[Footnote 3: I must re-mention that though this explanation is
made as simple as I possibly can make it, so far as words are
concerned, the figures present the result of an exact geometrical
investigation. Every dot, for instance, in Fig. 2, has had its
place separately determined by me.]

[Illustration: FIG. 2. ONE OF MARS'S LOOPS.]

This is one loop, you will understand, out of an immense number which
Mars makes in journeying round the earth, regarded as fixed. He
retreats to a great distance, swoops inward again toward the earth,
making a loop as in Fig. 2, and retreating again. Then he comes
again, makes another swoop, and a loop on another side, and so on.
He behaves, in fact, like that "little quiver fellow," a right
martialist, no doubt, who, as Justice Shallow tells us, "would about
and about, and come you in, and come you in,--and away again would a
go, and again would a come." The loops are not all of the same size.
The one shown in Fig. 2 is one of the smallest. I have before me a
picture which I have made of all this planet's loops from 1875 to
1892, and it forms the most curiously intertwined set of curves you
can imagine,--rather pretty, though not regular, the loops on one side
being much larger than those on the other. I would show the picture
here, but it is too large. One of these days, it will be given in a
book I am going to write about Mars, who is quite important enough to
have a book all to himself. I want you, now, to understand me that
Mars really does travel in a most complicated path, when you consider
the earth as at rest. If a perfect picture of all his loopings and
twistings since astronomy began could be drawn,--even on a sheet of
paper as large as the floor of a room,--the curves would so interlace