relation between the orbits E e and v v of the earth and Venus; s as before representing the line from which astronomers measure the motions of the planets round the Sun. It will be seen by comparing fig. 8 with fig. 10 that when the Earth and Venus are in conjunction as at E and v, the distance separating them bears a smaller proportion to the Earth's mean distance from the Sun, than that separating the Earth and Mars

when in conjunction. But then there is a circumstance in which Venus is less favourable for the purpose of astronomers than Mars. When the Earth and Mars are in conjunction as at E and M, fig. 8, the sun is on the opposite side of the earth, and so Mars is seen on a dark sky; but when Venus and the Earth are in con

this drawing, with additions derived from my charts of the planetary orbits, at page 185 of his Elementary Lessons in Astronomy. To this I of course make no objection whatever, as such drawings are meant to be copied. Only as he has omitted to refer the drawings to me, I am debarred from using my own work without showing my title to it. A similar remark applies to the figure at p. 184 of Mr. Lockyer's work, and to certain considerable portions of the letter-press.

junction, as in fig. 10, Venus lies directly towards the Sun, and even though visible (in powerful telescopes), yet is seen under very unfavourable conditions. The background of the sky is bright, and none but the chief stars can be discerned, unless the telescope is very large and powerful, in which case it is not so well adapted as a more manageable one would be, for the class of observations required. Even the leading stars are but faintly seen; and as Venus may not lie near any of them, the kind of measurement which was available in the case of Mars becomes too precarious for the purpose of determining any parallactic displacement of the planet.

Hence the direct observation of Venus, when nearest to us, after the manner applied to Mars, is not a very valuable method of determining the Sun's distance. It may yet be applied successfully (according to the plan proposed in 1848 by Dr. Gerling of Marburg); but even if it should, the method now to be considered is preferable.

When Venus and the Earth are in conjunction, she is not commonly on a direct line between the Earth and the Sun. She would be so, if the path v v' v lay in the same level with the path E Ee; but this is not the case. If we suppose the path E E' e to lie in the plane of the paper, then the path v v'v' must be supposed to intersect that plane in the line v' v' the half of v′ v v′ lying slightly below the plane of the paper, and the half v' v ' slightly above, the short lines near v and v showing the greatest amount of separation between the

two planes on account of this tilt. Hence, unless the conjunction happen to take place when Venus is at v or v', Venus will not be on a direct line joining the Earth and the Sun. But when Venus is within a certain distance of these two points,*—the nodes of her

It is easy to calculate how near Venus must be to a node in order that she may be visible on the Sun's disc. The extreme limits will be those corresponding to the case when the disc of Venus as seen from all the Earth save one point only, lies outside the Sun's disc, but as seen from that one point just touches the disc on the outside. Further, on account of the ellipticity of the two orbits, the exact extent of these limits will be different for conjunctions occurring when Venus is at vand at v". To estimate these limits strictly, the exact values of the distances of Venus and the Earth from the Sun when in longitudes corresponding to SE' and s e' may be taken from the Nautical Almanac, and combined with the estimated diameter of Venus. The formula to be used are sufficiently simple, and will occur at once to the mathematical reader,-or may be deduced from my paper on the transit of Venus in the Monthly Notices of the Astronomical Society for March 1869 (vol. xxix.). But for our present purpose such exactitude is not needed. It may be well, however, to determine in a general way the limits in question; for which purpose we may assume that no transit need be considered in which the line joining the centres of Venus and the Earth does not meet the Sun. Hence, Venus must not have a greater geocentric latitude than the Sun's apparent semi-diameter. Now the distance of Venus from the Earth, and when she is in inferior conjunction, is to the Earth's distance from the Sun as 277 to 1,000. So that when her apparent latitude is equal to 16' (which we may take as the Sun's semi-diameter) her heliocentric latitude is to 16' as 277 to 723, or is about 6'1. The inclination of her orbit being 3° 24', or 204', it is easily calculated that she must be within about 13° of a node, at the epoch of inferior conjunction, in order that a transit may occur. Thus at each node there is an arc of about 34° along any part of which Venus, if in inferior conjunction, will be projected on the Sun's disc; and as there are two nodes, the total range out of the 360° of her orbit along which transits can occur, is about 63°, or a 54th part of the whole. Hence, on the average there will be one transit in 54 conjunctions, and as conjunctions occur at average intervals of 583-920 days, there will be on the average one transit in 86 years. But this average would only correspond to the case where a very large number of conjunctions was considered.

orbit, as they are called,—she will be so nearly on the line joining the Earth and the Sun, that as seen from the Earth she will appear on the Sun's disc.

Now when a conjunction of this sort takes place, the observation of Venus's apparent place is rendered much easier, since the disc of the Sun forms a sort of indexplate, as it were, on which we can estimate her position. So that if Venus's distance is to be determined at all from observations of her parallactic displacement as seen from different parts of the Earth's surface, it is clear that the proper time for attacking the problem is when Venus is in transit.

FIG. 11.

S

Let us see, however, how this may be most effectually contrived.

Suppose that as seen from the point E (fig. 11) on the Earth at any moment Venus (v) is seen as at v on the Sun's disc; whereas as seen from E' she appears to be at v'. Then it seems at first sight as though nothing could be simpler than to determine the distance v v'; and then having the arc v v' and the length of the base line E E' we could at once determine the distance of Venus. For v v' is the arc between two lines from Venus containing the angle v v v', which is equal to EVE'. So that we know the angle E V E'. For example, if the arc v v' were shown to be thirty-five seconds as seen from the Earth, then the angle v v v

would be greater (because v is nearer to v v' than E is) in the proportion of E v to v V, or roughly as 7 to 5, so that the angle v v v′ (or E V E') is an angle of about forty-nine seconds. Now suppose that the stations E, E', are so placed on the Earth's surface that the base line E E' is known to be about 6,000 miles in length: then the distance E v exceeds 6,000 miles in the proportion that the radius of an arc of forty-nine seconds exceeds that arc, or roughly as 4,200 to 1. Hence E v is about 25,200,000 miles, and E s exceeds this distance in the proportion of about 7 to 2, or comes out equal to nearly 90,000,000 miles.

But for many reasons this direct method of solving the problem of the Sun's distance has not been hitherto applied. In the first place it is absolutely necessary that the observations made at E and E' should either be made exactly at the same moment, or that the difference of time should be exactly known, so that the two observations may be fairly compared together. But for this purpose we must know the exact position of the stations E and E' on the Earth, so as to be able from the apparent time at these stations to infer the true time at Greenwich or some other fixed station. It is easily seen that a very slight error in the determination of the longitude of either station would make the whole series of observations useless. For example, suppose the observer at E recorded the place of Venus on the Sun's disc at apparent noon for his station; and that the longitude of the station was supposed to be such that this epoch corresponded exactly with the epoch