discovering the true Amplitude of the Sun, and its Difference from the Magnetic Amplitude, which is what we call the Variation of the Compass. PROBLEM XVIII: P Given the Latitude of the Place, the Sun's Declination, and the Horizontal Refraction, to find the apparent Time of Sun's Rising and Setting, and also of his apparent Amplitude. PROBLEM V. Cafe 1. (The Latit. 50° 56' N. The Declin- Ær 90° 33'. N Point C below the Horizon 33' = BC; wherefore the Azimuth Circle Z B N, and the great Circle PC, being both drawn through the Point C, where the Sun really is ; there will be formed the Oblique Triangle Z PC, in which all the Sides are given ; viz. Z P = 39°04'; PC= 69° 26'; and Z C = 90° 33'; to find the Angle ZP C= Hour from Midnight of the Sun's rising. This I shall find by Theorem 39. Thus, CP-ZP = AM= 309 22'. VOL. II. Then Then the Sinc of C P = 69° 26' Co. Ar, 0.0286016 the Sine of ZP = 39°04' Co. Ar. 0.2005049 cZ+AM Add the Sine of = 60° 27' 9.939482 F CZ TAM the Sine of = 300 061 9.7002802 2 2 The Sum of all is 19.8688688 Pof which is the Sine of - ZPC=59° 18'= 9.9344344 Wherefore the Whole Angle Z PC = 118° 36', whose Complement to 180° is the Angle CPO= 619 24', which is the Hour from Midnight ; and reduced to Time, is 4 Hours, 5', 36", the apparent Time of the Sun's Rising on the first of May, A. De 1735. But the true Time of his rising was found to be 4 Hours 9' 52" at that time, by Prob. 9. Hence the Difference between the Angles APO and C PO, or the true and apparent Rising and Setting of the Sun on that Day is 4' 16', the apparent Rising being so much the fooneft ; so that the apparent Day is longer than the Astronomical Day by g' 32". Case 2.(The Latit. 50° 56' N. P 20° 34'S. IH Then the Scheme bein prepared, there will be formed the Oblique Triangle C Z P, in which the Side ZP = 39°04'; the Side ZC = 90° 33'; N the Side CP= 110 34'; to find the Angle CPZ? Here CP-ZP= AM = 71° 30'. Then Then ( the Sine of CP=110° 34' Co. Ar. 0.0286016 the Sine of ZP = 399 04' Co. Ar. 0.2005049 Add the Sine of CZ+AM = 81° oil 9.9946399 the Sine of CZ+AM = 9° 32' 9.2191164 2 2 - The Sum of all is 19.442 8628 of which is the Sine of P = 31° 461= 9.7214314 Wherefore the whole Angle Z PC is 63° 32', which reduced to Time, gives 4 Hours 14' 8", the Apparent Time of the Sun's Setting ; whose Complement to 12 Hours, viz. 7 Hours 45' 52", is the Apparent Time of his Rising. But by Prob. 9, the true Time of the Sun's Rising is 7 Hours 50' 81; and of his Setting 4 Hours 9' 52". Hence the apparent Rising is 4' 16'1 sooner than the True ; and the apparent Setting is 4' 1611 later than the True. Hence we observe 1. That by means of this Refraction, we have, on the two Days the Sun comes on this Parallel of North and South Declination, 16' 32" more real Day, or Sun-shine, than we should have were there no Refraction. 2. That since the Difference of the True and Apparent longest and shortest Day is g' 12", and of an Equinoctial Day is 3' 48"; therefore we may take 61 301 for a common Difference of the True and Apparent Day, one with another through the Year ; and thus upon the Account of this Refraction, we gain (at a Medium ) 397 Hours of Sun-shine, which is as good as the Addition of 34 of Equinoctial Artificial Days to the Year, which is almost the illuminated Part of one whole Year in every Century, which I think is a Consideration worthy the Notice of Divines. 3. That 3. That the Reason why the two great Luminaries, the Sun and the Moon, near the Horizon appear of an Oval Figure, is because their inferior Limbs are more refracted, and thereby raised higher than the superiour Limbs are ; and so those two Limbs will seem nearer to each other, and the Breadth of the Bodies contracted, while both Ends of the Horizontal Diameter being equally refracted and raised, keep the fame Distance and Magnitude. 4. That the Difference of the True and Apparent Time of the Sun and Moon's Rising and Setting is of very great Consequence to the Astronomer in the Business of the Eclipses ; For 'tis the Visible or Apparent Time we have regard to in computing how much of their Disks is obscured in Rising or Setting above or beneath the Horizon. 5: The Refraction also causech a Difference of the Amplitude of the Sun's Rising and Setting. For AO is the True Amplitude from the North ; but B() is the Visible or Apparent Amplitude, and is always less than the Former ; the Difference being the Arch AB; and this must be precisely heeded by the Mariner; for else he will never gain the true Variation of the Compass; the Visible, not the True Amplitude, being to be taken for that Purpose. PROBLEM. XIX. Given the Latitude of the Place, the Sun's Declination, with the 18 Degrees Depression below the Horizon, to find the Time of Break-a-day in the Morning, and End of Twilight in the Evening. PraEtice. By Observation, the Astronomers have found that the Crepusculum, or Twilight, begins in the Morn ing, and ends in the Evening, when the Sun is 180 below the Horizon ; which therefore I require for April the 5th, when the Sun has 10° 00' North Declination, and in the Latitude of Chichester 50° 56 ? Having prepared the Diagram with its proper Circles, and drawn the Parallel R R of 18° Depression, there will be conftituted the Oblique Tri Р angie CZP, in which all the three Sides are given. For the Side ZP = 39,9 04'; the Side PC=800 oo'; H Η and the Side ZP (is always) = 108° 00'; R R to find the Angle ZPC, or its Complement as being the Time from Midnight. Which is thus by Theorem 39. CP-ZP = AM = 40° 56'. Then (the Sine of Z P= 39° 04' Co. Ar. 0.2005049 the Sine of C P=802 00' Co. Ar. c.0066485 CZ +AM Add the Sine of 9.9838404 CZ-AM the Sine of 33° 32' 9.7422710 = 74° 28' 2 2 The Sum is 19.9332648 The : Sum is the Sine of ; P = 670 47': 670 47'= 9.9666324 The Double of which is the whole Angle Z PC = 135° 34 ; this reduced to Time, is 9 Hours, 2' 1611 the Evening Twilight ; the Complement of which to 12 Hours, |