increases continually its vertical Velocity, the Rays of Light therefore in their Paffage through the different Couches of the Atmotphere, whofe Density continually increafes in approaching the Earth, are more and more curved; in confequence of which the celeftial Objects appear more elevated than they really are, and that by how much the more their Rays are curved from their Entrance into the Atmofphere until they arrive to us, the Eye receiving the Impreffion of Light in the Direction which the Rays have when they enter the Eye. This apparent Elevation of the heavenly Bodies above their true Refraction Height, is called Astronomical Refraction, and is greatest near the Ho- increase Length of rizon, where repeated Obfervations prove, that it amounts to 33'; hence the Day. it is, that in our Climates, the Sun appears to rife 3 Minutes iooner, and fet 3 Minutes later than it really does, whereby the artificial Day is increafed 6 Minutes by the Effect of Refraction. This Effect gradually increases in advancing towards the Frigid-Zone, and at the Pole, by the Refraction alone, the Day becomes 36 Hours longer; hence it is alfo that the Sun and Moon at their rifing and fetting appear oval, the inferior Margin of thofe Luminaries being more refracted than the fuperior one, or appear higher in Proportion. Refraction Newton has fhewn how to determine the Law according to which Rule for Refraction varies from the Zenith to the Horizon; from his Theory it finding the refults, that the Radius (R) is to the Sine of 87d. as the Sine of (2) at any dif the Distance from the Zenith, to the Sine of (z-6r) of this fame Di- tance from stance diminished by fix Times the Refraction at this Distance, where- the Zenith. fore R-Sine 87 Sine 87 Sine z-Sine (2-6r) : Sine (z—6r); and R-Sine 87: Sine z-Sine (z-6r) Sine 87 Sine (z-6r); but R-Sine 87 is to Sine z-Sine (z-6r) as 3d.X Cof. 881 to 6rX Cos. (z-3r), Differences of the Arcs multiplied by the Cofines of the Arcs, which are the arithmetical Means between 90 and 87, and between z and z-6r. Therefore the Sine of 884., that is of 90d. diminished by the Triple of the horizontal Refraction, is to the Sine of the Distance z diminished by the Triple of the Refraction at that Distance, as the horizontal Refraction, is to the Refraction at the Distance z, and as the Cofine of 88d. to the Cofine of the Arc z diminished by the Triple of the Refraction; therefore the Refraction at the Distance z, is equal to the horizontal Refraction multiplied by the Tangent of z diminished by the Triple of its Refraction, the whole divided by the Tangent of 88d. 21. from whence it appears, that the Refractions are proportional to the Tangents of the Distances from the Zenith diminished by three Times the Refraction. Example. Let the Refraction at the Distance of 45 Degrees from the Zenith be required, which is known to be about 1, the Tangent of 884. 21m, is to the Tangent of 44d. 57m, as the horizontal Refraction 33m. is to 57', the Refraction at 45 Degrees Distance from the Zenith. By this Rule the following Table was conftructed. D. M. M. S. D. M.M. Table of App. Refrac. App. Refrac. cal Refrac tion. 033. 0,04 11.51,1 X. App Refrac. App Refrac APP Refrac. APP. Refrac. Alt. Ait. Alt. S. D. M.M. S. D. M. M. S. D. M.M. S. D. M.M. S. 8.30. 8,15.303.23,736 01.18,563 00.29,1 532.10,44 101.28,9 8.406. 1,516. 03.16,97 01.15,764 0.27,8 o 1031.22,24 2011. 7,9 8.505.54,816.30 3.10,8 01.13,065 00.26,5 o 1530.35,44 30.0.48,0 9. 05.48,517. 03, 4,539 01.10,466 c.25,3 o 2029.49,74 40 10.29,2 9.105.42,417.30 2.58,001. 7,967 c0.24,1 0 30 28.22,34 510.11,3 9.205.36,5 18. 02.53,641 0 5,568 0.22,9 0 32 28. 4,8 15 9.54,3 9.305.30,918.30 2.48,642 01. 3,369 0.21,7 0 36 27.30,35 10 9.38,2 9.45.25,419. 02.43,943 01. 1,170 0.20,6 0 4026.59,75 20 9.22,8 9.505.20,019.30 2.39,444 0.59,071 0.19,5 0 5025.41,85 30 9. 8,010. 05.14,820. 02.35,145 0.57,072 00.18,4 I 024.28,65 40 1 3021.14,76 10 2 8.54,0 10.155. 7,3 20.302.31,046 00.55,073 co.17,3 8.40,610.305. 0,121. 02.27,247 00.53,174 00.16,2 8.27,810.45 4.53,221.30 2.23,648 0.51,275 00.15,1 8.14,911. 04.46,6 22. 02.20.349 00.49,476 00.14,0 8. 2,811.154.40,3 23. 02.13,750 00.47,677 0.13,0 1 5019.24,86 30 7.51,111.304.34,324. 02. 7,451 00.45,978 00.12,0 018.35,06 40 7.40,3 11.45 4.28,625. 02. 1,652 00.44,279 00.11,0 12 1017.48,46 50 7.30,212.00 4.23,226. 01,56,253 00.42,680 00.10,0 2 2017. 4,57 o 7.20,5 12.204.16,127. 01.51,254 00.41,181 oo. 9,0 2 3016.23,87 10 7.11,112.404. 9,428. 01.46,655 00.39,682 oo. 8,0 2 40 15.45,47 20 7. 2,113. 04. 3,029. 01.42,456 00.38,283 00. 7,0 2 5015. 9,47 30 6.53,413.203.56,930. 01.38,457 00.36,884 00. 6,0 014.35,67 40 6.45,113.403.51,131. 01.34,658 00.35,585 00. 5,0 3 1014. 3,97 50 6.37,114. 03.45,532. 1.31,059 00.34,286 00. 4,0 3 2013.34,1 13 o 6.29,414.20 3.40,133. 01.27,660 00.33,087 00. 3,0 13 3 3013. 6,28 10 6.22,0 14.403.34,934. 01.24,461 00.31,788000. 2,0 3 40 12.39,68 20 6.14,815. 03.29.935. 01.21,462 00.30,489 oo. 1,0 3 5012.14,68 30 6. 8,015.303.23,736. 01.18,563 00.29,190 00. 0,0 TH THEORY of the SECONDARY PLANETS. I. HE firft Phenomenon which the Secondary Planets offer to natural Philofophers, is their Tendency towards their Primaries, in obferving the fame Law as the primary Planets towards the Sun. This Tendency has been fufficiently established in treating of the primary Planets, abstracting at first, as was necessary in order to fimplify the Question, from all the Irregularities which the Planets, by their mutual Attractions. produce in each others Motions, or which arife from the Action of the Sun. Having afterwards examined the Irregularities in the Motions of the primary Planets; but the Irregularities in the Motions of the fecondary Planets deserve particularly to be confidered, in order to fhew after a more fatisfactory Manner, the Univerfality of the Principle of Attraction, and the Harmony of the Syftem to which it ferves as a Basis. The different Kinds of Motions obferved for many Ages in the Moon, and the Laws of thofe Motions discovered by eminent Aftronomers, furnished Newton the Means of applying his Theory with Succefs to this Planet. This great Man, who had made fo many Difcoveries in the other Parts of the Syftem of the World, was refolved not to leave this Part unexamined; and though the Method he has purfued on this Occafion, is lefs evident, and less fatisfactory than the Method he employed in explaining the other Phenomena; we are however much indebted to him for having made it the Object of his Inquiry. II. gard to the Force of It is easy to perceive, that if the Distance of the Sun from the Earth Manner of and the Moon, was infinite with the refpect to their Distance from each having reother, the Sun would not disturb the Motion of the Moon about the Earth; Inequality because equal Forces, whofe Directions are parallel, which act on any of the two Bodies, cannot affect their relative Motions. But as the Angle the Sun, on formed by the Lines drawn from the Moon and the Earth to the Sun, the Earth though very small, cannot be esteemed as having no Quantity, from this and the Angle therefore is to be deduced the Inequality of the Action of the Sun on these two Bodies. Moon. is refolved in the Moon towards the Taking therefore, as Newton has done, (Propof. 66.) in the straight The Force Line drawn from the Moon to the Sun, a Line to exprefs the Force of the Sun with which the Sun attracts it; let this Line be confidered as the Dia- two others. gonal of a Parallelogram, one of whofe Sides will be in the ftraight Line drawn from the Moon to the Earth, and the other a Line drawn from the Moon parallel to the ftraight Line which joins the Sun and the Earth, One urges it is evident, that thofe two Sides of the fame Parallelogram will exprefs two Forces which might be fubftituted for the Force of the Sun Earth. on the Moon; and that the first of thofe two Forces which urges the Moon towards the Earth, will neither accelerate nor retard the Defcription of the Areas, nor confequently prevent her from obferving the Law of Kepler, viz. the Areas proportional to the Times, but will only change acts in the the Law of the Force with which the Moon tends towards the Earth, Direction of and confequently will alter the Form of her Orbit. As to the fecond the Line Force, that which acts in a Direction parallel to the Ray of the Orbit the Earth of the Earth, if it was equal to the Force with which the Sun acts on to the Sun. the Earth, it is easy to perceive that it would produce no Irregularity in the Motion of the Moon; but thofe Forces are only equal in thofe The other drawn from Points of the Moon's Orbit, where her Distance from the Sun becomes equal to the Diftance of the Earth from the Sun at the fame Time, which happens in the Quadratures; in every other Point of her Orbit thofe two Quantities being unequal, their Difference expreffes the perturbating Force of the Sun on the Moon, not only preventing her from defcribing equal Areas in equal Times, but also from moving always in the fame Plane. III. We find in Prop. 66 of the first Book, only the general Exposition of the Manner of eftimating the perturbating Forces of the Sun on the Moon: But in Prop. 25 of the third Book, we find the Calculation which determines their Quantity; we learn that the Part of the Force of the Sun which urges the Moon towards the Earth, is in its mean perturbating Quantity, the of the Force with which the Earth acts on her Measure of the Forces of the Sun. Accelera tion of the Areas defcri bed by the Moon pro duced by this Force. The Action of the Sun renders the ず 48 when she is in her mean Distance. The other Part of the fame Force of the Sun which acts in a Direction parallel to the Ray of the Orbit of the Earth, is to the first Part, as the Triple of the Cofine of the Angle formed by the ftraight Lines drawn from the Moon and the Earth to the Sun. IV. Newton employs this Determination of the perturbating Forces (Prop. 26, 27, 28, 29.) for computing the monthly Inequality in the Moon's Motion, called her Variation, whereby the moves swifter in the first and third Quarter, and flower in the Second and Fourth, and which becomes moft fenfible in the O&tants or 45 Degrees from the Syfigies. Newton, to determine this Inequality, abftracts from all the reft; he further fuppofes the Moon's Orbit to be circular, if the Sun was away, and he investigates the Acceleration in the Area which the Moon defcribes, produced by that one of the two perturbating Forces which acts in a Direction parallel to the Ray drawn from the Earth to the Sun. He found that the Area defcribed by the Moon in fmall equal Portions of Time, to be nearly as the Sum of the Number 219,46, and the verfed Sine of double of the Moon's Distance from the nearest Quadrature, (the Radius being Unity); fo that the greatest Inequality in the Areas defcribed by the Moon, arrives in the Octants or 45 Degrees from the Syfigies, where this verfed Sine is in its Maximum. V. To determine afterwards the Equation or Correction in the mean Motion of the Moon arifing from this Acceleration of the Area defcribCrbit of the ed by the Moon, he has Regard to the Change in the Form of the lunar Moon more Orbit, produced by the perturbating Force. He inveftigates the Quanbetween the tity which the perturbating Force would render the Line paffing through contracted the Quadratures longer than that which traverses the Syfigies. The tween the Data which he employs in folving this Problem, are the Velocities of Syfigies the Moon in those two Points, which he had fhewn how to determine than bein the foregoing Propofition, and the centripetal Forces correfponding Quadrato the fame Points, which are both one and the other compounded of tures. the Force with which the Moon tends towards the Earth, and the perturbating Forces of the Sun, which in the Syfigies and Quadratures act in the Direction of the Ray of the Orbit of the Moon. Now the Curvatures in those Points, being in the direct Proportion of the Attractons, and in the Inverfe of the Squares of the Velocities, by this Means he obtaines the Ratio of the Curvatures, and from thence deduces the Ratio of the Axes of the Orbit, affuming for Hypothefis, that this Curve is an Ellipfe, having its Centre in the Centre of the Earth, if the Sun be supposed to have no apparent Motion round the Earth; but when Regard is had to this Motion of the Sun, because the leffer Axe of the Ellipfe is also carried about the Earth with the fame Motion, as being always directed towards the Sun, that it is a Curve whofe Rays are the fame as thofe of the Ellipfe, but the Angles they form are increased in the Ratio of the periodic Motion of the Moon to its fynodical Motion. The first of those Motions being that in which the Moon is referred to a fixed Point in the Heavens; the other in which she is compared with the Sun. By the Means of thofe Suppofitions, Newton found that the Axe which paffes through the Quadratures, is greater than that which paffes through the Syfigies by VI. Variation He afterwards computes in the fame Hypothefis of the Moon's Or- Computabit being circular, if the Sun was away, by the Principle of the Areas tion of the proportional to the Times, the Equation or Correction in the mean Mo- of the tion of the Moon refulting not only from the Acceleration found in the Moon. foregoing Problem, her Orbit being fuppofed circular, but from the new Form of this Orbit. From the Combination of thofe two Caufes, he finds an Equation or Correction which becomes most confiderable in the Octants, and then amounts to 35m. 10' when the Earth is in its mean Distance; and in the other Points of the Earth's Orbit, is to 35m. 10', in the inverse Ratio of the Cube of the Distance from the Sun, because the Expreffion of the perturbating Force of the Sun, which is the Caufe of all thefe Irregularities of the Moon, is divided by the Cube of the Earth's Distance from the Sun. This Correction in the other Points of the Moon's Orbit, is proportional to the Sine of double of the Distance of the Moon from the nearest Quadrature. VII. Newton paffes from the Examination of the Variation of the Moon, Computato that of the Motion of the Nodes, (Prop. 30, 31.) In this Irquiry Motion he supposes the Moon's Orbit to be circular if the Sun was away, and of the attributes to the Force of this Luminary no other Effect than to change Nodes. |