Fig. 45.
Thus, if D a is twice as great as D A, as in [fig. 43], then a b is twice as great as A B, the length which the observers know; and if D a is only equal to half D A, as in [fig. 44], then a b is only equal to half the known length A B. In every possible case the length of a b is known. Take one other case in which the proportion is not quite so simple:—Suppose that D a is greater than D A in the proportion of 18 to 7, as in [fig. 45]; then b a is greater than A B in the same proportion; so that, for instance, if A B is a length of 7 inches, b a is a length of 18 inches.
We see from these simple cases how the actual size of a distant object can be learned by two observers who do not leave their room, so long only as they know the relative distances of that object and of another which comes: between it and them. We need not specially concern ourselves by inquiring how they could determine this last point: it is enough that it might become known to them in many ways. To mention only one. Suppose the sun was shining so as to throw the shadow of the dove on a uniformly paved court between the house and the dovecot, then it is easy to conceive how the position of the shadow on the uniform paving would enable the observers to determine (by counting rows) the relative distances of dove and dovecot.
Now, Venus comes between the earth and sun precisely as the dove in [fig. 45] comes between the window A B and the dovecot b a. The relative distances are known exactly, and have been known for hundreds of years. They were first learned by direct observation; Venus going round and round the sun, within the path of the earth, is seen now on one side (the eastern side) of the sun as an evening star, and now on the other side (the western side) as a morning star, and when she seems farthest away from the sun in direction E V (fig. 46) in one case, or E v in the other case, we know that the line E V or E v, as the case may be, must just touch her path; and perceiving how far her place in the heavens is from the sun's place at those times, we know, in fact, the size of either angle S E V or S E v, and, therefore, the shape of either triangle S E V or S E v. But this amounts to saying that we know what proportion S E bears to S V—that is, what proportion the distance of the earth bears to the distance of Venus.[17]
Fig. 46.
This proportion has been found to be very nearly that of 100 to 72; so that when Venus is on a line between the earth and sun, her distances from these two bodies are as 28 to 72, or as 7 to 18.