the same thing for Mars; with _?ME_ for an
alternative hypothesis that perhaps Mars' equant was on line joining
_EC_ with _MC_.]

I need not say that all these attempts and gropings, thus briefly
summarized, entailed enormous labour, and required not only great
pertinacity, but a most singularly constituted mind, that could thus
continue groping in the dark without a possible ray of theory to
illuminate its search. Grope he did, however, with unexampled diligence.

At length he hit upon a point that seemed nearly right. He thought he
had found the truth; but no, before long the position of the planet, as
calculated, and as recorded by Tycho, differed by eight minutes of arc,
or about one-eighth of a degree. Could the observation be wrong by this
small amount? No, he had known Tycho, and knew that he was never wrong
eight minutes in an observation.

So he set out the whole weary way again, and said that with those eight
minutes he would yet find out the law of the universe. He proceeded to
see if by making the planet librate, or the plane of its orbit tilt up
and down, anything could be done. He was rewarded by finding that at any
rate the plane of the orbit did not tilt up and down: it was fixed, and
this was a simplification on Copernicus's theory. It is not an absolute
fixture, but the changes are very small (see Laplace, page 266).

[Illustration: FIG. 30.--Excentric circle supposed to be divided into
equal areas. The sun, _S_, being placed at a selected point, it was
possible to represent the varying speed of a planet by saying that it
moved from _A_ to _B_, from _B_ to _C_, and so on, in equal times.]

At last he thought of giving up the idea of _uniform_ circular motion,
and of trying _varying_ circular motion, say inversely as its distance
from the sun. To simplify calculation, he divided the orbit into
triangles, and tried if making the triangles equal would do. A great
piece of luck, they did beautifully: the rate of description of areas
(not arcs) is uniform. Over this discovery he greatly rejoices. He feels
as though he had been carrying on a war against the planet and had
triumphed; but his gratulation was premature. Before long fresh little
errors appeared, and grew in importance. Thus he announces it himself:--

"While thus triumphing over Mars, and preparing for him, as for one
already vanquished, tabular prisons and equated excentric fetters, it is
buzzed here and there that the victory is vain, an