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The Doctrine of 'parallaxes is of the utmoft Importance in Agronomy; both for the determining of the Diftances of the Planets, Comets, and other Pht£no?'Mna of the Heavens; for the Calculation of Eclipfes; and tor determining the Longitude. See Planet, Distance, Longitude, and Eclipse.

Methods of finding the 'Parallaxes of the Celeftial tPtoentfi- mena arc various : Some of the principal and eaiier follow. To obferve the Parallax of a CeleftialVhienomenon.

Obfervc when the Phenomenon is in the fame Vertical wit" a fix'd Star which is near it 5 andmeafureits apparent Difiance from this Star. Obferve, again, when the phenomenon and fix'd Star are in equal Altitudes from the Horizon; and again meafure their Diitance : the Difference of thofe Diftances will be, very nearly, the 'Parallax of the Star.

The Parallax of a PbtSmmenmtsLzy be likewife found by obferving its Azimuth and Altitude; and by marking the Time, between the Observation and its Arrival at the Meridian.

All, requir'd to find the Parallax of the Moon, is the Pa- rallax of right Afcenfion : i. e. to find the EfFeclof the Magni- tude of the Semi-diameter of the Earth, with regard to the Phenomena of its Motion, 'tis fulficient to know how far the Meridian, to which the Eye refers it, deviates from the true Meridian. This is what M. CaJJini found and praftifed, with regard to Man; and which M. Maral.ii has fince practifed, with regard to the Moon. The whole Myitery here confiltsin having the Moon's true Motion, which refers to the Centre of the Earth; and its apparent Motion, which refers to the Place of Obfervatjon. The Difference of thefe, which is greateft in the Horizon, or Horary Circle of 6 0' Clock, gives the Horizon- tal parallax, for that Latitude whence the general 'Parallax, or that under the Equator is eafily found: The parallax of any Parallel being to that of the Eqiiator,as the Semi-diameter of this Parallel is to that of the Equator. See the Practice of this Method exemplify 'd in finding the Parallax of Mars.

The popular Method for that of the Moon, Wolfius gives us as follows;

To obferve the Moons Parallax.

Obferve the Moon's Meridian Altitude, with the greateftAc- curacy^&e Altitude,) and mark the Moment of Time: This Time being equated; (See Equation,) compute her true Lon- gitude and Latitude; and from thefe find her Declination, (See Declination.) and from her Declination and the Elevation of theEquator find her true Meridian Altitude. If the obferv'd Altitude be not meridian, reduce it to the true Altitude for the Time of Obfervation. Take the Refraction from the obferv'd Altitude, and fub tract the Remainder from the true Altitude : the Remainder is the Moon's Parallax.

By this means Tycho in 1583. Orf. 12. Her. 5'. it), from the Moon's Meridian Altitude obferv'd, 13 . 3.8 her Parallax 54 Minutes. See Moon.

To obferve the Moon's Parallax in an Bdipfe.

In an Eclipfe of the Moon, obferve when both Horns are in the fame Vertical Circle; in that Moment take the Altitudes of both Horns; the Difference of the two being halved and added to the leaft, or fubrracled from the greater!, gives nearly the vifible Altitude of the Moon's Centre. But the true Altitude is nearly equal to the Altitude of the Centre of the Shadow at that Time. Now we know the Altitude of the Centre of the Shadow; becaufe we know the Sun's Place in the Ecliptic, and its Depreifion under the Horizon, which is equal to the Altitude of the oppofite Point of the Ecliptic in which the Centre of the Shadow is. Thus have we both the true and apparent Altitude; the Difference whereof is the Parallax.

Front the Moon's Parallax AS 7 'Fig. 30. and Mitude SR, to find her Di fiance from the Earth.

By her apparent Altitude giren, we have her apparent Di- fiance from the Zenith, i. e. the Angle Z A S; or by her true Altitude the Angle ATS. Wherefore, fince, atthe fameTime, we have the ParallaBic Angle S; and the Semi-diameter of the Earth is reputed. By plain Trigonometry we mail have the Moon's Dillance in Semi-diameters of theEarth; thus : as the Sine of the Angle S is to rhe oppofite Side given, fo is the Sine the other Angle T, to the Side required TS.

Hence, according to Tycho's Obfervation, the Moon's Di- itance at that Time from the Earth was 6z Semi-diameters of theEarth. Hence alfo, fince, from the Moon's Theory, we have the Ratio ot her Diftances from the Earth in the feveral Degrees of her Anomaly; thofe Diftances being found by the Rule of Three in Semi-diameters ot theEarth, the Parallax is thence determined to the feveral Degrees of the true Ano- maly.

2)e leHire makes the greateft Horizontal parallax i°. 1'. 15". the {mailed 54' 5". The Moon's Diftance, therefore, when in her Perigee is y.jj&j that is, almoft 56 Semi-diameters 5 in her Apogee <J3f 7 o» that is, 6^~ Semi -diameters.

To obferve the Parallax o/'Mars.

i. Suppofe Mars'w the Meridian and Equator,in H. T'ab.A- ftronomy Fig, 31. and that the Gbferver under theEquator in A 3

obferves him culminating with fomefixM Star. i. If now the Obferver were m the Centre of theEarth, he wou'd ke Man conftantly in the fame Point of the Heaven with the Star; and therefore,together with it in the Pl ane of the Horizon.or of the fixth Horary. Em fince Mars, here, has fome fenfible Paral- lax, and the fix d Star none Mars will be feen in the Hori- zon, when in P the Plane of the fenfible Horizon; and the Star, when in the Plane ot the true Horizon : obferve,therefore,the lime between the Jranfus o£ Mars and of the Star thro' the Plane of the fixth Hour. 3. Convert this Time into Minutes ot the Equator; by this means we mail have the Arch P M to which the Angle P AM, and confequently the Angle AMD is nearly equal, which is the Horizontal parallax ot Man

If the Obferver were not under the Equator, but in a Pa- rallel, I CL that Difference will be a lefs Arch QM. Where- fore, fince the little Arches, Q^M and P M, are .-.s their Sines A D and I D; and fince A D G is equal to the Diitance of the Place from the Equator, i. e. to the Elevation of the Pole; and therefore, AD to ID, as the whole Sine to the Co-fine of the Elevation of the Pole; fay, as the Co-fine of the Elevation of the Pole I D is to the whole Sine AD; fo is the Parallax ob- ferv'd in I, to the Parallax to be obferv'd under the Equator.

Since Mars and the fix'd Star cannot be commodiouily ob- ferv'd in the Horizon; let 'em be obferv'd in the Circle of the third Hour : And fince the Parallax there obferv'd, T O, is to the Horizontal one, P M, as IS to I D; Say, as the Sine of the Angle IDS, or 45 ■ (fince the Plane D O is in the Middle between the Meridian DHand the true Horizon DMJ to the whole Sine, fo is the Parallax TO to the Horizontal Paral- lax P M.

If Mars be likewife cut of the Plane of the Equator; the Parallax found will be an Arch of a Parallel; which muft: therefore, be reduced, as above, to an Arch of the Equator. '

Lafily, if Mars be not ftationary, but either direct, or retro- grade; by Obfervations for feveral Days, find out what his Mo- tion is every Hour, that his true Place from the-Centre may beaftign'd for any given Time.

By this Method, Cafjini, to whom we owe this noble Inven- tion, obferv'd the greateft Horizontal Parallax of Mars to be 25 Seconds, or a little lefs. Ey the fame Method Mr. Flam- fiead found it near thirty Seconds.

By the fame Method the fame Author Cafjini obferv'd the 'Parallax of Venus.

It muft be here noted, that the Obfervation is to be made with a Telefcope, in whofe Focus are ftrain'd A B

four Threads cutting each other at right Angles A, B, C, D. The Telefcope to be turn'd about, till fome Star near Mars be feen to pafs over fome of the Threads; AB and CD being pa- rallel to the Equator; and therefore, AC and B D reprefenting Circles of Declination. Thus, by means of the perpendicular Threads, the Situations of the Star, and oi Mars in the Me- ridian, will be determined.

To find the Sun s Parallax.

The great Diflance of the Sun renders its Parallax too fmall to fall under even the niceft immediate Obfervation: Indeed- many Attempts have been made both by the Antients and Mo- derns; and many Methods invented for that Purpofe. Thefirft, that of Hipparchm, follow'd by Ptokmy, &c. was founded 011 the Obfervation of Lunar Eclipfes;the fecond, wasthat ot A- riftarchus, whereby the Angle fubtended by the Semi-diameter of the Moon's Orbit, feen from the Sun, was fought from the Lunar Phafcs: But, thefe both proving deficient, Aflronomers are forced to have Recourfe to the Parallaxes of the Planets nearer us. as. Mars and Venus; forfrom their Parallaxes known that of the Sun, which is inacceffible by any direct. Obferva- tion, is eafily deduced.

For from the Theory of the Motions of the Earth and Pla- nets, we know at any Time the Proportion of the Diftances of the Sun and Planets from us; and the Horizontal Parallaxes are in a reciprocal Proportion to thofe Diftances : Knowing, therefore, the Parallax of a Planet, that of the Sun may be found from it.

Thus, Mars, when oppofite to the Sun, is twice as near as the Sun is, his Parallax; therefore, will be twice as great as that of the Sun : and Venus, when in her inferior Conjunction with the Sun; is fometimes nearer us than he is; her Parallax, there- fore, is greater in the fame Proportion.

Thus, from the Parallaxes of Mars and Venus, the fame Cafjini found the Sun's Parallax to be ten Seconds, which im- plies his Diftance to be22;c(J2 Semi-diameters.

In an Obfervation of the Tranfit of Venus over the Sun, which will be feen in May, 1 7C1. Dr. Ualky has fhewri a Me- thod of finding the Sun's Parallax and Diftance to a great Nicety, viz. to a five hundredth Part of the Whole. See Sun.

The Parallax of the Stars, with regard to the Earth's annual Orbit. The Stars have no Parallax, with regard to the Earths Se- mi-diameter; yet, with regard to the Earth's annual Orbit,' 'tis ijuftly expected that fome Parallax be found. See Okbit. 3 Tits