d that the war is
raging anew as violently as before. For the enemy left at home a
despised captive has burst all the chains of the equations, and broken
forth from the prisons of the tables."

Still, a part of the truth had been gained, and was not to be abandoned
any more. The law of speed was fixed: that which is now known as his
second law. But what about the shape of the orbit--Was it after all
possible that Aristotle, and every philosopher since Aristotle, had been
wrong? that circular motion was not the perfect and natural motion, but
that planets might move in some other closed curve?

Suppose he tried an oval. Well, there are a great variety of ovals, and
several were tried: with the result that they could be made to answer
better than a circle, but still were not right.

Now, however, the geometrical and mathematical difficulties of
calculation, which before had been tedious and oppressive, threatened to
become overwhelming; and it is with a rising sense of despondency that
Kepler sees his six years' unremitting labour leading deeper and deeper
into complication.

One most disheartening circumstance appeared, viz. that when he made the
circuit oval his law of equable description of areas broke down. That
seemed to require the circular orbit, and yet no circular orbit was
quite accurate.

While thinking and pondering for weeks and months over this new dilemma
and complication of difficulties, till his brain reeled, an accidental
ray of light broke upon him in a way not now intelligible, or barely
intelligible. Half the extreme breadth intercepted between the circle
and oval was 429/100,000 of the radius, and he remembered that the
"optical inequality" of Mars was also about 429/100,000. This
coincidence, in his own words, woke him out of sleep; and for some
reason or other impelled him instantly to try making the planet
oscillate in the diameter of its epicycle instead of revolve round it--a
singular idea, but Copernicus had had a similar one to explain the
motions of Mercury.

[Illustration: FIG. 31.--Mode of drawing an ellipse. The two pins _F_
are the foci.]

Away he started through his calculations again. A long course of work
night and day was rewarded by finding that he was now able to hit off
the motions better than before; but what a singularly complicated motion
it was. Could it be expressed no more simply? Yes, the curve so
described by the planet is a comparatively simple one: it is a specia