way round the planet, the direct satellite will describe a curve
which will be less than half way round the planet: that is a span
due to a retrograde satellite will be more than 180 degrees,
while the span due to a direct satellite will be less than 180
degrees upon the planet's equator.
I would draw your attention to the fact that what the span will
be does not depend upon how much the orbit of the satellite is
inclined to the equator. This only decides how far the curve
marked out by the satellite will recede from the equator.
We find then, so far, that it is easy to distinguish between the
direct and the retrograde curves. The span of one is less, of the
other greater, than 180 degrees. The number of degrees which
either sort of curve subtends upon the equator entirely depends
upon the velocity of the satellite and the axial velocity of the
planet.
But of these two velocities that of the satellite may be taken as
sensibly invariable, when close enough to use his pencil. This
depends upon the law of centrifugal force, which teaches us that
the mass of the planet alone decides the velocity of a satellite
in its orbit at any fixed distance from the planet's centre. The
other velocity—that of the planet upon its axis—was, as we have
seen, not in the past what it is now. If then Mars, at various
times in his past history, picked up satellites, these satellites
will describe curves round him having different spans which will
depend upon the velocity of axial rotation of Mars at the time
and upon this only.
191
In what way now can we apply this knowledge of the curves
described by a satellite as a test of the lunar origin of the
lines on Mars?
To do this we must apply to Lowell's map. We pick out preferably,
of course, the most complete and definite curves. The chain of
canals of which Acheron and Erebus are members mark out a fairly
definite curve. We produce it by eye, preserving the curvature as
far as possible, till it cuts the equator. Reading the span on
the equator we find' it to be 255 degrees. In the first place we
say then that this curve is due to a retrograde satellite. We
also note on Lowell's map that the greatest rise of the curve is
to a point about 32 degrees north of the equator. This gives the
inclination of the satellite's orbit to the plane of Mars'
equator.
With these data we calculate the velocity which the planet must
have possessed at the time the canal was formed on the hypothesis
that the curve was indeed the work of a satellite. The final
question now remains If we determine the curve due to this
velocity of Mars on its axis, will this curve fit that one which
appears on Lowell's map, and of which we have really availed
ourselves of only three points? To answer this question we plot
upon a sphere, the curve of a satellite, in the manner I have
described, assigning to this sphere the velocity derived from the
span of 255 degrees. Having plotted the curve on the sphere it
only remains to transfer it to Lowell's map. This is easily
done.
192
This map (Pl. XXII) shows you the result of treating this, as
well as other curves, in the manner just described. You see that
whether the fragmentary curves are steep and receding far from
the equator; or whether they are flat and lying close along the
equator; whether they span less or more than 180 degrees; the
curves determined on the supposition that they are the work of
satellites revolving round Mars agree with the mapped curves;
following them with wonderful accuracy; possessing their
properties, and, indeed, in some cases, actually coinciding with
them.