Nov. 17, 18S2.] 



o KNOWI^EDGE 



405 



sun's disc at the moment of conjunction. But although 

 the path of Venus lies nmrhj, it does not lie exadUj, in 

 the same level as the earth's. Thus, if EE'e (Fig. 2) 

 represent the earth's path, while Wr represents that of 

 Venus (E being the place occupied by the earth at the 

 Autumnal Equinox, so that SV is directed to the point in 

 the heavens called the First Point of Aries), then the planes 

 of the two orbits intersect in the line E'e'. And if we regard 

 the earth's orbit as lying in the plane of the paper, then the 

 orbit of Venus must be regarded as crossing the plane of 

 the paper at V, passing to its greatest distance above (or 

 north of) that plane at v (the actual distance being repre- 

 sented by the short white line at r), passing down to the 

 paper again, which it crosses at c', and then passing to its 

 greatest distance below, or south of the plane of the paper, 

 where a short white line is seen near V, and so onwards to 

 its crossing place at V. 



It is manifest, then, that only when Venus is in con- 

 junction with the earth at or near V (which is called her 

 ascendiiuj nodv), or c' (which is called her drscrndin(/ node), 

 ■will she be seen upon the sun's face, crossing or transitiiiy 

 it as she passes onwards from conjunction. If she is in 

 conjunction when near i; .she will be above, or north of the 

 sun's disc, by her greatest possible amount ; if she is in 

 conjunction when near V (close to the short white line), 

 she will be below or south of the sun's disc by her greatest 

 possible amount. But when in conjunction anywhere on 

 the arc V r v', she will be more or less above the ccw/ve 

 of the sun's disc ; and when in conjunction anywhere on 

 the arc v' V V, she will be more or less below the vrutn' 

 of the Sim's disc ; and she will only transit the disc when 

 she is so near V or v\ as to be nut more than tlie tiiin's 

 fe mi-diameter above or below his centre. 



Let us then inquire within what distance from V or i' 

 Venus must be, so that, if in conjunction at the moment, 

 she would be seen upon some part of the solar disc. The 

 inquiry is a very simple one, in any case ; but to reduce 

 it to its simplest form, we will make the following assump- 

 tions : — First, the earth and Venus shall be assumed to be 

 at their mean distances ; next, we shall consider only the 

 centres of the two planets, inquiring under what circum- 

 stances Venus's centre would be on the sun's face, as seen 

 from the earth's centre at the moment of conjunction. 



We have, then, the following relations : — Let E (Fig. 3) 

 be the earth's centre, S the sun's, V the centre of Venus, 

 when in absolute conjunction, that is on the line E S. Let 

 lines E s and E s' be drawn from E, to touch the solar 

 globe ; then, owing to the sun's enormous distance, .s- IS ,s' 

 may be regarded as a straiglit line, squart; to E S. Draw 

 another line c V v, also square to E S. Then if, when in 

 conjunction, the centre of Venus is anywhere between 

 r and v', the centre of the disc of Venus will be on the 

 sun's face. 



Now s S s', the sun's diameter, contains 8.")2,90S miles, 

 if E S be assumed equal to 91,430,000 miles, as throughout 



this paper* ; and on the same assumption the distance, 

 E V, is 25,296,000 miles. But manifestly r V bears to 

 E V the same proportion that »• S bears to E S. Hence 

 we have the simple rule-of-three statement : — 



91,430,000 : 25,296,000 : : S .*, or 426,454 : V v. 

 Whence V r or V v' is equal to 117,987 miles. 



It follows that if Venus, when in conjunction, has her 

 centre not more than 117,987 miles above or below the line 

 joining the centres of the earth and sun, then Venus's 

 centre, as supposed to be seen from the earth's centre, will 

 be on the sun's face. 



Now it is easy to find how near Venus must be to her 

 node in order just to have this limiting distance 117,987 

 miles from the level of the earth's track. Venus's orbit 

 plane is inclined 3° 23' 30 8" to the plane in which the 

 earth travels. Let us suppose that the orbits are looked 

 at from a point in the prolongation of their common line of 

 intersection, so that they appear like two lines, as VSV 

 and ESE', in Fig. 4. Then the points V and V, where 



A'enus has her greatest distance from the plane of the 

 earth's orbit, are easily shownf to be 3,912,807 miles above 

 and below that plane. And when Venus is at any other 

 point of her orbit, as X — corresponding to either P or P' 

 if the orbit be supposed opened out to the circle VPV — 

 then her distance from the plane of the earth's orbit is less 

 than 3,912,807 miles in precisely the same proportion that 

 SX is less than SV orSV. But what we want is to 

 determine where Venus must 1)6 so that this distance may 

 be 117,987 miles, neither more nor less. If r and r' on 

 either side of S be the required points (corresponding to 

 the four points n, n', and m, m' in the orbit, where Si and ?$ 

 are the nodes) we manifestly have the following rule-of- 

 three sums for determining S r or Sc' : — 

 3,912,807 :1 17,987 ::SV or 66,134,000 : Sc or Sr' whence 

 we find that S r or S r' is equal to 1,994,212 miles. Thus 

 r 1-' is about 4,000,000 miles. And it is clear that each of 

 the arcs n >i' and m m' is very nearly the same in length as 

 V v' ; for where the arc is small it differs very little from 

 the chord. Hence we have this simple result : If Venus 

 is within 2,000,000 of miles of either node when in con- 

 junction, there is a transit; otherwise not 



* The result is not nt nil nffectrd by this assumption, which is 

 only introduced in order to make Venus's displacement a qnostiou 

 of miles, instead of heliocentric latitude. 



+ Of course, this distance is equal to Venus's mean distance x 

 sine of her inclination — on our assumption of mean distances. 

 Owing to the small eccentricity of Venus's orbit, this aesumption is 

 not far from absolnte correctness. 



