Note 17 — XIII. TRANSITS OF VENUS.

There is, however, a much more perfect way of determining this proportion, by applying the law which Kepler found to connect the distances of the planets from the sun with the times in which they complete the circuits of their orbits. The law is that, if we take any two planets, and write down the numbers expressing their periods of circuit (say in days), and the numbers expressing their distances from the sun (say in miles) in the same order; then if we multiply each number of the first pair into itself, and each number of the second pair twice into itself, the four numbers thus obtained will be proportional; that is to say, as the first is to the second, so will the third be to the fourth. Now, as every one knows who has worked sums in the rule of three, when any three are given out of four proportionals, the fourth can always be found; but we know the periods of circuit both of the earth and Venus (365·2564 days and 224·7008 days respectively) very exactly indeed, because they have traversed their orbits so many times since they began to be observed by astronomers. We can call the earth's distance 100, and then applying the rule just stated, we get Venus' distance relatively to the earth's. The reader who cares to work out this little sum will find no difficulty whatever—if at least he is able to extract the cube roots of any number. The proportion runs thus:—

365·2564 × 365·2564 : 224·7008 × 224·7008 :: 100 × 100 × 100 : (Venus' distance cubed.)

Work out this sum and we get for Venus' distance 72·333. The ratio of Venus' distance to the earth's is almost exactly expressed by the numbers 217 and 300.