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Not in Twenty-Four Hours!
by Roger M. Firestone, MPS

Most of the statements in the many
Masonic degrees are accurate in the sci-
entific sense, at least as far as the knowl-
edge of the day extended when they were
written. With today's insights, we can
find errors of various sorts in the works
of most of the nineteenth-century Ma-
sonic authors, but we should hardly fault
them for this. After all, Pike and others
sought to illuminate philosophical prin-
ciples by drawing on material from nat-
ural philosophy, as science was known
until only century ago.

However, there is one striking inaccu-
racy in a degree that is familiar to us all.
That error is the statement, " . . . the
tide ebbs and flows twice in twenty-four
hours." It does not! Moreover, the fact
that it does not must have been known to
mariners from the most ancient of times,
since the difference is easily measurable.
Today, however, most of us have so little
involvement with the maritime world
that we do not give the phrase a second
thought when we hear the degree con-
ferred. An examination of this Masonic
error is a useful topic for instruction in
some of the seven liberal arts and sci-
ences familiar to Fellow Crafts, notably
those of astronomy and mathematics
(arithmetic and geometry).

Most people are aware, at least to some
limited extent, that the moon is respon-
sible for the oceanic tides. This is an
oversimplification. The moon is the
cause of both tides in the ocean in a
corresponding, but far smaller, rise and
fall in the elevation of land masses as
well. Both land and water are subject
equally to the acceleration of gravity; it
is just that water is more able to move
under its influence.

However, the sun is also responsible for
considerable tidal effects. The moon is
some 387.5 times closer to the earth, on
the average, than the sun. The force of
gravity is inversely proportional to the
square of the distance, which means that
the a factor 387.72 = 150,156.25 reduces
the effects of the sun's mass relative to the
moon in terms of gravity. But tidal effects
are determined, not by the force of grav-
ity, but by the local difference in the force
of gravity. This means that the cube, not
the square of the distance is the deter-
mining factor. For the sun vs. the moon,
the actual reduction factor is
58,185,546.88. This enormous factor
would appear to reduce the influence of
the sun to insignificance. But we have
not reckoned in the relative masses of the
two bodies. In fact, the sun is so much
more massive than the moon, that the
factor of almost 60 million is just about
cancelled out. In everyday terms, the
moon is responsible for about 55 % of the
effects of tides observed, and the sun
produces 45 % .

As the moon orbits the earth, its posi-
tion with respect to the sun changes con-
stantly. Twice a month, at new moon and
full moon, the moon, earth, and sun are
nearly in a straight line. (When they are
in a truly straight line, we get an eclipse.
This happens about two months a year,
when the moon is at a node of its orbit.
Otherwise the moon lies above or below
the plane of the earth' s orbit around the
sun--the ecliptic--and the shadows do
not line up for an eclipse.) Whether the
moon and sun are in the same direction
or pulling in opposite directions, the ef-
fect on the tides is the same: the size of
the tides is larger. See Figure 1. These
tides are known as spring tides.

When the moon is at first or last quar-
ter, its pull is at right angles to that of the
sun, and the tides are correspondingly
smaller. See Figure 2. These tides are
known as neap tides. The distances from
the earth to the sun and moon also vary
somewhat, and these have a lesser effect
on the heights of the tides on annual and
monthly bases, respectively. Local geog-
raphy also affects the height of the tides;
in some locations, the variation is but a
few inches, while in others it is substan-
tial. Canada's Bay of Fundy is famous
for tides in excess of fifty feet.

Because the tidal shape produce;l by
the sun and moon has two high points
and two low points, a given location on
the earth will experience two high and
two low tides during approximately one
revolution--but not exactly one revolu-
tion.

To see why, we will visit an old paradox
due to the Stoic philosopher Zeno of an-
cient Greece. To his disciples, Zeno once
propounded the problem of a race be-
tween Achilles and a tortoise. The leg-
endary hero could outrun ordinary
human beings--how much more so a
lowly tortoise. So Zeno proposed to give
the tortoise a head start. Let us say that
the tortoise began 100 yards ahead of
Achilles, and let us also assume that
Achilles is ten times faster than his torpid
competitor. The starter gives a signal
and our contestants begin to race. Achil-
les covers the 100 yards of the head start
in only a few seconds. But the tortoise
has not been idle; while Achilles was
rushing forward, the tortoise has also
stuck his neck out and covered ten yards
in the same time. Now Achilles runs ten
yards, but in the same time, the tortoise
moves ahead one yard. Achilles covers
the one yard, but the tortoise, deter-
mined to maintain the dignity of the
reptile race, advances one-tenth yard.
And so it goes. No matter how much
distance Achilles travels, the tortoise
seems always to be ahead of him. Yet we
know that any of us, not just demigods
of Greek myth, can outdistance a tor-
toise. What is wrong with Zeno's reason-
ing?

In fact, solving Zeno's paradox re-
quires knowledge of branches of mathe-
matics that was well beyond what the
Greeks had developed. The Greeks had
mastered arithmetic and geometry well
enough, but this problem requires
knowledge of infinite series or at least
decimal fractions, and those did not be-
come common knowledge until modern
times.

For this simple case, we may look at the
various distances that Achilles travels in
the description above. First he covers 100
yards, then 10, then 1, then 0.1, then
0.01 + . . . and if we sum up quite a
number of these, we get a result like
111.11111 .... The part to the right of
the decimal point looks a great deal like
the decimal expansion of 1/9. This guess
is exactly correct. Achilles will pass the
tortoise after running 111 1/9 yards,
when the tortoise has covered just 11 1/9
yards. It is clear that the difference be-
tween these two distances is exactly the
100-yard head start that Zeno gave
Achilles.

This problem is a special case of the
summation of a particular infinite series.
Summing an infinite series in general
may not be possible for the most brilliant
mathematician, but some such series are
quite easy to deal with. The series we are
examining here is the simple one
1 +x +x2 +x3 +x4 +

Although there are a number of ways
to sum this series in mathematics, one of
the simplest is to assume that it has a
sum, which we will call L.  We may then
observe that S = 1 +(1 +x +x2 +x3
+x4 + ....) = 1 +xS. This is an
equation we can solve by algebra, and it
is not hard to see that the value of S is
1/1-x. Advanced mathematics tells us
this is true only if x lies between the
values of -1 and + 1. For Zeno's race, x
has the value of 0.1 (and we have to
multiply S by the amount of the head
start). The value of S is then 1/1-0.1 =
1-1/9, just as we have seen before.

The result can be applied to other prob-
lems. For example, how many times does
the minute hand of a clock pass the hour
hand in 12 hours? The answer is not 12.
(if you have a digital watch, I don't want
to know about it!) The two hands are
performing a constant version of Zeno's
race, in fact. Start at noon, or high
twelve, since this is a Masonic article. An
hour later, the minute hand points to 12
again, but the hour hand has moved to
the 1. By the time the minute hand
reaches the 1, the hour hand has ad-
vanced 1/12 of the way to the 2 and so
on. which is another way of saying 1/11
hours elapse between two times when the
minute hand has passes the hour hand.
Dividing the 12 hours by this time, we
see that the minute hand passes the hour
hand only 11 times in 12 hours.

Now we can solve the problem of the
tides. Relative to the sun, the moon re-
volves around the earth in 29 1/2 days.
(Relative to the fixed stars, the time is
shorter, but it is the motion with respect
to the sun that matters for the tides.) This
is very much like the motion of two hands
of the watch, except that the ratio of the
speeds is different. Instead of one going
12 times faster, the earth spins 29.5 times
faster than the moon moves around it.
But the principle is the same. We have to
plug in only the value 1/29.5 in for x in
the formula and see what it gives us.

The answer is 1.035, approximately.
This is the interval in days between the
times when the moon appears, let us say,
due south. The actual times of high and
low tides has much to do with the shape
of land surfaces and so on, but this inter-
val is the period that encompasses a com-
ple cycle of two high and two low tides.
If we multiply 1.035 by 24 hours, we get
a period of 24 hours and 50.5 minutes.
In other words, "the tide ebbs and flows
twice in twenty-five hours, not the num-
ber we are used to hearing.

If this simple phrase can lead us into an
exploration of astronomy and higher
mathematics, how much else of wisdom
can we find in the Masonic degrees? And
can we also learn the lesson of not taking
the simplest and most obvious facts for
granted. The progressiveness of the sci-
ence of Masonry is here illustrated by the
need for the leaming of the Fellow Craft
to be applied to the incomplete or simpli-
fied instruction given to an entered Ap-
prentice.

Philalethes, October 1993
