173 



PARALLAX. 



PARALLAX. 



to the centre of the sun, and the change arising from excentric 

 position in each case is called parallax. 



From the effects of parallax we derive all our knowledge of the 

 distance and magnitude of the bodies which are visible in the 

 heavens. 



Let A B be any line the length of which is accurately measured, and 

 let the angles CAB, c B A, be observed, then the distances c A and c B 

 can be computed. In this way trigonometrical surveys are made, with 

 the further precaution that the angle A c B is observed when this is 

 possible, and c is to be fixed with great nicety. The angle A c B is 

 known, since it = 1 80 (BAC + ABC), and we have A B : B c : : sin 

 A c B : sin c A B. In the above figure, let A be the position of a spec- 

 tator on the earth's surface, B the centre of the earth, c the moon, and 

 z (in B A produced) the geocentric zenith. Then z A c is the apparent 

 geocentric zenith distance at A ; z B c the true geocentric zenith dis- 

 tance, that is, that which would be seen from the centre of the earth ; 

 and ACB = ZAC ZBC, the moon's parallax : also 



sin parallax = x sin Appt. geocent. zen. dist. 



B c 



When A c is at right angles (DBA, this sine = 1, and the moon is in 

 the horizon. This value of the parallax is called the /iv';/i'"' 

 parallax ; naming this P, and any other value of the parallax p, wu have 



AB 



sin r = ,and 

 u c 



sin / = sin r x sin appt. geocent. zen. dist. 



It is evident that if r can be measured, the distance R of the moon's 

 centre from the centre of the earth can be found, for the other quan- 

 tity A B or r is the radius of the earth at the place of observation, 

 which is known from terrestrial measurement. Now, suppose a second 

 spectator on the same meridian at A', whose geocentric zenith is '/.', and 

 that the two observers each observe the moon upon the meridian at 

 the same moment : then, if ; and - be two observed geocentric zenith 



distances, and p andj/ the parallaxes, t= =L, p'=?A = L,aiid 



Be B BO u 



/ABC=2 p 

 i. A' BC = //'', 



and adding AB A'=Z + / (p+p'), where ABA' is 

 the sum or difference of the geocentric latitudes of A and A', and 2 and 

 =', are known by observation; hence the value of p+j>', or AC A is 

 found. From the equations 



r r 1 



j>+ji'=z + t' ABA'; niup = - sin 2; sin ;>' = - sin z' ; 



R H 



it U easy to find the value of R. 



In practice, though the process is leas simple, the principle remains 

 the same. Two distant observatories can scarcely be found exactly 

 on the same meridian ; but the tables of the moon enable us to reduce 

 the observation at one of the observatories to exactly icltat it would 

 hate keen if it had been made under the meridian of the other. When 

 the parallax is small, it is advisable to compare the planet by the 

 micrometer with stara which are nearly in its parallel. When the 

 parallax, and consequently the distance, for any given time is known, 

 the distance and parallax for any other time can be found from theory. 

 By observations of this kind, combining the observations of La Caille 

 at the Cape of Good Hope with other observations made in Europe, 

 the parallax of the moon and of Mars were fixed with great accuracy. 

 The late Professor Henderson investigated the value of the moon's 

 parallax from a comparison of his own observations at the Cape with 

 those made at Greenwich and Cambridge. (' Mem. Ast. Soc.,' vol. x., 

 p. 283.) 



It will be seen that the point from which the moon's zenith 

 distances are to be measured is in the prolongation of a line drawn 

 from the centre of the earth, and not in the prolongation of a line in 

 the direction of gravity, which is pointed out by a plumb-line. The 

 correction which is to be applied to the astronomical zenith, in order 

 to find out the geocentric zenith, is given in many collections of tables 

 for a certain hypothesis of the figure of the earth. The horizontal 

 parallax given in the ' Nautical Almanac ' is that which belongs to the 

 where the earth's radius is largest. A second table for 

 reducing this equatorial horizontal parallax to the parallax proper to 

 the place of observation (namely, log. rad., supposing the equatorial rad. 

 = 1 ) always accompanies that above referred to. 



'1/a.c of the AM/I. The first attempt to determine the sun's 

 distance seems to be due to Aristarchus of Samoa, and presupjxMos the 

 knowledge of the moon's parallax. On drawing a figure, it will imme- 

 diately be en that when the moon has completed her first quarter 

 (she is then said to be dichotomizett, or cut exactly in two), the sun, 

 moon, and spectator form a triangle, which U right-angled at the moon. 

 Now the angle which separates the nun from the moon can be observed 

 at the same instant : suppose it = K, we have 



Distance earth ft sun = distance earth & moon x sec. K. 

 The exact moment of dichotomy cannot be noted with much accu- 

 yet repeated observations would show that the sun was far more 

 distant than the moon. The ancient astronomers seem to have esti- 

 mated the sun's parallax to be from 2' to V, which minVml a gradual 



i"U inii'iMVcd. The parallax 

 ABT A.ND SCI. DIV. VOL. VI. 



sun might with modern instruments be measured in the same way as 

 that of the moon or planets above described, but not so well, as a longer 

 time must elapse between the passage of the sun and that of a star 

 nearly in the same parallel. Ptolemy says that Hipparchua computed 

 the moon's parallax from the phenomena of solar eclipses ; that is, he 

 deduced the value of the moon's parallax from the phenomena of solar 

 eplipses on two suppositions of the sun's parallax : namely, that it was 

 = ; and, again, that it was a definite small quantity. As the circum- 

 stances of a solar eclipse vary from the effects of parallax, it is clear 

 that in this way Hipparchus would get something like equations of 

 condition involving the parallaxes of the sun and of the moon, which 

 could be solved as soon as, by the problem of Aristarchus or by any 

 other method, he could determine the relation between these two 

 quantities. 



We have thus shown that an approximate knowledge of the distances 

 of the moon, sun, and planets, in terms of the magnitude of the earth, 

 requires nothing more than observation and the solution of a triangle 

 one side of which and the two including angles arc known. The magni- 

 tudes of these bodies can be immediately calculated from their apparent 

 diameters and true distance ; so that up to this point there is no room 

 for scepticism, if it be granted that the angles of a triangle equal two 

 right angles. 



There is a method of ascertaining the parallax by one observer. Let 

 Mars in opposition be the object, and compare it in right ascension 

 with a neighbouring star in the same parallel on the meridian, and also 

 several hours before and again after his transit. The parallax, being 

 wholly in a vertical circle, will not affect the right ascension in the 

 meridian : hence the meridian comparison will give the true difference 

 of right ascension between the planet and star. The other observations 

 (after correcting the place of Mars by his hourly motion, which is 

 known either from the tables or from observations on preceding and 

 succeeding days) present right ascensions of the planet affected by 

 parallax in different ways, and from these effects it is easy to compute 

 the actual value of the horizontal piirallax, and consequently the 

 distance of the planet in terms of the earth's radius at the place. This 

 method has recently been proposed by the Astronomer-Royal, as well 

 adapted for determining the parallax of the star from observations 

 made during the oppositions of 1 862 and 1864. 



Kepler's discoveries * that the planets move in ellipses round the 

 sun in the focus ; that the area swept by eaoh radius vector in a given 

 time is a constant quantity for the same planet ; and lastly, that the 

 squares of the periodic times are as the cubes of the mean distance 

 have supplied means for a much more accurate determination of the 

 sun's parallax. Assuming these laws, the forms of the orbits of the 

 earth and planets, and their relalicc distances, can be determined from 

 observation : hence, if the parallax of any one planet can be found, the 

 parallaxes of the sun and of all the other planets can be computed. 

 Observations of Mare, for instance, at his opposition, made at the Cipe 

 of Good Hope and at Greenwich, will afford a very tolerable value of 

 his parallax, and hence of his distance. Again, as the proportion 

 between the distances of Mars and the earth from the sun at any time 

 is known from the form of their orbits and their periodic times, and 

 the angle between the sun and Mars at the earth can be observed, the 

 triangle between the sun, earth, and Mars can be completely solved, 

 and hence the distance of the sun and his parallax be computed. 

 These observations can be repeated at every opposition of Mars ; and 

 if Mars be compared by the micrometer with stars near the samu 

 parallel, there is scarcely a limit to the possible accuracy of the 

 observations. 



The observation however by which the parallax of the sun is 

 determined with the greatest certainty, is that of the passage of Venus 

 over the sun's disc, commonly called the transit of Venus. [TUANSIT 

 US.] 



In the figure let s be the gun, E0 the earth, and v and v' two 

 positions of Venus, which is supposed to be moving in the direction 

 v v'. To make the figure simple, we suppose the earth to be at rest, 

 and that v v' represents the excess of the angular motion of Venus 

 above that of the earth. A spectator at E will see the commencement 

 of the transit when Venus is at v, but a spectator at e will only 

 begin to see it when Venus is at v'. The time which must elapse 



* Though thc!c discoveries are due to Kepler, the satisfactory pi oot of th-ir 

 truth waj givin by Newton. 



