Lecture fhall have an infinite Diverfity of Distances of the XXI. Sun from the Earth: And confequently, the true Distance of the Sun from the Earth cannot be obtained by this Method. There is no need to confine this Me. thod to the Phafis of a SINCE the Moment in which the true Dichotomy happens is uncertain, but it is certain that it happens before the Quadrature; Ricciolus takes that Point of Time which is in the Middle, between the Time that the Phafis begins to be doubtful whether it be bifected or not, and the Time of Quadrature: But he had done righter, if he had taken the middle Point between the Time when it becomes doubtful whether the Moon's Side is concave or ftreight, and the Time again when it is doubtful whether it is ftreight or convex; which Point of Time is after the Quadrature: And if he had done this, he would have found the Sun's Distance a great deal bigger than he has made it. THERE is now no need to confine this Method to, the Phafis of a Dichotomy or Bifection, for it can be as well perform'd when the Moon has any other Phafis bigger or lefs than a Dichotomy: For obferve by a very good Telescope, with a Micrometer, the Phafis of the Moon, that is, the Proportion of the illuminated Part of the Diameter to the whole; and at the fame Moment of Time take her Elongation from the Sun: The illuftrated Part of the Diameter, if it be lefs than the Semidiameter, is to be fubducted from the Semidiameter; but if it be greater, the Semidiameter is to be fubducted from it, and mark the Refidue: Then fay, as the Semidiameter of the Moon is to the Refidue, fo is the Radius to the Sine of an Angle, which is therefore found: This Angle added to, or fubftracted from a right Angle, gives the exterior Angle of the Triangle at the Moon: But we have the Angle at the Earth, which is the Elongation obferved; which therefore being fubducted from the exterior Angle, leaves the Angle at the Sun. And in the Triangle SLT, having all the Angles and one Side LT, we can find the other Side ST, the Distance of the Sun from Lecture XXI. Since therefore the The Sun's Parallax from the Earth. But it is almost impoffible to determine accurately the Quantity of the Lunar Phafis, fo that there may not be an Error of a few Seconds committed; and confequently, we cannot by this Method find precisely enough the true Distance of the Sun. However, from fuch Obfervations, we are fure, that the Sun is above 7000 Semidiameters of the Earth diftant from us. true Distance of the Sun can neither be found by Distance and Eclipfes, nor by the Phafes of the Moon, the may be deAftronomers are forced to have Recourse to the Par-duced from allaxes of the Planets that are next to us, as Mars the Parallax and Venus, that are fometimes much nearer to us of the Pla than the Sun is: Their Parallaxes they endeavour to find by fome of the Methods above explained: And if these Parallaxes were known, then the Parallax and Distance of the Sun, which cannot directly by any Obfervations be attained, would eafily be deduced from them. For from the Theory of the Motions of the Earth and Planets, we know at any Time the Proportion of the Distances of the Sun and Planets from us; and the horizontal Parallaxes are in a reciprocal Proportion to thefe Distances. Wherefore, knowing the Parallax of a Planet, we may from thence find the Parallax of the Sun. nets. MARS, when he is in an Acronychal Pofition, Particularly that is, oppofite to the Sun, is twice as near to us by Mars in an Acrony as the Sun is; and therefore his Parallax will be chal Pofitwice as great. But Venus, when she is in her infe- tion. rior Conjunction with the Sun, is four-times nearer to us than he is, and her Parallax is greater in the fame Proportion: Therefore, tho' the extreme Smallness of the Sun's Parallax renders it unobfervable by our Senfes, yet the Parallaxes of Mars or Venus, which are twice or four-times greater, may become fenfible. The Aftronomers have beftowed much Pains in finding out the Parallax of Mars; but of late, Mars was in his Oppofition to the Sun, and alfo in his Perihelion, and confequently, in his nearest Approach to the Earth: And Lecture And then he was most accurately observed by two XXI. of the most eminent Aftronomers of our Age, who have determined his Parallax to have been scarce 30 Seconds; from whence we can easily collect, that the Parallax of the Sun is fcarce 11 Seconds, and his Distance about 19000 Semidiameters of the Earth. The Paral By an Obfervation of the Body of Venus, seen lax of the paffing over the Body of the Sun, which will Sun found happen in May, 1761, Dr. Halley has fhewed a by obferving Venus in the Method of finding the Parallax and Distance of Body of the the Sun to a great Nicety, viz. within a five hundredth Part of the whole; and we must wait till then, before it can be determin'd to fo great an Exactness. Sun, How the allax is to BECAUSE the Method whereby the Aftronomers foretel Eclipfes of the Sun, requires that the Moon's Parallax both as to Longitude and Latitude should be known by Calculation: And also, as often as the Moon's Place, in the Heavens is to be observed, that it may be compared to the Place found by Aftronomical Tables, in order to establish her Theory; it will be neceffary to reduce the true Place found by the Tables to her apparent Place, which cannot be done without the Calculation of the Parallax It will be convenient to explain the Method by which the Moon's Parallax for any Point of Time is to be calculated. . FIRST, by Aftronomical Tables compute the Place Moon's Par- of the Moon in the Ecliptick and her Latitude, for be found for the given Time. In the Figure suppose HO_the any Time by Horizon, HZO the Meridian, Z the Vertex, EC Calculation, the Ecliptick, in which let L be the Place of the Moon, found by the Tables. And firft, let us fuppose the Moon to be without Latitude. From the Vertex Z let fall upon the Ecliptick the Perpendicular Zn A, which will be therefore a Circle of Latitude; and the Point z will be the 90th Degree of the Ecliptick. From the Time given we have the right-Afcenfion of the Sun, and his Equatorial Distance from the Meridian: From thence we shall Table XXI. Fig. 6. find find the Point of the Equator culminating, which Lecture is the right Afcenfion of Mid-Heaven, or of that XXI. Point of the Ecliptick which culminates : And therefore we know that Point which is then in the Meridian, as alfo the Angle Z En of the Ecliptick and Meridian, This is either found by the Calculation we explained in the spherical Doctrine, or by Tables of Aftronomy: By this Means we find the Arch EL; but we have the Arch EÆ the Declination of the Point E, and consequently the Arch ZE will be known. Therefore in the right-angled Triangle Zn E, we have the Side ZE, and the Angle ZEN. Hence we can find En and the Point z or the Point of the 90th Degree, and the Arch Zn, its Distance from the Vertex; whofe Complement » A is the Measure of the Angle that the Horizon and the Ecliptick make: And because we have the Place of the Moon, we must have the Arch z L. Therefore in the right-angled Triangle Zn L, having the Sides Zn and n L, we fhall have from them the Angle ZL2, which is called the Paralladick Angle, as likewife the Side Z L, the The ParalDistance of the Moon from the Vertex. Let the lactick AnRadius be to the Sine of the Arch Z L, as the horizontal Parallax of the Moon taken from the Tables to its Parallax in L, which therefore is found. Let it be o L. From o on the Ecliptick let fall the Perpendicular om. And in the Triangle o L m (which being very small, may be taken for a rectilinear one) we have befides the right Angle, the Side Lo and the Angle L M equal to Z Ln; wherefore we fhall find out the Arch Lm, the Parallax of Longitude, and om, the Parallax of Latitude, which were to be found. SUPPOSE now the Moon has fome Latitude, and its Place in the Ecliptick be the Point L, but let it be placed in the Circle of Latitude LP at P. And because the Angle n LP is right, and we have the Angle LZ, and its Complement ZLP; in the Triangle Z LP we have two Sides, Z L, which was found before, and LP, the Moon's Latitude, and gle. the Lecture the Angle ZLP, whereby we can find out the Side XXII, ZP, and the Angle ZPL. Let the Radius be to the Sine of the Arch ZP, as the horizontal Parallax of the Moon to a Fourth, which will be Pq: This will be the Parallax of the Moon in the Circle of Altitude. Let qd be an Arch parallel to the Ecliptick; and in the finall Triangle Pdq, which may be taken as a right-angled Triangle, we have the Angle dPq, which is the Complement of the Angle ZPL to two right Angles, and the Side Pq: Therefore we fhall have Pd the Parallax of Latitude, and qd the Parallax of Longitude: For because the Latitude of the Moon is but fmall, the Arch of the Parallel d q is nearly equal to the Arch of the Ecliptick which is correspondent to it. H the EARTH. ITHERTO we have given an Ac count of the general Affections of the Planets Motions, and have explained the Appearances which arise from their Motions and the Motions of the Earth together. We will now come to their particular Theories, in which the Period of each, its Distance from the Sun, the Form of its Orbit, and its Polition are determined; which |