e planet, which shone at its brightest about
September 5, gradually grew less and less bright as it traveled off,
after rounding the station near October 5 (really on Oct. 7), toward
the east. He observed, then, that the seeming loop followed by the
planet was a real looped track (so far, at least, as our observer on
the earth was concerned). Fig. 2 shows the apparent shape of Mars's
loop, the dates corresponding to those shown in Fig. 1. Only it does
not lie flat, as shown on the paper, but must be supposed to lie
somewhat under the surface of the paper, as shown by the little
upright _a, b,_ which, indeed, gives the distance under the paper at
which the part of the loop is supposed to lie where lowest at _m_. The
other similar uprights at M_1, M_2, and M_3 show the depression at
these places. You perceive that the part M_1, M_2, lies higher than
the part M_2, M_3. If the loop were flat, and, like E, the earth,
were in the level of the paper, it would be seen edgewise, and the
advancing, receding, and advancing parts of the planet's course would
all lie on the same line upon the sky. But being thus out of the
level, we see through the loop, so to speak, and it has the seeming
shape shown in Fig. 1.[3]

[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
bein