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By Trigonometrical Calculation.
The Globe being in the former Pofition, you will find upon it an Oblique-angled Spherical Triangle, compofed of P L, an Arch of the Brafs Meridian: OfP H, an Arch of the Meridian that paf feth over Hierufalem; and of L H, an Arch of the Quadrant of Altitude. In which Triangle you have given, (1.) PL, the Com plement of the Latitude of London, 38 Deg. 30 Min. (2.) The Side P H (the Complement of the Latitude of Hierufalem 59: Deg. 20 Min. (3.) The Angle L PH, the Difference of Longi tude of the Two Places, 46 Deg. To find the Two Angles of Pofition at L and H: And, (2.) The Side L H, the Diffance of the. Two Places.
The Canons for Calculation.
By CASE III. and CA SE IX. of A.O. S. T..
(1.) As the Sine of half the Sum of the Sides P L and PH, 48 Deg. 25 Min.
Is to the Sine of half their Difference, 9 Dég. 55 Min..
So is the Co-tangent of half the Difference of Longitude (i. e) half the Angle LPH, 23 Deg.
To the Tangent of 28 Deg. 29 Min.
Which is the half Difference of the Angles PLH and PH L. (2.) As the Co-fine of half the Sum of the Sides P L and P_H, 41 Deg. 35 Min.
Is to the Co-fine of half their Difference, 80 Deg. 2 Min. So is the Co-tangent of half the Difference of Longitude, i. e.. half the Angle LPH 23 Deg.
To the Tangent of 74 Deg. 2 Min.
Which is the Tangent of half the Sum of the Two Angles
(3.) For the Side L H, which is the Diffance,
As the Sine of PH L, 45 Deg. 33 Min..
I's to the Sine of PL, 38 Deg. 30 Min.
So is the Sine of L PH, the Differ. of Longitude, 46 Deg.
Which 38 Deg. 31 Min. reduced in Miles, makes 2331 Miles
3. If the Two Places propofed fhould be fo fituate, that one have North, and the other South, Latitude, and under different Meridians. As fuppofe Conftantinople, lying in North Latitude 47 D. at C. and the Cape of Good Hope, lying in 35 Deg. of South Latitude, at V, and differing in Longitude 59 Deg. To find the Di ftance of thefe Two Places
Upon the Terreftrial Globe.
Bring one of the Places (as Conftantinople) to the Brafs Meridian, and there keep the Globe; then apply the Quadrant of Altitude (or rather a thin Plate of Brafs divided as that is) to the Two Places, and you shall find 97 Deg. 42 Min. of the Quadrant (or Brafs Plate) to be contained between them, and that is their Distance..
By Trigonometrical Calculation..
The Globe refting in its former Pofition, you will discover upon it an Oblique-angled Spherical Triangle, compofed of P C, an Arch of the Brass Meridian: PV, an Arch of a Meridian, paffing through the Place in South Latitude: And of C V, the Quadrant (or Brass Plate) representing the Arch of a Great Circle paffing through both the Places:-In which Triangle you have given, (1.) The Side PC, the Complement of the Latitude of Conftanti nople, 43 Deg. (2.) The Side PV, the South Latitude of the Cape of Good Hope, with 90 Deg. added, which make 125 Deg. And, (3.) The Angle V PC, the Difference of Longitude, 59 Deg. To find, (1.) The Angles of Pofition P V C and PCV: And, (2.) The third Side VC, the Distance of the Two Places.
Fig. The Canons for Calculations, as in the laft, by CASE III and XXVI. IX. of A. O. S. T.
(1.) As the Sine of the Half Sum of the Sides PV and PC, 84 D. Ís to the Sine of half their Difference, 41 Deg.
So is the Co-tangent of half the Difference of Longitude, i. e. (half the Angle V PC) 29 Deg. 30 Min.
To the Tangent of 49 Deg. 23 Min.
Which is the Half Difference of the Angles PVC and PCV.. (2.) As the Co-fine of half the Sum of the Sides, PV and PC, 6 Deg.
Is to the Co-fine of half their Difference, 49 Deg.
So is the Co-tangent of half the Difference of Longitude, 61 Deg.
Which is half the Sum of the Two Angles, PCV and PVC.
The Two Angles of Pofition.
(3.) For the Side V C, which is the Diftance of the Two Places.
As the Sine of PVC, 36 Deg. 9 Min.
Is to the Sine of P C, 43 Deg.
So is the Sine of V PC, 59 Deg.
To the Sine of 82 Deg. 18 Min.
Whose Complement to 180 Deg. is 97 Deg. 42 Min. for the
And these are all the Varieties of Pofitions that any Two
Hereas, for the refolving of the following Aftronomical Fig. Problems by Trigonometrical Calculation, it is abfolutely XXVII. neceffary, that the true Place (or Longitude) of the Sun in the Écliptick be first known; I have therefore inferted Aftronomical Tables of the Sun's Mean Longitude and Anomaly in Tears, Months, Days and Hours; and alfo of Equation, whereby the Sun's true. Place in the Ecliptick may be found at any time. It is alfo to be noted, That the feveral Problems in this Section, being wrought firft by the Cæleftial Globe, in order to their Trigonometrical Čalculation; I have, therefore, to inform the Fancy, and ease the Memory, of the Reader (in all the Schemes relating to them, (which confift of Great Circles of the Sphere, and reprefent the Triangle to the Eye, which the Problem is to be refolved by) noted the Circles, Lines and Points, in all the Schemes, with the fame Letters or Characters; and prefixed Them, and their Significations, next before the Problems.