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 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 that you would not be able to track them out, but be always leaving the true track and getting upon one crossing it slightly aslant,—­just like the lines by which trains are made to run easily off one track on to another.