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oppofite angle. Therefore diftance C P: diftance CG:: S.CGP: S. C P G or C PA; that is, the fines of the parallaxes are reciprocally as the diftances from the earth's centre.

(525.) III. Let S be the star or planet, whofe parallax is fought. See Plate XXIX. fig. 4. Obferve it when it is in the fame vertical circle with any two fixed ftars, A, B. Obferve again when the fame two stars come into a pofition parallel to the horizon at a and b; and let the planet be come tos. Then with an inftrument meature the altitude of a or b, and likewife the altitude of s; and the difference of these altitudes is the parallax. For the real place of the ftar S is fomewhere in the line A B, and therefore it is also somewhere in the line a b, and therefore its altitude is the fame as that of a or b. Therefore the parallax is the difference of the altitudes of a and s, or of b and s. (526.) IV. Let S be the star or planet; obferve its diftance from any fixed star B, which is in the fame vertical circle ZS B; and measure the diftance SB with an inftrument. Then obferve again when the fame two ftars have equal altitudes above the horizon, at b and s, and then take the diftance bs. This distance will be very near the true distance of the ftars B and S; therefore the firft diftance BS fubtracted from the latter diftance bs, when B is below S, gives the parallax; or the latter diftance fubtracted from the former, when B is above S, gives the parallax.

(527) V. The parallax may be found by obferving the azimuth and altitude of the star or planet. Let H Z O, Plate XXIX. fig. 5. be the meridian, EQ the equinoctial, H O the horizon, Z the ze nith, P the pole, S the ftar, ZSB a vertical circle paffing through it. Obferve the altitude BS, and the azimuth B O, and mark the moment of time when there observations are made; then obferve the moment of time that the ftar comes to the meridian, and then you have the diftance of time from the obfervations. Convert this into degrees, allowing only 23 hours 56 minutes to 360 degrees (which is the time of the earth's rotation to the fame ftar), and you have the arch ED or angle EP A, fuppofing P A D an hour circle. Therefore in the spherical triangle Z PA, we have the angle Z PA, and angle P Z A equal to B O, and the fide Z P the co-latitude, to find the fide Z A the complement of the altitude; this fubtracted from Z S, known by obfervation, the remainder AS is the parallax.

(528.) VI. Another method is performed by a telescope, with crofs hairs in the focus. Direct the telescope to the planet, and turn it round till its motion is along one of the crof's hairs, which represents part of the planet's parallel circle; and the other hair perpendicular to it, will reprefent its hour circle. Obferve the time when the planet comes to this hour circle, there fix the telescope, and then take its altitude; then observe the time when fome fixed ftar, whofe right afcenfion is known, comes to the fame hour circle. The diference of time between the planet and ftar coning to this hour circle, turned into degrees (alowing 360 degrees to 23 hours. 56 minutes), gives he difference of right afcenfions of the planet and tar; and fo the apparent right afcenfion of the lanet is known. VOL. II. PART II.

(529.) When the planet comes to the meridian, obferve it with the telescope, and note the time ; and when the star comes to the meridian, note the time of that: then the difference of the times

reduced to degrees as before, gives the true diffe rence of right afcenfions, fo the true right afcenfion of the planet will be known. Therefore let HO, Plate XXIX. fig. 6. be the horizon, H Z O the meridian, 7 the zenith, P the pole,. A the true place of the planet, S its apparent place, ZSB a vertical circle; then in the triangle Z PS, we have Z P, Z S, and angle ZPS to find the angle P Z S. In the triangle Z PA, we have 'Z P, angles Z PA, PZA; to find Z A, which taken from Z S, gives A S the parallax.

(530.) If the planet have a proper motion of its own, its true place will be always changing; and therefore the change of place must be computed for the time of the obfervations. This is done by obferving its place when in the meridian, twice; and thence the change of place is had for 24 hours: and therefore the place at the times of obfervations will be had by proportioning the motion according to the times. Here the angle Z PS fhould be about 90°, to have APS the greateft poffible.

(531.) VII. The operation reprefented in Plate XXIX. fig. 7. requires two obfervers in different places of the earth, and can be applied to none of the planets but MARS in oppofition to the Sun, or to VENUS on the Sun's disk. It is beft perform ed when the Sun is about the equinox. Let PERQ be the earth, PR its axis, E Q the equinoctial, S the planet Mars in oppofition to the Sun, and if near the perihelion, it is better. Let two places, F, G, be taken, the one in N. lat. the other in S. lat. the further from the equinoc. tial the better; and nearly in the fame meridian, or rather fo placed, that the line F G, drawn from the one to the other, may be nearly perpendicular to the orbit of Mars. By this there is a greater bafe to work upon. Then let the two obfervers pitch upon fome fixed ftar as A, which Mars comes very near at that time; and the nearer the better. Having two good inftruments perfectly alike, furnifhed with micrometers, and being h tuated at F and G; let them obferve for several nights fucceffively about midnight, the places of Mars at B and C, as he paffes by the star A; and take the distances A B and A C every night, during his tranfit by this ftar. Thefe obfervations are to be continued till the diftances begin to increase, and no longer; for then he is patt the ftar.

(532.) From these obfervations, the nearest diftance of Mars from the ftar A may be found, as obferved from the places F and G; at least they may be found by interpolation. Let these nearest diftances be A B and A C; then we have the difference BC, or the angle BSC or FS G. And from the fituation of the places F and G, the length and position of F G will be known; and by these, FS may be found. And laftly, the angle which the radius of the earth fubtends at the distance FS, or the horizontal parallax of Mars will be known. If, inftead of Mars in oppofition, VENUS be obferved on the body of the Sun; then her neareft diftances from either limb of the Sun muft be taken, whofe difference will give the angle at Venus, fubtended by F G; the reft as before. Uuuu

Thus

Thus the parallax of Venus will be obtained. the parallax of the fixed ftars that we fhould be The parallax of Mars, when nearest the earth, has beft able to determine their distance. The mebeen found 25", 27", and 30" at different times. thod pointed out by Galileo, and first attempted (533) A STAR or PLANET appears LOWER by Hooke, Flamstead, Molineux, and Bradley, of than it really is, by the QUANTITY of the PARAL- taking the diftances of flars from the zenith that LAX, which is greater the lower the ftar is; and pafs very near it, has given us a much juster idea of therefore the HORIZONTAL PARALLAX is the the immense distance of the stars, and furnished greatest. The parallaxes of two planets are as the us with an approximation to the knowledge of cofines of the apparent altitudes directly, and their their parallax, that is much nearer the truth than distances from the earth's centre reciprocally. we ever had before. For when the distance is given, the parallax is as the fine of the zenith distance (by method 1.) and if the apparent altitude be given, the parallax is reciprocally as the diftance, by method 2, and therefore is in a compound ratio, when neither is given. Here the parallax being very fmall, one may take the parallax itself for the fine of the pa

rallax.

(534) The PARALLAX of a PLANET being known, its DISTANCE may be found. For this is only working backward, faying, As fine of the parallax, to the earth's radius; fo S. zenith diftance to the planet's distance.

(535) Having the parallax of ANY of the planets, the distances of ALL the planets from the fun may be known, in diameters of the earth, or any fort of measure. For the diftances of the planets from the fun and from one another are known in fome affumed measure; and by the parallax of a planet, the, true diftance of the earth from it is known: and therefore all the other diftances will be known by proportion.

(536.) The 7th of thefe methods has been practifed in determining the parallax of Venus, from obfervations made at different parts of the earth, upon what is called her TRANSIT over the fun's dik, a phænomenon, that rarely happens: but when it does happen, it affords the best; and indeed the only accurate method of determining that most important problem in aftronomy, THE SUN'S PARALLAX, or the angle under which the earth's femidiameter appears from the fun.

TREE.

(537.) The firft tranfit, or paffage of Venus over the fun's disk, that ever was obferved, happened in 1639, but perhaps the only mortals who faw it were Mr HORROX and his friend Mr CRABTwo tranfits have happened fince; the first in 1761, and the last in 1769. There will be no more before 1874, and the next to that will happen in 1996. The two laft tranfits were carefully obferved. From the firft of thefe Mr SHORT has computed the fun's parallax to be 8.69; and from the latt, the beft aftronomers have concluded it to be 86. This is an obfervation of the greatest confequence, because it is only by a knowledge of the fun's diftance from the earth, in fome known meafure, that we can acquire a knowledge of the true dimenfions of the folar fyftem.

(538.) As to the FIXED STARS, no method of afcertaining their distance has hitherto been found out. Those who have formed conjectures concerning them, have thought that they were at leaft 400,000 times farther from us, than we are

from the fun.

(539) Dr HERSCHEL has propofed a method of afcertaining the PARALLAX of the fixed fars, fomething fimilar, but more complete, than that mentioned by GALILEO and others; for it is by

(540.) But Herfchel mentions the infafficiency of their inftruments, which were fimilar to the prefent zenith sectors, the method of zenith distances being liable to confiderable errors on account of refraction, the change of position of the earth's axis arifing from nutation, preceffion of the equinoxes, and other caufes, and the aberration of light. The method of his own is by means of double ftars; which is exempted from these errors, and of fuch a nature that the annual parallax, even if it fhould not exceed the tenth part of a fecond, may still become more vifible, and be af certained, at least to a much greater degree of approximation than it has ever been done.

(541.) This method is capable of every im
provement which the telescope and mechanism of
micrometers can furnish; but as it goes on pre-
fumptions which can hardly lead to any firm con-
viction, we are not likely to gain any farther
knowledge, than that the ftars are at too great
diftance to be subjected as yet to our calculations.
He fuppofes that the stars are, one with another,
about the fize of the fun; and that the differen
of their apparent magnitudes is owing to ther
apparent diftances; both of which fuppofitions
being only hypothetical, it it evident that the co-
clufions founded on them cannot be depended on
with abfolute certainty.

SECT. IV. Of the DIVISIONS of the STARRY
HEAVENS.

(542.) The STARS, from their apparently var ous magnitudes, have been diftributed into fiveral claffes, or orders. Thote which appear fr geft, are called STARS of the FIRST MAGNITUDE; the next to them in luftre, STARS of the SECOND MAGNITUDE; and fo on to the SIXTH, which are the smallest that are vifible to the bare eye. Th diftribution having been made long before the i vention of telescopes, the ftars which cannot be feen without the affiftance of thefe inftruments, are diftinguished by the name of TELESCOPE

STARS.

(543.) The ancients divided the ftarry spher into particular coNSTELLATIONS, or clutters of ftars, according as they lay near one another, * as to occupy thofe fpaces, which the figures it different forts of animals or things would take up, if they were there delineated. And that ftars, which could not be brought into any part. cular conftellation, were called unformed fart.

(544.) By this divifion, the ftars are fo die guifhed from one another, that any particular la may be readily found in the heavens, by meas of a celeftial globe; on which the confolat are fo delineated, that the most remarkable dis are placed in fuch parts of the figures,

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moft easily distinguifhed. See Plates XXX. and XXXI. fig. 1. of each.

(546.) The starry heavens are also divided into three parts: viz. I. The ZODIAC, which extends (545.) The number of the ancient conftellations quite round the heavens; is about 16 degrees is 48, and upon our prefent globes about 70. On broad, fo that it takes in the orbits of all the plaSenex's globes are inferted Bayer's letters; the nets, as well as that of the moon; and along the first in the Greek alphabet being put to the big- middle of which is the ecliptic. II. All that region geft ftar in each conftellation, the fecond to the of the heavens which is on the NORTH fide of the next, and fo on; by which means every ftar is as zodiac, containing 21 conftellations: And, III. eafily found as if a name were given to it. Thus, That on the soUTH fide, contsining 15. if the ftar y in the conftellation of the ram be mentioned, every aftronomer knows as well what ftar is meant, as if it were pointed out to him in the heavens.

(547.) The following TABLES exhibit the names of the ancient and modern conftellations, and the number of stars obferved in each of them by different aftronomers.

(548.) TABLE I. THE ANCIENT CONSTELLATIONS. ·

NUMBER of STARS in EACH, according to

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(550) TABLE III. EVELIUS'S CONSTELLATIONS made out of the UNFORMED STARS,

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SECT. V. Of CALCULATING the PERIODICAL TIMES, PLACES, &c. of the CELESTIAL BODIES; CONSTRUCTING ASTRONOMICAL TA

ELES, and DELINEATING the PHASES of the

MOON.

(551.) This fection, if treated fully, would comprehend almost the whole of what may be called PRACTICAL ASTRONOMY. But as it is by

inveftigation of it would neceffarily lead us into far the most difficult and abftrufe, fo the thorough very deep geometrical demonftrations. The great labours of former aftronomers have left little, indeed, to do in this refpect. Tables of the motions of all the celeftial bodies have been made long a go, the periodical times, excentricities, &c. of the planets determined; and, as we fuppofe, few will defire to repeat thefe laborious operations, we fhall here only give fome general hints of the methods by which they have been originally accom: plified, that the operations of the young aftronomer, who makes ufe of tables already formed, may not be merely mechanical.

(552.) It has been already obferved, that the foundation of all aftronomical operations is the drawing a meridian line. This being done, the next thing is, to find out the latitude of the place where the obfervations are to be made, and for which the meridian line is drawn. Now, the latitude of a place is equal to the elevation either of the north or fouth pole above the horizon; but as there is no ftar exactly in either of the celeftial poles, to find the altitude of that invifible point, Called the POLE of the beavers, we must choofe fore ftar near it, which does not fet; and, having found its greatcft and leaft altitudes, divide their difference by 2; and half that difference, added to the leaft, or fubtracted from the greatest, alti. tude of the ftar, gives the exact altitude of the

pole, or latitude of the place. Thus, fuppof the greatest altitude of the ftar observed is 6c2, and its leaft 50°, we then know, that the latitud of the place, where the obfervation was made, a exactly 55°,

(553.) Having found the latitude, the OBLIQUI fun's annual path, with the earth's equator, s TY of the ECLIPTIC, or the angle made by the dian diftance from the zenith, which is ey eafily obtained by the following method. O ferve, about the fummer folftice, the fun's meri done by a quadrant, with a moveable indes, furnithed with fights; if this distance is fubtracted from the latitude of the place, provided the is nearer the equator than the place of oblerva tion, the remainder will be the obliquity of the ecliptic: but, if the place of obfervation is near er the equator than the fun at that time, the ze nith diftance must be added. By this method, the obliquity of the ecliptic hath been determined to be 23° 29'.

(554) Here it is proper to obferve, that the obliquity of the ecliptic to the equinoctial, found, at prefent, to be above the third part of a degree lefs than Ptolemy found it. And mck of the obfervers after him, found it to decrease gri dually, down to Tycho Brahe's time. If t objected, that we cannot depend on the obferva tions of the ancients, because of the incorrectnes of their inftruments, we have to answer, that botà Brahe and Flamstead are allowed to have bee ftead makes this obliquity 24 minutes of a decre very good obfervers; and yet we find, that Fl is than Brake did, about 100 years before b and as Ptolemy was 1324 years before Ty Brahe, fo the gradual decreafe anfwers nearly to the difference of time between these three an nomers. If we confider, that the earth is not a partid

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perfect sphere, but an oblate spheroid, having its axis fhorter than its equatorial diameter; and that the fun and moon are conftantly acting obliquely, upon the greater quantity of matter, about the equator, pulling it, as it were, towards a nearer and nearer coincidence with the ecliptic; it will not appear improbable, that these actions should gradually diminish the angle between thefe planes. Nor is it lefs probable, that the mutual attractions of all the planets fhould have a tendency to bring their orbits to a coincidence: but this change is too fmall to become obfervable in many ages.

(555) The DECLINATION of the SUN from the equator, for any day, may be found, by the fame method by which we find the obliquity of the ecliptic: and thus, a table of his declination for every day in the year, might be conftructed: Thus, alfo, the declination of the stars might be found. Having found the declination of the fun, his right afcention, and place in the ecliptic, may be geometrically found, by the olution of a cafe, in fpherical trigonometry. For let EQ, in Plate XXIX. fig. 8. represent the celestial equator, y the fun, and X the ecliptic; then, in the right angled spherical triangle ECy, we have the fide Ey, equal to the fun's declination: the angle ECy is always 23 29', being the angle of the ecliptic with the equator; and angle yEC is 90°, or a right angle. From thefe data, we can find the fide EC, the right afcenfion; and Cy, the fun's place in the ecliptic, or his diftance from the equinoctial point; and thus, a table of the fun's place, for every day in the year, anfwerable to his declination, may be formed.

(556.) The RIGHT ASCENSION of the STARS may be found by the fun's place in the ecliptic, and a good pendulum clock: for which purpose, the motion of the clock must be so adjusted, that the hand may run through the 24 hours, in the time that a star, leaving the meridian, will arrive at it again; which, time is fomewhat shorter than the natural day, because of the space the fun moves through in the mean time eastward. The clock being thus adjusted, when the fun is in the meridian, fix the hand to the point, from whence we are to begin to reckon our time; and then obferve when the ftar comes to the meridian, and mark the hour and minute that the hand then thews: The hours and minutes defcribed by the index, turned into degrees and minutes of the equator, will give the difference between the right afcenfion of the fun and ftars; which difference, being added to the right afcenfion of the fun, will give the right afcenfion of the star.

(557.) If we know the right afcenfion of any ONE ftar, we may from it, find the right afcenfion of ALL the others which we fee, by mark ing the time upon the clock, between the arrival of a ftar, whole right afcenfion we know to the meridian, and another ftar, whofe afcenfion is to be found. This time, converted into hours and minutes of the equator, will give the difference of right afcenfions; from whence, by addition, we collect the right afcenfion of the star which was to be found out.

(558.) When the right afcenfion and declination of a star is found, its longitude and latitude,

or diftance from the firft ftar of Aries, and north or fouth from the ecliptic, may thence be eafily found; and the places of the fixed ftars being ail marked in a catalogue, according to their longitudes and latitudes, it may be thence conceived, how the longitude and latitude of a planet or comet may be found for any particular time, by comparing its diftance from them, and its apparent path may thus be traced; and thus, the paths of Mercury and Venus were traced by M. Caflini. (559.) To find out the PERIODICAL TIMES OF the PLANETS, we muft obferve when they have no latitude. At that time the planet is in the ecliptic, and confequently, in one of its nodes; fo that, by waiting till it returns to the fame node again, and keeping an exact account of the time, the periodical time of its revolution round the fun, may be known pretty exactly. By fimilar obfervations, from the theory of the earth's motion, we can find the pofition of the line of the nodes; and when once the pofition of this line is found, the angle of inclination of the planet's orbit to the earth may alfo be known.

(560.) The EXCENTRICITY of the EARTH'S ORBIT may be determined, by observing the apparent diameters of the fun at different times: when the fun's diameter is least, the earth is at the greateft diftance; and, when this diameter is greateft, the earth is at its leaft diftance from him. But, as this method muft neceffarily be precarious, another is recommended by Keil, by obferviug the velocity of the earth in its orbit, or the apparent velocity of the fun, which is demonftrated to be always reciprocally as the fquare of the distance.

(561.) The excentricities of the orbits of the other planets may be likewise found, by obferving their velocities at different times; for all of them obferve the fame proportions, with regard to the increase or decrease of their velocity, that the earth does; only, in this cafe, care must be taken, to obferve the real, not the apparent, velocities of the planets, the laft depending on the motion of the earth at the fame time. Their aphelia, or points of their orbits, where they are farthest from the fun, may be known, by making several obfervations of their distances from him, and thus perceiving, when these diftances ceafe to increase.

(562.) The pofition of the aphelion being determined, the planet's diftance from it, at any time, may also be found by obfervation, which is called its true or coequated anomaly; but, by fuppofing the motion of the planet to be regular and uniform, tables of that motion may easily be conftructed. From thence, the planet's mean place in its orbit, may be found for any moment of time; and one of these moments being fixed upon, as an epocha or beginning of the table, it is easy to understand, that from thence, tables of the planet's place in its orbit, for any number of years, either preceding, or fubfequent to that period, may be conftructed.

(563.) ASTRONOMICAL TABLES are to be conftructed, according to the meridian of equal time, and not true or apparent time, because of the inequalities of the earth's motion, as well as that of the planet, and equations must be made, to be added to, or fubtracted from the mean motion

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