcenfion given being 303 d. 14 m. are found in the Fourth Quadrant, and therefore must be fubftra&ed from 360 d. and the Remainder will be 56 d. 46 m. which must be made ufe of in the Calculation, inftead of 303 d. 14 m. And then in the Triangle OR, r. By the Right Afcenfion given. As the Tangent of the Side fion, R, 56 d. 46 m. the Right Afcen Is to the Sine of 90 d. or Radius: So is the Co-fine of the Angle at 66 d. 29 m. To the Co-tangent of, 59 d. eo m. Sun's Longitude. Now, because the Right Afcenfion given was in the Fourth Quadrant, the Longitude 56 d. 46 m. thus found, must be in the Fourth Quadrant alfo, which will be in 1 d. of Aquarius. 2. By the Declination given. As the Sine of the Sun's greatest Declination, 23 d. 31 m. the Is to the Sine of the Side clination given : So is the Radius, Sine go d. R, 20 d. the Sun's prefent De To the Sine of the Side O, 59 d. from Libra And being in the Fourth Quadrant, it gives the Longitude of the Sun to be in 1 d. of Aquarius, as before.. PROB.. |